Theorem ecoprdi 3257

Description: Lemma used in proving distributive laws via equivalence classes.

Hypotheses

Ref	Expression
ecoprdist.1	⊢ D = ((S × S) / R)
ecoprdist.2	⊢ (((z ∈ S ∧ w ∈ S) ∧ (v ∈ S ∧ u ∈ S)) → ([⟨z, w⟩]RF[⟨v, u⟩]R) = [⟨M, N⟩]R)
ecoprdist.3	⊢ (((x ∈ S ∧ y ∈ S) ∧ (M ∈ S ∧ N ∈ S)) → ([⟨x, y⟩]RG[⟨M, N⟩]R) = [⟨H, J⟩]R)
ecoprdist.4	⊢ (((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S)) → ([⟨x, y⟩]RG[⟨z, w⟩]R) = [⟨W, X⟩]R)
ecoprdist.5	⊢ (((x ∈ S ∧ y ∈ S) ∧ (v ∈ S ∧ u ∈ S)) → ([⟨x, y⟩]RG[⟨v, u⟩]R) = [⟨Y, Z⟩]R)
ecoprdist.6	⊢ (((W ∈ S ∧ X ∈ S) ∧ (Y ∈ S ∧ Z ∈ S)) → ([⟨W, X⟩]RF[⟨Y, Z⟩]R) = [⟨K, L⟩]R)
ecoprdist.7	⊢ (((z ∈ S ∧ w ∈ S) ∧ (v ∈ S ∧ u ∈ S)) → (M ∈ S ∧ N ∈ S))
ecoprdist.8	⊢ (((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S)) → (W ∈ S ∧ X ∈ S))
ecoprdist.9	⊢ (((x ∈ S ∧ y ∈ S) ∧ (v ∈ S ∧ u ∈ S)) → (Y ∈ S ∧ Z ∈ S))
ecoprdist.10	⊢ H = K
ecoprdist.11	⊢ J = L

Assertion

Ref	Expression
ecoprdi	⊢ ((A ∈ D ∧ B ∈ D ∧ C ∈ D) → (AG(BFC)) = ((AGB)F(AGC)))

Distinct variable group(s):   x,y,z,w,v,u,A   x,B,y,z,w,v,u   x,C,y,z,w,v,u   x,F,y,z,w,v,u   x,R,y,z,w,v,u   x,S,y,z,w,v,u   x,G,y,z,w,v,u   z,D,w,v,u

