:: RMOD_3 semantic presentation  Show TPTP formulae Show IDV graph for whole article:: Showing IDV graph ... (Click the Palm Trees again to close it)

definition
let R be Ring;
let V be RightMod of R;
let W1, W2 be Submodule of V;
func W1 + W2 -> strict Submodule of V means :Def1: :: RMOD_3:def 1
the carrier of it = { (v + u) where v, u is Vector of V : ( v in W1 & u in W2 ) } ;
existence
ex b1 being strict Submodule of V st the carrier of b1 = { (v + u) where v, u is Vector of V : ( v in W1 & u in W2 ) }
proof end;
uniqueness
for b1, b2 being strict Submodule of V st the carrier of b1 = { (v + u) where v, u is Vector of V : ( v in W1 & u in W2 ) } & the carrier of b2 = { (v + u) where v, u is Vector of V : ( v in W1 & u in W2 ) } holds
b1 = b2
by RMOD_2:37;
end;

:: deftheorem Def1 defines + RMOD_3:def 1 :
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for b5 being strict Submodule of V holds
( b5 = W1 + W2 iff the carrier of b5 = { (v + u) where v, u is Vector of V : ( v in W1 & u in W2 ) } );

definition
let R be Ring;
let V be RightMod of R;
let W1, W2 be Submodule of V;
func W1 /\ W2 -> strict Submodule of V means :Def2: :: RMOD_3:def 2
the carrier of it = the carrier of W1 /\ the carrier of W2;
existence
ex b1 being strict Submodule of V st the carrier of b1 = the carrier of W1 /\ the carrier of W2
proof end;
uniqueness
for b1, b2 being strict Submodule of V st the carrier of b1 = the carrier of W1 /\ the carrier of W2 & the carrier of b2 = the carrier of W1 /\ the carrier of W2 holds
b1 = b2
by RMOD_2:37;
end;

:: deftheorem Def2 defines /\ RMOD_3:def 2 :
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for b5 being strict Submodule of V holds
( b5 = W1 /\ W2 iff the carrier of b5 = the carrier of W1 /\ the carrier of W2 );

theorem :: RMOD_3:1  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: RMOD_3:2  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: RMOD_3:3  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: RMOD_3:4  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th5: :: RMOD_3:5  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for x being set holds
( x in W1 + W2 iff ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )
proof end;

theorem :: RMOD_3:6  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for v being Vector of V st ( v in W1 or v in W2 ) holds
v in W1 + W2
proof end;

theorem Th7: :: RMOD_3:7  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for x being set holds
( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )
proof end;

Lm1: for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds W1 + W2 = W2 + W1
proof end;

Lm2: for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds the carrier of W1 c= the carrier of (W1 + W2)
proof end;

Lm3: for R being Ring
for V being RightMod of R
for W1 being Submodule of V
for W2 being strict Submodule of V st the carrier of W1 c= the carrier of W2 holds
W1 + W2 = W2
proof end;

theorem :: RMOD_3:8  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W being strict Submodule of V holds W + W = W by Lm3;

theorem :: RMOD_3:9  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds W1 + W2 = W2 + W1 by Lm1;

theorem Th10: :: RMOD_3:10  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V holds W1 + (W2 + W3) = (W1 + W2) + W3
proof end;

theorem Th11: :: RMOD_3:11  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds
( W1 is Submodule of W1 + W2 & W2 is Submodule of W1 + W2 )
proof end;

theorem Th12: :: RMOD_3:12  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1 being Submodule of V
for W2 being strict Submodule of V holds
( W1 is Submodule of W2 iff W1 + W2 = W2 )
proof end;

theorem Th13: :: RMOD_3:13  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W being strict Submodule of V holds
( ((0). V) + W = W & W + ((0). V) = W )
proof end;

Lm4: for R being Ring
for V being RightMod of R
for W, W', W1 being Submodule of V st the carrier of W = the carrier of W' holds
( W1 + W = W1 + W' & W + W1 = W' + W1 )
proof end;

