Theorem ecovcom 8861

Description: Lemma used to transfer a commutative law via an equivalence relation. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Hypotheses

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

Assertion

Ref	Expression
ecovcom	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝐴   𝑧,𝐵,𝑤   𝑥, + ,𝑦,𝑧,𝑤   𝑥, ∼ ,𝑦,𝑧,𝑤   𝑥,𝑆,𝑦,𝑧,𝑤   𝑧,𝐶,𝑤

Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦,𝑧,𝑤)   𝐺(𝑥,𝑦,𝑧,𝑤)   𝐻(𝑥,𝑦,𝑧,𝑤)   𝐽(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem ecovcom

Step	Hyp	Ref	Expression
1		ecovcom.1	. 2 ⊢ 𝐶 = ((𝑆 × 𝑆) / ∼ )
2		oveq1 7437	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 + [⟨𝑧, 𝑤⟩] ∼ ))
3		oveq2 7438	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑥, 𝑦⟩] ∼ ) = ([⟨𝑧, 𝑤⟩] ∼ + 𝐴))
4	2, 3	eqeq12d 2750	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → (([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) = ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑥, 𝑦⟩] ∼ ) ↔ (𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) = ([⟨𝑧, 𝑤⟩] ∼ + 𝐴)))
5		oveq2 7438	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 + 𝐵))
6		oveq1 7437	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ([⟨𝑧, 𝑤⟩] ∼ + 𝐴) = (𝐵 + 𝐴))
7	5, 6	eqeq12d 2750	. 2 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ((𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) = ([⟨𝑧, 𝑤⟩] ∼ + 𝐴) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴)))
8		ecovcom.4	. . . 4 ⊢ 𝐷 = 𝐻
9		ecovcom.5	. . . 4 ⊢ 𝐺 = 𝐽
10		opeq12 4879	. . . . 5 ⊢ ((𝐷 = 𝐻 ∧ 𝐺 = 𝐽) → ⟨𝐷, 𝐺⟩ = ⟨𝐻, 𝐽⟩)
11	10	eceq1d 8783	. . . 4 ⊢ ((𝐷 = 𝐻 ∧ 𝐺 = 𝐽) → [⟨𝐷, 𝐺⟩] ∼ = [⟨𝐻, 𝐽⟩] ∼ )
12	8, 9, 11	mp2an 692	. . 3 ⊢ [⟨𝐷, 𝐺⟩] ∼ = [⟨𝐻, 𝐽⟩] ∼
13		ecovcom.2	. . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) = [⟨𝐷, 𝐺⟩] ∼ )
14		ecovcom.3	. . . 4 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑥, 𝑦⟩] ∼ ) = [⟨𝐻, 𝐽⟩] ∼ )
15	14	ancoms 458	. . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑥, 𝑦⟩] ∼ ) = [⟨𝐻, 𝐽⟩] ∼ )
16	12, 13, 15	3eqtr4a 2800	. 2 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) = ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑥, 𝑦⟩] ∼ ))
17	1, 4, 7, 16	2ecoptocl 8846	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ⟨cop 4636   × cxp 5686  (class class class)co 7430  [cec 8741   / cqs 8742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fv 6570  df-ov 7433  df-ec 8745  df-qs 8749

This theorem is referenced by:  addcomsr  11124  mulcomsr  11126  axmulcom  11192