:: SETWOP_2 semantic presentation

Lm1: for i being Nat holds Seg i is Element of Fin NAT
by FINSUB_1:def 5;

Lm2: now

let i be Nat; :: thesis:
assume i + 1 in Seg i ; :: thesis:
then ( i + 1 <= i & i < i + 1 ) by FINSEQ_1:3, XREAL_1:31;
hence contradiction ; :: thesis:

end;

theorem :: SETWOP_2:1

canceled;

theorem :: SETWOP_2:2

canceled;

theorem Th3: :: SETWOP_2:3

for C, D being non empty set
for c1, c2 being Element of C
for F being BinOp of D
for f being Function of C,D st F is commutative & F is associative & c1 <> c2 holds
F $$ {c1,c2},f = F . (f . c1),(f . c2)

proof end;

theorem Th4: :: SETWOP_2:4

for C, D being non empty set
for c being Element of C
for B being Element of Fin C
for F being BinOp of D
for f being Function of C,D st F is commutative & F is associative & ( B <> {} or F has_a_unity ) & not c in B holds
F $$ (B \/ {c}),f = F . (F $$ B,f),(f . c)

proof end;

theorem :: SETWOP_2:5

for C, D being non empty set
for c1, c2, c3 being Element of C
for F being BinOp of D
for f being Function of C,D st F is commutative & F is associative & c1 <> c2 & c1 <> c3 & c2 <> c3 holds
F $$ {c1,c2,c3},f = F . (F . (f . c1),(f . c2)),(f . c3)

proof end;

theorem :: SETWOP_2:6

for C, D being non empty set
for B1, B2 being Element of Fin C
for F being BinOp of D
for f being Function of C,D st F is commutative & F is associative & ( ( B1 <> {} & B2 <> {} ) or F has_a_unity ) & B1 misses B2 holds
F $$ (B1 \/ B2),f = F . (F $$ B1,f),(F $$ B2,f)

proof end;

theorem Th7: :: SETWOP_2:7

for C, C', D being non empty set
for B being Element of Fin C
for A being Element of Fin C'
for F being BinOp of D
for f being Function of C,D
for g being Function of C',D st F is commutative & F is associative & ( A <> {} or F has_a_unity ) & ex s being Function st
( dom s = A & rng s = B & s is one-to-one & g | A = f * s ) holds
F $$ A,g = F $$ B,f

proof end;

theorem :: SETWOP_2:8

for C, D, E being non empty set
for B being Element of Fin C
for f being Function of C,D
for H being BinOp of E
for h being Function of D,E st H is commutative & H is associative & ( B <> {} or H has_a_unity ) & f is one-to-one holds
H $$ (f .: B),h = H $$ B,(h * f)

proof end;

theorem :: SETWOP_2:9

for C, D being non empty set
for B being Element of Fin C
for F being BinOp of D
for f, f' being Function of C,D st F is commutative & F is associative & ( B <> {} or F has_a_unity ) & f | B = f' | B holds
F $$ B,f = F $$ B,f'

proof end;

theorem :: SETWOP_2:10

for C, D being non empty set
for B being Element of Fin C
for e being Element of D
for F being BinOp of D
for f being Function of C,D st F is commutative & F is associative & F has_a_unity & e = the_unity_wrt F & f .: B = {e} holds
F $$ B,f = e

proof end;

theorem Th11: :: SETWOP_2:11

for C, D being non empty set
for B being Element of Fin C
for e being Element of D
for F, G being BinOp of D
for f, f' being Function of C,D st F is commutative & F is associative & F has_a_unity & e = the_unity_wrt F & G . e,e = e & ( for d1, d2, d3, d4 being Element of D holds F . (G . d1,d2),(G . d3,d4) = G . (F . d1,d3),(F . d2,d4) ) holds
G . (F $$ B,f),(F $$ B,f') = F $$ B,(G .: f,f')

proof end;