Lm5: for R being Ring
for V being RightMod of R
for W being Submodule of V holds W is Submodule of (Omega). V
proof end;

theorem Th14: :: RMOD_3:14  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being strict RightMod of R holds
( ((0). V) + ((Omega). V) = V & ((Omega). V) + ((0). V) = V ) by Th13;

theorem Th15: :: RMOD_3:15  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W being Submodule of V holds
( ((Omega). V) + W = RightModStr(# the carrier of V,the add of V,the Zero of V,the rmult of V #) & W + ((Omega). V) = RightModStr(# the carrier of V,the add of V,the Zero of V,the rmult of V #) )
proof end;

theorem :: RMOD_3:16  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being strict RightMod of R holds ((Omega). V) + ((Omega). V) = V by Th15;

theorem :: RMOD_3:17  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W being strict Submodule of V holds W /\ W = W
proof end;

theorem Th18: :: RMOD_3:18  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds W1 /\ W2 = W2 /\ W1
proof end;

theorem Th19: :: RMOD_3:19  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V holds W1 /\ (W2 /\ W3) = (W1 /\ W2) /\ W3
proof end;

Lm6: for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds the carrier of (W1 /\ W2) c= the carrier of W1
proof end;

theorem Th20: :: RMOD_3:20  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds
( W1 /\ W2 is Submodule of W1 & W1 /\ W2 is Submodule of W2 )
proof end;

theorem Th21: :: RMOD_3:21  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W2 being Submodule of V holds
( ( for W1 being strict Submodule of V st W1 is Submodule of W2 holds
W1 /\ W2 = W1 ) & ( for W1 being Submodule of V st W1 /\ W2 = W1 holds
W1 is Submodule of W2 ) )
proof end;

theorem :: RMOD_3:22  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
W1 /\ W3 is Submodule of W2 /\ W3
proof end;

theorem :: RMOD_3:23  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W3, W2 being Submodule of V st W1 is Submodule of W3 holds
W1 /\ W2 is Submodule of W3
proof end;

theorem :: RMOD_3:24  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 & W1 is Submodule of W3 holds
W1 is Submodule of W2 /\ W3
proof end;

theorem Th25: :: RMOD_3:25  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W being Submodule of V holds
( ((0). V) /\ W = (0). V & W /\ ((0). V) = (0). V )
proof end;

theorem :: RMOD_3:26  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th27: :: RMOD_3:27  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W being strict Submodule of V holds
( ((Omega). V) /\ W = W & W /\ ((Omega). V) = W )
proof end;

theorem :: RMOD_3:28  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being strict RightMod of R holds ((Omega). V) /\ ((Omega). V) = V by Th27;

Lm7: for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds the carrier of (W1 /\ W2) c= the carrier of (W1 + W2)
proof end;

theorem :: RMOD_3:29  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds W1 /\ W2 is Submodule of W1 + W2
proof end;

Lm8: for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2
proof end;

theorem Th30: :: RMOD_3:30  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1 being Submodule of V
for W2 being strict Submodule of V holds (W1 /\ W2) + W2 = W2
proof end;

Lm9: for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds the carrier of (W1 /\ (W1 + W2)) = the carrier of W1
proof end;

theorem Th31: :: RMOD_3:31  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W2 being Submodule of V
for W1 being strict Submodule of V holds W1 /\ (W1 + W2) = W1
proof end;

Lm10: for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V holds the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))
proof end;

theorem :: RMOD_3:32  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V holds (W1 /\ W2) + (W2 /\ W3) is Submodule of W2 /\ (W1 + W3)
proof end;

Lm11: for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
the carrier of (W2 /\ (W1 + W3)) = the carrier of ((W1 /\ W2) + (W2 /\ W3))
proof end;

theorem Th33: :: RMOD_3:33  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
W2 /\ (W1 + W3) = (W1 /\ W2) + (W2 /\ W3)
proof end;

Lm12: for R being Ring
for V being RightMod of R
for W2, W1, W3 being Submodule of V holds the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))
proof end;

theorem :: RMOD_3:34  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W2, W1, W3 being Submodule of V holds W2 + (W1 /\ W3) is Submodule of (W1 + W2) /\ (W2 + W3)
proof end;

