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Theorem ecovcom 7806
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 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐷, 𝐺⟩] )
ecovcom.3 (((𝑧𝑆𝑤𝑆) ∧ (𝑥𝑆𝑦𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = [⟨𝐻, 𝐽⟩] )
ecovcom.4 𝐷 = 𝐻
ecovcom.5 𝐺 = 𝐽
Assertion
Ref Expression
ecovcom ((𝐴𝐶𝐵𝐶) → (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
StepHypRef Expression
1 ecovcom.1 . 2 𝐶 = ((𝑆 × 𝑆) / )
2 oveq1 6617 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = (𝐴 + [⟨𝑧, 𝑤⟩] ))
3 oveq2 6618 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = ([⟨𝑧, 𝑤⟩] + 𝐴))
42, 3eqeq12d 2636 . 2 ([⟨𝑥, 𝑦⟩] = 𝐴 → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) ↔ (𝐴 + [⟨𝑧, 𝑤⟩] ) = ([⟨𝑧, 𝑤⟩] + 𝐴)))
5 oveq2 6618 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 + [⟨𝑧, 𝑤⟩] ) = (𝐴 + 𝐵))
6 oveq1 6617 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → ([⟨𝑧, 𝑤⟩] + 𝐴) = (𝐵 + 𝐴))
75, 6eqeq12d 2636 . 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 4377 . . . . 5 ((𝐷 = 𝐻𝐺 = 𝐽) → ⟨𝐷, 𝐺⟩ = ⟨𝐻, 𝐽⟩)
1110eceq1d 7735 . . . 4 ((𝐷 = 𝐻𝐺 = 𝐽) → [⟨𝐷, 𝐺⟩] = [⟨𝐻, 𝐽⟩] )
128, 9, 11mp2an 707 . . 3 [⟨𝐷, 𝐺⟩] = [⟨𝐻, 𝐽⟩]
13 ecovcom.2 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐷, 𝐺⟩] )
14 ecovcom.3 . . . 4 (((𝑧𝑆𝑤𝑆) ∧ (𝑥𝑆𝑦𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = [⟨𝐻, 𝐽⟩] )
1514ancoms 469 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = [⟨𝐻, 𝐽⟩] )
1612, 13, 153eqtr4a 2681 . 2 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ))
171, 4, 7, 162ecoptocl 7790 1 ((𝐴𝐶𝐵𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cop 4159   × cxp 5077  (class class class)co 6610  [cec 7692   / cqs 7693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fv 5860  df-ov 6613  df-ec 7696  df-qs 7700
This theorem is referenced by:  addcomsr  9860  mulcomsr  9862  axmulcom  9928
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