Lm3: for D being non empty set
for F being BinOp of D st F is commutative & F is associative holds
for d1, d2, d3, d4 being Element of D holds F . (F . d1,d2),(F . d3,d4) = F . (F . d1,d3),(F . d2,d4)

proof end;

theorem :: SETWOP_2:12

for C, D being non empty set
for B being Element of Fin C
for F being BinOp of D
for f, f' being Function of C,D st F is commutative & F is associative & F has_a_unity holds
F . (F $$ B,f),(F $$ B,f') = F $$ B,(F .: f,f')

proof end;

theorem :: SETWOP_2:13

for C, D being non empty set
for B being Element of Fin C
for F, G being BinOp of D
for f, f' being Function of C,D st F is commutative & F is associative & F has_a_unity & F has_an_inverseOp & G = F * (id D),(the_inverseOp_wrt F) holds
G . (F $$ B,f),(F $$ B,f') = F $$ B,(G .: f,f')

proof end;

theorem Th14: :: SETWOP_2:14

for C, D being non empty set
for B being Element of Fin C
for e, d being Element of D
for F, G being BinOp of D
for f being Function of C,D st F is commutative & F is associative & F has_a_unity & e = the_unity_wrt F & G is_distributive_wrt F & G . d,e = e holds
G . d,(F $$ B,f) = F $$ B,(G [;] d,f)

proof end;

theorem Th15: :: SETWOP_2:15

for C, D being non empty set
for B being Element of Fin C
for e, d being Element of D
for F, G being BinOp of D
for f being Function of C,D st F is commutative & F is associative & F has_a_unity & e = the_unity_wrt F & G is_distributive_wrt F & G . e,d = e holds
G . (F $$ B,f),d = F $$ B,(G [:] f,d)

proof end;

theorem :: SETWOP_2:16

for C, D being non empty set
for B being Element of Fin C
for d being Element of D
for F, G being BinOp of D
for f being Function of C,D st F is commutative & F is associative & F has_a_unity & F has_an_inverseOp & G is_distributive_wrt F holds
G . d,(F $$ B,f) = F $$ B,(G [;] d,f)

proof end;

theorem :: SETWOP_2:17

proof end;

theorem Th18: :: SETWOP_2:18

for C, E, D being non empty set
for B being Element of Fin C
for F being BinOp of D
for f being Function of C,D
for H being BinOp of E
for h being Function of D,E st F is commutative & F is associative & F has_a_unity & H is commutative & H is associative & H has_a_unity & h . (the_unity_wrt F) = the_unity_wrt H & ( for d1, d2 being Element of D holds h . (F . d1,d2) = H . (h . d1),(h . d2) ) holds
h . (F $$ B,f) = H $$ B,(h * f)

proof end;

theorem :: SETWOP_2:19

for C, D being non empty set
for B being Element of Fin C
for F being BinOp of D
for u being UnOp of D
for f being Function of C,D st F is commutative & F is associative & F has_a_unity & u . (the_unity_wrt F) = the_unity_wrt F & u is_distributive_wrt F holds
u . (F $$ B,f) = F $$ B,(u * f)

proof end;

theorem :: SETWOP_2:20

proof end;

theorem :: SETWOP_2:21

for C, D being non empty set
for B being Element of Fin C
for F being BinOp of D
for f being Function of C,D st F is commutative & F is associative & F has_a_unity & F has_an_inverseOp holds
(the_inverseOp_wrt F) . (F $$ B,f) = F $$ B,((the_inverseOp_wrt F) * f)

proof end;

definition

let D be non empty set ;
let p be FinSequence of D;
let d be Element of D;
func [#] p,d -> Function of NAT ,D equals :: SETWOP_2:def 1
(NAT --> d) +* p;
coherence ;

end;

:: deftheorem defines [#] SETWOP_2:def 1 :

theorem Th22: :: SETWOP_2:22

for D being non empty set
for d being Element of D
for i being Nat
for p being FinSequence of D holds
( ( i in dom p implies ([#] p,d) . i = p . i ) & ( not i in dom p implies ([#] p,d) . i = d ) )

proof end;

theorem :: SETWOP_2:23

for D being non empty set
for d being Element of D
for p being FinSequence of D holds ([#] p,d) | (dom p) = p

proof end;

theorem :: SETWOP_2:24

for D being non empty set
for d being Element of D
for p, q being FinSequence of D holds ([#] (p ^ q),d) | (dom p) = p

proof end;

theorem :: SETWOP_2:25

for D being non empty set
for d being Element of D
for p being FinSequence of D holds rng ([#] p,d) = (rng p) \/ {d}

proof end;

theorem :: SETWOP_2:26

for E, D being non empty set
for d being Element of D
for h being Function of D,E
for p being FinSequence of D holds h * ([#] p,d) = [#] (h * p),(h . d)

proof end;