Lm13: for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3))
proof end;

theorem :: RMOD_3:35  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
W2 + (W1 /\ W3) = (W1 + W2) /\ (W2 + W3)
proof end;

theorem Th36: :: RMOD_3:36  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W3, W2 being Submodule of V
for W1 being strict Submodule of V st W1 is Submodule of W3 holds
W1 + (W2 /\ W3) = (W1 + W2) /\ W3
proof end;

theorem :: RMOD_3:37  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being strict Submodule of V holds
( W1 + W2 = W2 iff W1 /\ W2 = W1 )
proof end;

theorem :: RMOD_3:38  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1 being Submodule of V
for W2, W3 being strict Submodule of V st W1 is Submodule of W2 holds
W1 + W3 is Submodule of W2 + W3
proof end;

theorem :: RMOD_3:39  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2, W3 being Submodule of V st W1 is Submodule of W2 holds
W1 is Submodule of W2 + W3
proof end;

theorem :: RMOD_3:40  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W3, W2 being Submodule of V st W1 is Submodule of W3 & W2 is Submodule of W3 holds
W1 + W2 is Submodule of W3
proof end;

theorem :: RMOD_3:41  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds
( ( W1 is Submodule of W2 or W2 is Submodule of W1 ) iff ex W being Submodule of V st the carrier of W = the carrier of W1 \/ the carrier of W2 )
proof end;

definition
let R be Ring;
let V be RightMod of R;
func Submodules V -> set means :Def3: :: RMOD_3:def 3
for x being set holds
( x in it iff ex W being strict Submodule of V st W = x );
existence
ex b1 being set st
for x being set holds
( x in b1 iff ex W being strict Submodule of V st W = x )
proof end;
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff ex W being strict Submodule of V st W = x ) ) & ( for x being set holds
( x in b2 iff ex W being strict Submodule of V st W = x ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines Submodules RMOD_3:def 3 :
for R being Ring
for V being RightMod of R
for b3 being set holds
( b3 = Submodules V iff for x being set holds
( x in b3 iff ex W being strict Submodule of V st W = x ) );

registration
let R be Ring;
let V be RightMod of R;
cluster Submodules V -> non empty ;
coherence
not Submodules V is empty
proof end;
end;

theorem :: RMOD_3:42  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: RMOD_3:43  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: RMOD_3:44  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being strict RightMod of R holds V in Submodules V
proof end;

definition
let R be Ring;
let V be RightMod of R;
let W1, W2 be Submodule of V;
pred V is_the_direct_sum_of W1,W2 means :Def4: :: RMOD_3:def 4
( RightModStr(# the carrier of V,the add of V,the Zero of V,the rmult of V #) = W1 + W2 & W1 /\ W2 = (0). V );
end;

:: deftheorem Def4 defines is_the_direct_sum_of RMOD_3:def 4 :
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds
( V is_the_direct_sum_of W1,W2 iff ( RightModStr(# the carrier of V,the add of V,the Zero of V,the rmult of V #) = W1 + W2 & W1 /\ W2 = (0). V ) );

Lm14: for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds
( W1 + W2 = RightModStr(# the carrier of V,the add of V,the Zero of V,the rmult of V #) iff for v being Vector of V ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )
proof end;

Lm15: for R being Ring
for V being RightMod of R
for v, v1, v2 being Vector of V holds
( v = v1 + v2 iff v1 = v - v2 )
proof end;

theorem :: RMOD_3:45  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th46: :: RMOD_3:46  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V st V is_the_direct_sum_of W1,W2 holds
V is_the_direct_sum_of W2,W1
proof end;

theorem :: RMOD_3:47  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being strict RightMod of R holds
( V is_the_direct_sum_of (0). V, (Omega). V & V is_the_direct_sum_of (Omega). V, (0). V )
proof end;

theorem Th48: :: RMOD_3:48  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for C1 being Coset of W1
for C2 being Coset of W2 st C1 meets C2 holds
C1 /\ C2 is Coset of W1 /\ W2
proof end;

theorem Th49: :: RMOD_3:49  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V holds
( V is_the_direct_sum_of W1,W2 iff for C1 being Coset of W1
for C2 being Coset of W2 ex v being Vector of V st C1 /\ C2 = {v} )
proof end;

theorem :: RMOD_3:50  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being strict RightMod of R
for W1, W2 being Submodule of V holds
( W1 + W2 = V iff for v being Vector of V ex v1, v2 being Vector of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ) by Lm14;

theorem Th51: :: RMOD_3:51  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for v, v1, v2, u1, u2 being Vector of V st V is_the_direct_sum_of W1,W2 & v = v1 + v2 & v = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 )
proof end;