Proof of Theorem ecoprdi

Step	Hyp	Ref	Expression
1		ecoprdist.1	. 2 ⊢ D = ((S × S) / R)
2		opreq1 3006	. . 3 ⊢ ([⟨x, y⟩]R = A → ([⟨x, y⟩]RG([⟨z, w⟩]RF[⟨v, u⟩]R)) = (AG([⟨z, w⟩]RF[⟨v, u⟩]R)))
3		opreq1 3006	. . . 4 ⊢ ([⟨x, y⟩]R = A → ([⟨x, y⟩]RG[⟨z, w⟩]R) = (AG[⟨z, w⟩]R))
4		opreq1 3006	. . . 4 ⊢ ([⟨x, y⟩]R = A → ([⟨x, y⟩]RG[⟨v, u⟩]R) = (AG[⟨v, u⟩]R))
5	3, 4	opreq12d 3014	. . 3 ⊢ ([⟨x, y⟩]R = A → (([⟨x, y⟩]RG[⟨z, w⟩]R)F([⟨x, y⟩]RG[⟨v, u⟩]R)) = ((AG[⟨z, w⟩]R)F(AG[⟨v, u⟩]R)))
6	2, 5	cleq12d 1115	. 2 ⊢ ([⟨x, y⟩]R = A → (([⟨x, y⟩]RG([⟨z, w⟩]RF[⟨v, u⟩]R)) = (([⟨x, y⟩]RG[⟨z, w⟩]R)F([⟨x, y⟩]RG[⟨v, u⟩]R)) ↔ (AG([⟨z, w⟩]RF[⟨v, u⟩]R)) = ((AG[⟨z, w⟩]R)F(AG[⟨v, u⟩]R))))
7		opreq1 3006	. . . 4 ⊢ ([⟨z, w⟩]R = B → ([⟨z, w⟩]RF[⟨v, u⟩]R) = (BF[⟨v, u⟩]R))
8	7	opreq2d 3013	. . 3 ⊢ ([⟨z, w⟩]R = B → (AG([⟨z, w⟩]RF[⟨v, u⟩]R)) = (AG(BF[⟨v, u⟩]R)))
9		opreq2 3007	. . . 4 ⊢ ([⟨z, w⟩]R = B → (AG[⟨z, w⟩]R) = (AGB))
10	9	opreq1d 3012	. . 3 ⊢ ([⟨z, w⟩]R = B → ((AG[⟨z, w⟩]R)F(AG[⟨v, u⟩]R)) = ((AGB)F(AG[⟨v, u⟩]R)))
11	8, 10	cleq12d 1115	. 2 ⊢ ([⟨z, w⟩]R = B → ((AG([⟨z, w⟩]RF[⟨v, u⟩]R)) = ((AG[⟨z, w⟩]R)F(AG[⟨v, u⟩]R)) ↔ (AG(BF[⟨v, u⟩]R)) = ((AGB)F(AG[⟨v, u⟩]R))))
12		opreq2 3007	. . . 4 ⊢ ([⟨v, u⟩]R = C → (BF[⟨v, u⟩]R) = (BFC))
13	12	opreq2d 3013	. . 3 ⊢ ([⟨v, u⟩]R = C → (AG(BF[⟨v, u⟩]R)) = (AG(BFC)))
14		opreq2 3007	. . . 4 ⊢ ([⟨v, u⟩]R = C → (AG[⟨v, u⟩]R) = (AGC))
15	14	opreq2d 3013	. . 3 ⊢ ([⟨v, u⟩]R = C → ((AGB)F(AG[⟨v, u⟩]R)) = ((AGB)F(AGC)))
16	13, 15	cleq12d 1115	. 2 ⊢ ([⟨v, u⟩]R = C → ((AG(BF[⟨v, u⟩]R)) = ((AGB)F(AG[⟨v, u⟩]R)) ↔ (AG(BFC)) = ((AGB)F(AGC))))
17		ecoprdist.2	. . . . . . . 8 ⊢ (((z ∈ S ∧ w ∈ S) ∧ (v ∈ S ∧ u ∈ S)) → ([⟨z, w⟩]RF[⟨v, u⟩]R) = [⟨M, N⟩]R)
18	17	opreq2d 3013	. . . . . . 7 ⊢ (((z ∈ S ∧ w ∈ S) ∧ (v ∈ S ∧ u ∈ S)) → ([⟨x, y⟩]RG([⟨z, w⟩]RF[⟨v, u⟩]R)) = ([⟨x, y⟩]RG[⟨M, N⟩]R))
19	18	adantl 305	. . . . . 6 ⊢ (((x ∈ S ∧ y ∈ S) ∧ ((z ∈ S ∧ w ∈ S) ∧ (v ∈ S ∧ u ∈ S))) → ([⟨x, y⟩]RG([⟨z, w⟩]RF[⟨v, u⟩]R)) = ([⟨x, y⟩]RG[⟨M, N⟩]R))
20		ecoprdist.3	. . . . . . 7 ⊢ (((x ∈ S ∧ y ∈ S) ∧ (M ∈ S ∧ N ∈ S)) → ([⟨x, y⟩]RG[⟨M, N⟩]R) = [⟨H, J⟩]R)
21		ecoprdist.7	. . . . . . 7 ⊢ (((z ∈ S ∧ w ∈ S) ∧ (v ∈ S ∧ u ∈ S)) → (M ∈ S ∧ N ∈ S))
22	20, 21	sylan2 346	. . . . . 6 ⊢ (((x ∈ S ∧ y ∈ S) ∧ ((z ∈ S ∧ w ∈ S) ∧ (v ∈ S ∧ u ∈ S))) → ([⟨x, y⟩]RG[⟨M, N⟩]R) = [⟨H, J⟩]R)
23	19, 22	eqtrd 1128	. . . . 5 ⊢ (((x ∈ S ∧ y ∈ S) ∧ ((z ∈ S ∧ w ∈ S) ∧ (v ∈ S ∧ u ∈ S))) → ([⟨x, y⟩]RG([⟨z, w⟩]RF[⟨v, u⟩]R)) = [⟨H, J⟩]R)