Lm4: for D being non empty set
for e being Element of D
for F being BinOp of D
for p, q being FinSequence of D st len p = len q & F . e,e = e holds
F .: ([#] p,e),([#] q,e) = [#] (F .: p,q),e

proof end;

Lm5: for D being non empty set
for e, d being Element of D
for F being BinOp of D
for p being FinSequence of D st F . e,d = e holds
F [:] ([#] p,e),d = [#] (F [:] p,d),e

proof end;

Lm6: for D being non empty set
for d, e being Element of D
for F being BinOp of D
for p being FinSequence of D st F . d,e = e holds
F [;] d,([#] p,e) = [#] (F [;] d,p),e

proof end;

definition

let i be Nat;
:: original: Seg
redefine func Seg i -> Element of Fin NAT ;
coherence by Lm1;

end;

definition

let f be FinSequence;
:: original: dom
redefine func dom f -> Element of Fin NAT ;
coherence

proof end;

end;

notation

let D be non empty set ;
let p be FinSequence of D;
let F be BinOp of D;
synonym F $$ p for F "**" p;

end;

definition

let D be non empty set ;
let p be FinSequence of D;
let F be BinOp of D;
assume A1: ( ( F has_a_unity or len p >= 1 ) & F is associative & F is commutative ) ;
redefine func F "**" p equals :Def2: :: SETWOP_2:def 2
F $$ (dom p),([#] p,(the_unity_wrt F));
compatibility by A1, FINSOP_1:4;

end;

:: deftheorem Def2 defines $$ SETWOP_2:def 2 :

theorem :: SETWOP_2:27

canceled;

theorem :: SETWOP_2:28

canceled;

theorem :: SETWOP_2:29

canceled;

theorem :: SETWOP_2:30

canceled;

theorem :: SETWOP_2:31

canceled;

theorem :: SETWOP_2:32

canceled;

theorem :: SETWOP_2:33

canceled;

theorem :: SETWOP_2:34

canceled;

theorem Th35: :: SETWOP_2:35

for D being non empty set
for F being BinOp of D
for i being Nat st F has_a_unity holds
F "**" (i |-> (the_unity_wrt F)) = the_unity_wrt F

proof end;

theorem :: SETWOP_2:36

canceled;

theorem Th37: :: SETWOP_2:37

for D being non empty set
for d being Element of D
for F being BinOp of D
for i, j being Nat st F is associative & ( ( i >= 1 & j >= 1 ) or F has_a_unity ) holds
F "**" ((i + j) |-> d) = F . (F "**" (i |-> d)),(F "**" (j |-> d))

proof end;

theorem :: SETWOP_2:38

for D being non empty set
for d being Element of D
for F being BinOp of D
for i, j being Nat st F is commutative & F is associative & ( ( i >= 1 & j >= 1 ) or F has_a_unity ) holds
F "**" ((i * j) |-> d) = F "**" (j |-> (F "**" (i |-> d)))

proof end;

theorem Th39: :: SETWOP_2:39

for E, D being non empty set
for F being BinOp of D
for H being BinOp of E
for h being Function of D,E
for p being FinSequence of D st F has_a_unity & H has_a_unity & h . (the_unity_wrt F) = the_unity_wrt H & ( for d1, d2 being Element of D holds h . (F . d1,d2) = H . (h . d1),(h . d2) ) holds
h . (F "**" p) = H "**" (h * p)

proof end;

theorem :: SETWOP_2:40

for D being non empty set
for F being BinOp of D
for u being UnOp of D
for p being FinSequence of D st F has_a_unity & u . (the_unity_wrt F) = the_unity_wrt F & u is_distributive_wrt F holds
u . (F "**" p) = F "**" (u * p)

proof end;