theorem :: RMOD_3:52  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V st V = W1 + W2 & ex v being Vector of V st
for v1, v2, u1, u2 being Vector of V st v = v1 + v2 & v = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 ) holds
V is_the_direct_sum_of W1,W2
proof end;

definition
let R be Ring;
let V be RightMod of R;
let v be Vector of V;
let W1, W2 be Submodule of V;
assume A1: V is_the_direct_sum_of W1,W2 ;
func v |-- W1,W2 -> Element of [:the carrier of V,the carrier of V:] means :Def5: :: RMOD_3:def 5
( v = (it `1 ) + (it `2 ) & it `1 in W1 & it `2 in W2 );
existence
ex b1 being Element of [:the carrier of V,the carrier of V:] st
( v = (b1 `1 ) + (b1 `2 ) & b1 `1 in W1 & b1 `2 in W2 )
proof end;
uniqueness
for b1, b2 being Element of [:the carrier of V,the carrier of V:] st v = (b1 `1 ) + (b1 `2 ) & b1 `1 in W1 & b1 `2 in W2 & v = (b2 `1 ) + (b2 `2 ) & b2 `1 in W1 & b2 `2 in W2 holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines |-- RMOD_3:def 5 :
for R being Ring
for V being RightMod of R
for v being Vector of V
for W1, W2 being Submodule of V st V is_the_direct_sum_of W1,W2 holds
for b6 being Element of [:the carrier of V,the carrier of V:] holds
( b6 = v |-- W1,W2 iff ( v = (b6 `1 ) + (b6 `2 ) & b6 `1 in W1 & b6 `2 in W2 ) );

theorem :: RMOD_3:53  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: RMOD_3:54  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: RMOD_3:55  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: RMOD_3:56  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: RMOD_3:57  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for v being Vector of V st V is_the_direct_sum_of W1,W2 holds
(v |-- W1,W2) `1 = (v |-- W2,W1) `2
proof end;

theorem :: RMOD_3:58  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R
for W1, W2 being Submodule of V
for v being Vector of V st V is_the_direct_sum_of W1,W2 holds
(v |-- W1,W2) `2 = (v |-- W2,W1) `1
proof end;

definition
let R be Ring;
let V be RightMod of R;
func SubJoin V -> BinOp of Submodules V means :Def6: :: RMOD_3:def 6
for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
it . A1,A2 = W1 + W2;
existence
ex b1 being BinOp of Submodules V st
for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b1 . A1,A2 = W1 + W2
proof end;
uniqueness
for b1, b2 being BinOp of Submodules V st ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b1 . A1,A2 = W1 + W2 ) & ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b2 . A1,A2 = W1 + W2 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def6 defines SubJoin RMOD_3:def 6 :
for R being Ring
for V being RightMod of R
for b3 being BinOp of Submodules V holds
( b3 = SubJoin V iff for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b3 . A1,A2 = W1 + W2 );

definition
let R be Ring;
let V be RightMod of R;
func SubMeet V -> BinOp of Submodules V means :Def7: :: RMOD_3:def 7
for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
it . A1,A2 = W1 /\ W2;
existence
ex b1 being BinOp of Submodules V st
for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b1 . A1,A2 = W1 /\ W2
proof end;
uniqueness
for b1, b2 being BinOp of Submodules V st ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b1 . A1,A2 = W1 /\ W2 ) & ( for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b2 . A1,A2 = W1 /\ W2 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def7 defines SubMeet RMOD_3:def 7 :
for R being Ring
for V being RightMod of R
for b3 being BinOp of Submodules V holds
( b3 = SubMeet V iff for A1, A2 being Element of Submodules V
for W1, W2 being Submodule of V st A1 = W1 & A2 = W2 holds
b3 . A1,A2 = W1 /\ W2 );

theorem :: RMOD_3:59  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: RMOD_3:60  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: RMOD_3:61  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: RMOD_3:62  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th63: :: RMOD_3:63  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is Lattice
proof end;

theorem Th64: :: RMOD_3:64  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is 0_Lattice
proof end;

theorem Th65: :: RMOD_3:65  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is 1_Lattice
proof end;

theorem :: RMOD_3:66  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is 01_Lattice
proof end;

theorem :: RMOD_3:67  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for R being Ring
for V being RightMod of R holds LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) is M_Lattice
proof end;