Theorem ecoprcom 3255

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

Hypotheses

Ref	Expression
ecoprcom.1	⊢ C = ((S × S) / R)
ecoprcom.2	⊢ (((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S)) → ([⟨x, y⟩]RF[⟨z, w⟩]R) = [⟨D, G⟩]R)
ecoprcom.3	⊢ (((z ∈ S ∧ w ∈ S) ∧ (x ∈ S ∧ y ∈ S)) → ([⟨z, w⟩]RF[⟨x, y⟩]R) = [⟨H, J⟩]R)
ecoprcom.4	⊢ D = H
ecoprcom.5	⊢ G = J

Assertion

Ref	Expression
ecoprcom	⊢ ((A ∈ C ∧ B ∈ C) → (AFB) = (BFA))

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

Proof of Theorem ecoprcom

Step	Hyp	Ref	Expression
1		ecoprcom.1	. 2 ⊢ C = ((S × S) / R)
2		opreq1 3006	. . 3 ⊢ ([⟨x, y⟩]R = A → ([⟨x, y⟩]RF[⟨z, w⟩]R) = (AF[⟨z, w⟩]R))
3		opreq2 3007	. . 3 ⊢ ([⟨x, y⟩]R = A → ([⟨z, w⟩]RF[⟨x, y⟩]R) = ([⟨z, w⟩]RFA))
4	2, 3	cleq12d 1115	. 2 ⊢ ([⟨x, y⟩]R = A → (([⟨x, y⟩]RF[⟨z, w⟩]R) = ([⟨z, w⟩]RF[⟨x, y⟩]R) ↔ (AF[⟨z, w⟩]R) = ([⟨z, w⟩]RFA)))
5		opreq2 3007	. . 3 ⊢ ([⟨z, w⟩]R = B → (AF[⟨z, w⟩]R) = (AFB))
6		opreq1 3006	. . 3 ⊢ ([⟨z, w⟩]R = B → ([⟨z, w⟩]RFA) = (BFA))
7	5, 6	cleq12d 1115	. 2 ⊢ ([⟨z, w⟩]R = B → ((AF[⟨z, w⟩]R) = ([⟨z, w⟩]RFA) ↔ (AFB) = (BFA)))
8		ecoprcom.2	. . . 4 ⊢ (((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S)) → ([⟨x, y⟩]RF[⟨z, w⟩]R) = [⟨D, G⟩]R)
9		ecoprcom.4	. . . . 5 ⊢ D = H
10		ecoprcom.5	. . . . 5 ⊢ G = J
11		opeq12 1878	. . . . . 6 ⊢ ((D = H ∧ G = J) → ⟨D, G⟩ = ⟨H, J⟩)
12		eceq2 3215	. . . . . 6 ⊢ (⟨D, G⟩ = ⟨H, J⟩ → [⟨D, G⟩]R = [⟨H, J⟩]R)
13	11, 12	syl 12	. . . . 5 ⊢ ((D = H ∧ G = J) → [⟨D, G⟩]R = [⟨H, J⟩]R)
14	9, 10, 13	mp2an 520	. . . 4 ⊢ [⟨D, G⟩]R = [⟨H, J⟩]R
15	8, 14	syl6eq 1140	. . 3 ⊢ (((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S)) → ([⟨x, y⟩]RF[⟨z, w⟩]R) = [⟨H, J⟩]R)
16		ecoprcom.3	. . . 4 ⊢ (((z ∈ S ∧ w ∈ S) ∧ (x ∈ S ∧ y ∈ S)) → ([⟨z, w⟩]RF[⟨x, y⟩]R) = [⟨H, J⟩]R)
17	16	ancoms 334	. . 3 ⊢ (((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S)) → ([⟨z, w⟩]RF[⟨x, y⟩]R) = [⟨H, J⟩]R)
18	15, 17	eqtr4d 1131	. 2 ⊢ (((x ∈ S ∧ y ∈ S) ∧ (z ∈ S ∧ w ∈ S)) → ([⟨x, y⟩]RF[⟨z, w⟩]R) = ([⟨z, w⟩]RF[⟨x, y⟩]R))
19	1, 4, 7, 18	2ecoptocl 3240	1 ⊢ ((A ∈ C ∧ B ∈ C) → (AFB) = (BFA))

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

This theorem is referenced by:  addcompq 3856  mulcompq 3858  addcomsr 3990  mulcomsr 3992  axaddcom 4070  axmulcom 4071

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-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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