theorem :: SETWOP_2:41

for D being non empty set
for d being Element of D
for F, G being BinOp of D
for p being FinSequence of D st F is associative & F has_a_unity & F has_an_inverseOp & G is_distributive_wrt F holds
(G [;] d,(id D)) . (F "**" p) = F "**" ((G [;] d,(id D)) * p)

proof end;

theorem :: SETWOP_2:42

for D being non empty set
for F being BinOp of D
for p being FinSequence of D st F is commutative & F is associative & F has_a_unity & F has_an_inverseOp holds
(the_inverseOp_wrt F) . (F "**" p) = F "**" ((the_inverseOp_wrt F) * p)

proof end;

theorem Th43: :: SETWOP_2:43

for D being non empty set
for e being Element of D
for F, G being BinOp of D
for p, q being FinSequence of D st F is commutative & F is associative & F has_a_unity & e = the_unity_wrt F & G . e,e = e & ( for d1, d2, d3, d4 being Element of D holds F . (G . d1,d2),(G . d3,d4) = G . (F . d1,d3),(F . d2,d4) ) & len p = len q holds
G . (F "**" p),(F "**" q) = F "**" (G .: p,q)

proof end;

theorem Th44: :: SETWOP_2:44

for D being non empty set
for e being Element of D
for F, G being BinOp of D
for i being Nat
for T1, T2 being Element of i -tuples_on D st F is commutative & F is associative & F has_a_unity & e = the_unity_wrt F & G . e,e = e & ( for d1, d2, d3, d4 being Element of D holds F . (G . d1,d2),(G . d3,d4) = G . (F . d1,d3),(F . d2,d4) ) holds
G . (F "**" T1),(F "**" T2) = F "**" (G .: T1,T2)

proof end;

theorem Th45: :: SETWOP_2:45

for D being non empty set
for F being BinOp of D
for p, q being FinSequence of D st F is commutative & F is associative & F has_a_unity & len p = len q holds
F . (F "**" p),(F "**" q) = F "**" (F .: p,q)

proof end;

theorem Th46: :: SETWOP_2:46

for D being non empty set
for F being BinOp of D
for i being Nat
for T1, T2 being Element of i -tuples_on D st F is commutative & F is associative & F has_a_unity holds
F . (F "**" T1),(F "**" T2) = F "**" (F .: T1,T2)

proof end;

theorem :: SETWOP_2:47

for D being non empty set
for d1, d2 being Element of D
for F being BinOp of D
for i being Nat st F is commutative & F is associative & F has_a_unity holds
F "**" (i |-> (F . d1,d2)) = F . (F "**" (i |-> d1)),(F "**" (i |-> d2))

proof end;

theorem :: SETWOP_2:48

for D being non empty set
for F, G being BinOp of D
for i being Nat
for T1, T2 being Element of i -tuples_on D st F is commutative & F is associative & F has_a_unity & F has_an_inverseOp & G = F * (id D),(the_inverseOp_wrt F) holds
G . (F "**" T1),(F "**" T2) = F "**" (G .: T1,T2)

proof end;

theorem Th49: :: SETWOP_2:49

for D being non empty set
for e, d being Element of D
for F, G being BinOp of D
for p being FinSequence of D st F is commutative & F is associative & F has_a_unity & e = the_unity_wrt F & G is_distributive_wrt F & G . d,e = e holds
G . d,(F "**" p) = F "**" (G [;] d,p)

proof end;

theorem Th50: :: SETWOP_2:50

for D being non empty set
for e, d being Element of D
for F, G being BinOp of D
for p being FinSequence of D st F is commutative & F is associative & F has_a_unity & e = the_unity_wrt F & G is_distributive_wrt F & G . e,d = e holds
G . (F "**" p),d = F "**" (G [:] p,d)

proof end;

theorem :: SETWOP_2:51

for D being non empty set
for d being Element of D
for F, G being BinOp of D
for p being FinSequence of D st F is commutative & F is associative & F has_a_unity & F has_an_inverseOp & G is_distributive_wrt F holds
G . d,(F "**" p) = F "**" (G [;] d,p)

proof end;

theorem :: SETWOP_2:52

proof end;