Theorem ecovicom 6609

Description: Lemma used to transfer a commutative law via an equivalence relation. (Contributed by Jim Kingdon, 15-Sep-2019.)

Hypotheses

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

Assertion

Ref	Expression
ecovicom	⊢ ((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 ecovicom

Step	Hyp	Ref	Expression
1		ecovicom.1	. 2 ⊢ 𝐶 = ((𝑆 × 𝑆) / ∼ )
2		oveq1 5849	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 + [⟨𝑧, 𝑤⟩] ∼ ))
3		oveq2 5850	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑥, 𝑦⟩] ∼ ) = ([⟨𝑧, 𝑤⟩] ∼ + 𝐴))
4	2, 3	eqeq12d 2180	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → (([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) = ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑥, 𝑦⟩] ∼ ) ↔ (𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) = ([⟨𝑧, 𝑤⟩] ∼ + 𝐴)))
5		oveq2 5850	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 + 𝐵))
6		oveq1 5849	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ([⟨𝑧, 𝑤⟩] ∼ + 𝐴) = (𝐵 + 𝐴))
7	5, 6	eqeq12d 2180	. 2 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ((𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) = ([⟨𝑧, 𝑤⟩] ∼ + 𝐴) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴)))
8		ecovicom.4	. . . 4 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝐷 = 𝐻)
9		ecovicom.5	. . . 4 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝐺 = 𝐽)
10		opeq12 3760	. . . . 5 ⊢ ((𝐷 = 𝐻 ∧ 𝐺 = 𝐽) → ⟨𝐷, 𝐺⟩ = ⟨𝐻, 𝐽⟩)
11	10	eceq1d 6537	. . . 4 ⊢ ((𝐷 = 𝐻 ∧ 𝐺 = 𝐽) → [⟨𝐷, 𝐺⟩] ∼ = [⟨𝐻, 𝐽⟩] ∼ )
12	8, 9, 11	syl2anc 409	. . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → [⟨𝐷, 𝐺⟩] ∼ = [⟨𝐻, 𝐽⟩] ∼ )
13		ecovicom.2	. . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) = [⟨𝐷, 𝐺⟩] ∼ )
14		ecovicom.3	. . . 4 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑥, 𝑦⟩] ∼ ) = [⟨𝐻, 𝐽⟩] ∼ )
15	14	ancoms 266	. . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑥, 𝑦⟩] ∼ ) = [⟨𝐻, 𝐽⟩] ∼ )
16	12, 13, 15	3eqtr4d 2208	. 2 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) = ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑥, 𝑦⟩] ∼ ))
17	1, 4, 7, 16	2ecoptocl 6589	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136  ⟨cop 3579   × cxp 4602  (class class class)co 5842  [cec 6499   / cqs 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fv 5196  df-ov 5845  df-ec 6503  df-qs 6507

This theorem is referenced by:  addcomnqg  7322  mulcomnqg  7324  addcomsrg  7696  mulcomsrg  7698  axmulcom  7812