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Theorem ecovcom 8009
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 6802 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = (𝐴 + [⟨𝑧, 𝑤⟩] ))
3 oveq2 6803 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = ([⟨𝑧, 𝑤⟩] + 𝐴))
42, 3eqeq12d 2786 . 2 ([⟨𝑥, 𝑦⟩] = 𝐴 → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) ↔ (𝐴 + [⟨𝑧, 𝑤⟩] ) = ([⟨𝑧, 𝑤⟩] + 𝐴)))
5 oveq2 6803 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 + [⟨𝑧, 𝑤⟩] ) = (𝐴 + 𝐵))
6 oveq1 6802 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → ([⟨𝑧, 𝑤⟩] + 𝐴) = (𝐵 + 𝐴))
75, 6eqeq12d 2786 . 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 4542 . . . . 5 ((𝐷 = 𝐻𝐺 = 𝐽) → ⟨𝐷, 𝐺⟩ = ⟨𝐻, 𝐽⟩)
1110eceq1d 7938 . . . 4 ((𝐷 = 𝐻𝐺 = 𝐽) → [⟨𝐷, 𝐺⟩] = [⟨𝐻, 𝐽⟩] )
128, 9, 11mp2an 672 . . 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 446 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = [⟨𝐻, 𝐽⟩] )
1612, 13, 153eqtr4a 2831 . 2 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ))
171, 4, 7, 162ecoptocl 7993 1 ((𝐴𝐶𝐵𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cop 4323   × cxp 5248  (class class class)co 6795  [cec 7897   / cqs 7898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fv 6038  df-ov 6798  df-ec 7901  df-qs 7905
This theorem is referenced by:  addcomsr  10113  mulcomsr  10115  axmulcom  10181
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