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Theorem ecovcom 8848
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 7433 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = (𝐴 + [⟨𝑧, 𝑤⟩] ))
3 oveq2 7434 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = ([⟨𝑧, 𝑤⟩] + 𝐴))
42, 3eqeq12d 2744 . 2 ([⟨𝑥, 𝑦⟩] = 𝐴 → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) ↔ (𝐴 + [⟨𝑧, 𝑤⟩] ) = ([⟨𝑧, 𝑤⟩] + 𝐴)))
5 oveq2 7434 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 + [⟨𝑧, 𝑤⟩] ) = (𝐴 + 𝐵))
6 oveq1 7433 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → ([⟨𝑧, 𝑤⟩] + 𝐴) = (𝐵 + 𝐴))
75, 6eqeq12d 2744 . 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 4880 . . . . 5 ((𝐷 = 𝐻𝐺 = 𝐽) → ⟨𝐷, 𝐺⟩ = ⟨𝐻, 𝐽⟩)
1110eceq1d 8770 . . . 4 ((𝐷 = 𝐻𝐺 = 𝐽) → [⟨𝐷, 𝐺⟩] = [⟨𝐻, 𝐽⟩] )
128, 9, 11mp2an 690 . . 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 457 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = [⟨𝐻, 𝐽⟩] )
1612, 13, 153eqtr4a 2794 . 2 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ))
171, 4, 7, 162ecoptocl 8833 1 ((𝐴𝐶𝐵𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cop 4638   × cxp 5680  (class class class)co 7426  [cec 8729   / cqs 8730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fv 6561  df-ov 7429  df-ec 8733  df-qs 8737
This theorem is referenced by:  addcomsr  11118  mulcomsr  11120  axmulcom  11186
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