% Mizar problem: t20_quantal1,quantal1,979,55 fof(t20_quantal1,conjecture,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v4_lattice3(A) & v4_group_1(A) & v7_quantal1(A) & v8_quantal1(A) & l1_quantal1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ( v17_quantal1(B,A) => ( k1_quantal1(A,C,D) = k1_quantal1(A,k4_monoid_0(A,C,k2_quantal1(A,D,B)),B) & k2_quantal1(A,C,D) = k2_quantal1(A,k4_monoid_0(A,k1_quantal1(A,D,B),C),B) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_finset_1,cc1_lattices,cc2_lattices,d15_quantal1,d16_quantal1,d17_quantal1,d4_group_1,dt_k15_lattice3,dt_k1_group_1,dt_k1_quantal1,dt_k1_xboole_0,dt_k2_quantal1,dt_k4_monoid_0,dt_l1_group_1,dt_l1_lattices,dt_l1_quantal1,dt_l1_struct_0,dt_l2_lattices,dt_l3_lattices,dt_m1_subset_1,dt_u1_struct_0,existence_l1_group_1,existence_l1_lattices,existence_l1_quantal1,existence_l1_struct_0,existence_l2_lattices,existence_l3_lattices,existence_m1_subset_1,fc1_struct_0,fc1_xboole_0,fraenkel_a_0_4_fraenkel,fraenkel_a_0_5_fraenkel,fraenkel_a_3_10_quantal1,fraenkel_a_3_11_quantal1,fraenkel_a_3_12_quantal1,fraenkel_a_3_15_quantal1,fraenkel_a_4_5_quantal1,fraenkel_a_4_6_quantal1,rc1_finset_1,rc1_xboole_0,rc2_quantal1,rc2_xboole_0,rc3_struct_0,redefinition_k4_monoid_0,redefinition_r3_lattices,reflexivity_r3_lattices,s3_fraenkel__e4_49__quantal1,s3_fraenkel__e7_49__quantal1,t11_quantal1,t12_quantal1,t1_subset,t2_subset,t2_tarski,t6_boole,t7_boole,t8_boole]), [file(quantal1,t20_quantal1)]). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc1_finset_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), []). fof(cc1_lattices,axiom,( ! [A] : ( l3_lattices(A) => ( ( ~ v3_struct_0(A) & v10_lattices(A) ) => ( ~ v3_struct_0(A) & v4_lattices(A) & v5_lattices(A) & v6_lattices(A) & v7_lattices(A) & v8_lattices(A) & v9_lattices(A) ) ) ) ), file(lattices,cc1_lattices), []). fof(cc2_lattices,axiom,( ! [A] : ( l3_lattices(A) => ( ( ~ v3_struct_0(A) & v4_lattices(A) & v5_lattices(A) & v6_lattices(A) & v7_lattices(A) & v8_lattices(A) & v9_lattices(A) ) => ( ~ v3_struct_0(A) & v10_lattices(A) ) ) ) ), file(lattices,cc2_lattices), []). fof(d15_quantal1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_quantal1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => k1_quantal1(A,B,C) = k15_lattice3(A,a_3_10_quantal1(A,B,C)) ) ) ) ), file(quantal1,d15_quantal1), []). fof(d16_quantal1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_quantal1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => k2_quantal1(A,B,C) = k15_lattice3(A,a_3_11_quantal1(A,B,C)) ) ) ) ), file(quantal1,d16_quantal1), []). fof(d17_quantal1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_quantal1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ( v17_quantal1(B,A) <=> ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ( k2_quantal1(A,k1_quantal1(A,C,B),B) = C & k1_quantal1(A,k2_quantal1(A,C,B),B) = C ) ) ) ) ) ), file(quantal1,d17_quantal1), []). fof(d4_group_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_group_1(A) ) => ( v4_group_1(A) <=> ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => k1_group_1(A,k1_group_1(A,B,C),D) = k1_group_1(A,B,k1_group_1(A,C,D)) ) ) ) ) ) ), file(group_1,d4_group_1), []). fof(dt_k15_lattice3,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l3_lattices(A) ) => m1_subset_1(k15_lattice3(A,B),u1_struct_0(A)) ) ), file(lattice3,k15_lattice3), []). fof(dt_k1_group_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_group_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k1_group_1(A,B,C),u1_struct_0(A)) ) ), file(group_1,k1_group_1), []). fof(dt_k1_quantal1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_quantal1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k1_quantal1(A,B,C),u1_struct_0(A)) ) ), file(quantal1,k1_quantal1), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k2_quantal1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_quantal