Theorem ecovcom 6701

Description: Lemma used to transfer a commutative law via an equivalence relation. Most uses will want ecovicom 6702 instead. (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 5929	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 + [⟨𝑧, 𝑤⟩] ∼ ))
3		oveq2 5930	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑥, 𝑦⟩] ∼ ) = ([⟨𝑧, 𝑤⟩] ∼ + 𝐴))
4	2, 3	eqeq12d 2211	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → (([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) = ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑥, 𝑦⟩] ∼ ) ↔ (𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) = ([⟨𝑧, 𝑤⟩] ∼ + 𝐴)))
5		oveq2 5930	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 + 𝐵))
6		oveq1 5929	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ([⟨𝑧, 𝑤⟩] ∼ + 𝐴) = (𝐵 + 𝐴))
7	5, 6	eqeq12d 2211	. 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 3810	. . . . 5 ⊢ ((𝐷 = 𝐻 ∧ 𝐺 = 𝐽) → ⟨𝐷, 𝐺⟩ = ⟨𝐻, 𝐽⟩)
11	10	eceq1d 6628	. . . 4 ⊢ ((𝐷 = 𝐻 ∧ 𝐺 = 𝐽) → [⟨𝐷, 𝐺⟩] ∼ = [⟨𝐻, 𝐽⟩] ∼ )
12	8, 9, 11	mp2an 426	. . 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 268	. . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑥, 𝑦⟩] ∼ ) = [⟨𝐻, 𝐽⟩] ∼ )
16	12, 13, 15	3eqtr4a 2255	. 2 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) = ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑥, 𝑦⟩] ∼ ))
17	1, 4, 7, 16	2ecoptocl 6682	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ⟨cop 3625   × cxp 4661  (class class class)co 5922  [cec 6590   / cqs 6591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-xp 4669  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fv 5266  df-ov 5925  df-ec 6594  df-qs 6598

This theorem is referenced by: (None)