24	23	3impb 610	. . . 4 ⊢ (((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S) ∧ (v ∈ S ∧ u ∈ S)) → ([⟨x, y⟩]RG([⟨z, w⟩]RF[⟨v, u⟩]R)) = [⟨H, J⟩]R)
25		ecoprdist.10	. . . . 5 ⊢ H = K
26		ecoprdist.11	. . . . 5 ⊢ J = L
27		opeq12 1878	. . . . . 6 ⊢ ((H = K ∧ J = L) → ⟨H, J⟩ = ⟨K, L⟩)
28		eceq2 3215	. . . . . 6 ⊢ (⟨H, J⟩ = ⟨K, L⟩ → [⟨H, J⟩]R = [⟨K, L⟩]R)
29	27, 28	syl 12	. . . . 5 ⊢ ((H = K ∧ J = L) → [⟨H, J⟩]R = [⟨K, L⟩]R)
30	25, 26, 29	mp2an 520	. . . 4 ⊢ [⟨H, J⟩]R = [⟨K, L⟩]R
31	24, 30	syl6eq 1140	. . 3 ⊢ (((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S) ∧ (v ∈ S ∧ u ∈ S)) → ([⟨x, y⟩]RG([⟨z, w⟩]RF[⟨v, u⟩]R)) = [⟨K, L⟩]R)
32		ecoprdist.4	. . . . . 6 ⊢ (((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S)) → ([⟨x, y⟩]RG[⟨z, w⟩]R) = [⟨W, X⟩]R)
33		ecoprdist.5	. . . . . 6 ⊢ (((x ∈ S ∧ y ∈ S) ∧ (v ∈ S ∧ u ∈ S)) → ([⟨x, y⟩]RG[⟨v, u⟩]R) = [⟨Y, Z⟩]R)
34	32, 33	opreqan12d 3015	. . . . 5 ⊢ ((((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S)) ∧ ((x ∈ S ∧ y ∈ S) ∧ (v ∈ S ∧ u ∈ S))) → (([⟨x, y⟩]RG[⟨z, w⟩]R)F([⟨x, y⟩]RG[⟨v, u⟩]R)) = ([⟨W, X⟩]RF[⟨Y, Z⟩]R))
35		ecoprdist.6	. . . . . 6 ⊢ (((W ∈ S ∧ X ∈ S) ∧ (Y ∈ S ∧ Z ∈ S)) → ([⟨W, X⟩]RF[⟨Y, Z⟩]R) = [⟨K, L⟩]R)
36		ecoprdist.8	. . . . . 6 ⊢ (((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S)) → (W ∈ S ∧ X ∈ S))
37		ecoprdist.9	. . . . . 6 ⊢ (((x ∈ S ∧ y ∈ S) ∧ (v ∈ S ∧ u ∈ S)) → (Y ∈ S ∧ Z ∈ S))
38	35, 36, 37	syl2an 349	. . . . 5 ⊢ ((((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S)) ∧ ((x ∈ S ∧ y ∈ S) ∧ (v ∈ S ∧ u ∈ S))) → ([⟨W, X⟩]RF[⟨Y, Z⟩]R) = [⟨K, L⟩]R)
39	34, 38	eqtrd 1128	. . . 4 ⊢ ((((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S)) ∧ ((x ∈ S ∧ y ∈ S) ∧ (v ∈ S ∧ u ∈ S))) → (([⟨x, y⟩]RG[⟨z, w⟩]R)F([⟨x, y⟩]RG[⟨v, u⟩]R)) = [⟨K, L⟩]R)
40	39	3impdi 630	. . 3 ⊢ (((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S) ∧ (v ∈ S ∧ u ∈ S)) → (([⟨x, y⟩]RG[⟨z, w⟩]R)F([⟨x, y⟩]RG[⟨v, u⟩]R)) = [⟨K, L⟩]R)
41	31, 40	eqtr4d 1131	. 2 ⊢ (((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S) ∧ (v ∈ S ∧ u ∈ S)) → ([⟨x, y⟩]RG([⟨z, w⟩]RF[⟨v, u⟩]R)) = (([⟨x, y⟩]RG[⟨z, w⟩]R)F([⟨x, y⟩]RG[⟨v, u⟩]R)))
42	1, 6, 11, 16, 41	3ecoptocl 3241	1 ⊢ ((A ∈ D ∧ B ∈ D ∧ C ∈ D) → (AG(BFC)) = ((AGB)F(AGC)))

Syntax hints:   → wi 2   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   × cxp 2408  (class class class)co 3001  [cec 3198   / cqs 3199

This theorem is referenced by:  distrpq 3861  distrsr 3994  axdistr 4074

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003  df-ec 3202  df-qs 3205

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