1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k2_quantal1(A,B,C),u1_struct_0(A)) ) ), file(quantal1,k2_quantal1), []). fof(dt_k4_monoid_0,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_group_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k4_monoid_0(A,B,C),u1_struct_0(A)) ) ), file(monoid_0,k4_monoid_0), []). fof(dt_l1_group_1,axiom,( ! [A] : ( l1_group_1(A) => l1_struct_0(A) ) ), file(group_1,l1_group_1), []). fof(dt_l1_lattices,axiom,( ! [A] : ( l1_lattices(A) => l1_struct_0(A) ) ), file(lattices,l1_lattices), []). fof(dt_l1_quantal1,axiom,( ! [A] : ( l1_quantal1(A) => ( l3_lattices(A) & l1_group_1(A) ) ) ), file(quantal1,l1_quantal1), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). fof(dt_l2_lattices,axiom,( ! [A] : ( l2_lattices(A) => l1_struct_0(A) ) ), file(lattices,l2_lattices), []). fof(dt_l3_lattices,axiom,( ! [A] : ( l3_lattices(A) => ( l1_lattices(A) & l2_lattices(A) ) ) ), file(lattices,l3_lattices), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). fof(existence_l1_group_1,axiom,( ? [A] : l1_group_1(A) ), file(group_1,l1_group_1), []). fof(existence_l1_lattices,axiom,( ? [A] : l1_lattices(A) ), file(lattices,l1_lattices), []). fof(existence_l1_quantal1,axiom,( ? [A] : l1_quantal1(A) ), file(quantal1,l1_quantal1), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). fof(existence_l2_lattices,axiom,( ? [A] : l2_lattices(A) ), file(lattices,l2_lattices), []). fof(existence_l3_lattices,axiom,( ? [A] : l3_lattices(A) ), file(lattices,l3_lattices), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(fc1_struct_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(u1_struct_0(A)) ) ), file(struct_0,fc1_struct_0), []). fof(fc1_xboole_0,axiom,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), []). fof(fraenkel_a_0_4_fraenkel,axiom,( ! [A] : ( r2_hidden(A,a_0_4_fraenkel) <=> ? [B] : ( m1_subset_1(B,f1_s3_fraenkel) & A = f2_s3_fraenkel(B) & p1_s3_fraenkel(B) ) ) ), file(fraenkel,a_0_4_fraenkel), []). fof(fraenkel_a_0_5_fraenkel,axiom,( ! [A] : ( r2_hidden(A,a_0_5_fraenkel) <=> ? [B] : ( m1_subset_1(B,f1_s3_fraenkel) & A = f2_s3_fraenkel(B) & p2_s3_fraenkel(B) ) ) ), file(fraenkel,a_0_5_fraenkel), []). fof(fraenkel_a_3_10_quantal1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(B) & l1_quantal1(B) & m1_subset_1(C,u1_struct_0(B)) & m1_subset_1(D,u1_struct_0(B)) ) => ( r2_hidden(A,a_3_10_quantal1(B,C,D)) <=> ? [E] : ( m1_subset_1(E,u1_struct_0(B)) & A = E & r1_lattices(B,k4_monoid_0(B,E,C),D) ) ) ) ), file(quantal1,a_3_10_quantal1), []). fof(fraenkel_a_3_11_quantal1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(B) & l1_quantal1(B) & m1_subset_1(C,u1_struct_0(B)) & m1_subset_1(D,u1_struct_0(B)) ) => ( r2_hidden(A,a_3_11_quantal1(B,C,D)) <=> ? [E] : ( m1_subset_1(E,u1_struct_0(B)) & A = E & r1_lattices(B,k4_monoid_0(B,C,E),D) ) ) ) ), file(quantal1,a_3_11_quantal1), []). fof(fraenkel_a_3_12_quantal1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(B) & v10_lattices(B) & v4_lattice3(B) & v4_group_1(B) & v7_quantal1(B) & v8_quantal1(B) & l1_quantal1(B) & m1_subset_1(C,u1_struct_0(B)) & m1_subset_1(D,u1_struct_0(B)) ) => ( r2_hidden(A,a_3_12_quantal1(B,C,D)) <=> ? [E] : ( m1_subset_1(E,u1_struct_0(B)) & A = E & r3_lattices(B,k4_monoid_0(B,C,E),D) ) ) ) ), file(quantal1,a_3_12_quantal1), []). fof(fraenkel_a_3_15_quantal1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(B) & v10_lattices(B) & v4_lattice3(B) & v4_group_1(B) & v7_quantal1(B) & v8_quantal1(B) & l1_quantal1(B) & m1_subset_1(C,u1_struct_0(B)) & m1_subset_1(D,u1_struct_0(B)) ) => ( r2_hidden(A,a_3_15_quantal1(B,C,D)) <=> ? [E] : ( m1_subset_1(E,u1_struct_0(B)) & A = E & r3_lattices(B,k4_monoid_0(B,E,C),D) ) ) ) ), file(quantal1,a_3_15_quantal1), []). fof(fraenkel_a_4_5_quantal1,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(B) & v10_lattices(B) & v4_lattice3(B) & v4_group_1(B) & v7_quantal1(B) & v8_quantal1(B) & l1_quantal1(B) & m1_subset_1(C,u1_struct_0(B)) & m1_subset_1(D,u1_struct_0(B)) & m1_subset_1(E,u1_struct_0(B)) ) => ( r2_hidden(A,a_4_5_quantal1(B,C,D,E)) <=> ? [F] : ( m1_subset_1(F,u1_struct_0(B)) & A = F & r3_lattices(B,k4_monoid_0(B,F,k4_monoid_0(B,D,k2_quantal1(B,E,C))),C) ) ) ) ), file(quantal1,a_4_5_quantal1), []). fof(fraenkel_a_4_6_quantal1,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(B) & v10_lattices(B) & v4_lattice3(B) & v4_group_1(B) & v7_quantal1(B) & v8_quantal1(B) & l1_quantal1(B) & m1_subset_1(C,u1_struct_0(B)) & m1_subset_1(D,u1_struct_0(B)) & m1_subset_1(E,u1_struct_0(B)) ) => ( r2_hidden(A,a_4_6_quantal1(B,C,D,E)) <=> ? [F] : ( m1_subset_1(F,u1_struct_0(B)) & A = F & r3_lattices(B,k4_monoid_0(B,k4_monoid_0(B,k1_quantal1(B,E,C),D),F),C) ) ) ) ), file(quantal1,a_4_6_quantal1), []). fof(rc1_finset_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), []). fof(rc1_xboole_0,axiom,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), []). fof(rc2_quantal1,axiom,( ? [A] : ( l1_quantal1(A) & ~ v3_struct_0(A) ) ), file(quantal1,rc2_quantal1), []). fof(rc2_xboole_0,axiom,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), []). fof(rc3_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). fof(redefinition_k4_monoid_0,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_group_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_monoid_0(A,B,C) = k1_group_1(A,B,C) ) ), file(monoid_0,k4_monoid_0), []). fof(redefinition_r3_lattices,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v6_lattices(A) & v8_lattices(A) & v9_lattices(A) & l3_lattices(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => ( r3_lattices(A,B,C) <=> r1_lattices(A,B,C) ) ) ), file(lattices,r3_lattices), []). fof(reflexivity_r3_lattices,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v6_lattices(A) & v8_lattices(A) & v9_lattices(A) & l3_lattices(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => r3_lattices(A,B,B) ) ), file(lattices,r3_lattices), []). fof(s3_fraenkel__e4_49__quantal1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v4_lattice3(A) & v4_group_1(A) & v7_quantal1(A) & v8_quantal1(A) & l1_quantal1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) & m1_subset_1(D,u1_struct_0(A)) ) => ( ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ( r3_lattices(A,k4_monoid_0(A,E,C),D) <=> r3_lattices(A,k4_monoid_0(A,E,k4_monoid_0(A,C,k2_quantal1(A,D,B))),B) ) ) => a_0_4_fraenkel = a_0_5_fraenkel ) ) ), file(quantal1,s3_fraenkel__e4_49__quantal1), []). fof(s3_fraenkel__e7_49__quantal1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v4_lattice3(A) & v4_group_1(A) & v7_quantal1(A) & v8_quantal1(A) & l1_quantal1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) & m1_subset_1(D,u1_struct_0(A)) ) => ( ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ( r3_lattices(A,k4_monoid_0(A,C,E),D) <=> r3_lattices(A,k4_monoid_0(A,k4_monoid_0(A,k1_quantal1(A,D,B),C),E),B) ) ) => a_0_4_fraenkel = a_0_5_fraenkel ) ) ), file(quantal1,s3_fraenkel__e7_49__quantal1), []). fof(t11_quantal1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v4_lattice3(A) & v4_group_1(A) & v7_quantal1(A) & v8_quantal1(A) & l1_quantal1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ( r3_lattices(A,k4_monoid_0(A,B,C),D) <=> r3_lattices(A,C,k2_quantal1(A,B,D)) ) ) ) ) ) ), file(quantal1,t11_quantal1), []). fof(t12_quantal1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v10_lattices(A) & v4_lattice3(A) & v4_group_1(A) & v7_quantal1(A) & v8_quantal1(A) & l1_quantal1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ( r3_lattices(A,k4_monoid_0(A,B,C),D) <=> r3_lattices(A,B,k1_quantal1(A,C,D)) ) ) ) ) ) ), file(quantal1,t12_quantal1), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t2_subset,axiom,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), []). fof(t2_tarski,axiom,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []).