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Theorem ecovicom 6416
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 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐷, 𝐺⟩] )
ecovicom.3 (((𝑧𝑆𝑤𝑆) ∧ (𝑥𝑆𝑦𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = [⟨𝐻, 𝐽⟩] )
ecovicom.4 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → 𝐷 = 𝐻)
ecovicom.5 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → 𝐺 = 𝐽)
Assertion
Ref Expression
ecovicom ((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 ecovicom
StepHypRef Expression
1 ecovicom.1 . 2 𝐶 = ((𝑆 × 𝑆) / )
2 oveq1 5675 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = (𝐴 + [⟨𝑧, 𝑤⟩] ))
3 oveq2 5676 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = ([⟨𝑧, 𝑤⟩] + 𝐴))
42, 3eqeq12d 2103 . 2 ([⟨𝑥, 𝑦⟩] = 𝐴 → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) ↔ (𝐴 + [⟨𝑧, 𝑤⟩] ) = ([⟨𝑧, 𝑤⟩] + 𝐴)))
5 oveq2 5676 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 + [⟨𝑧, 𝑤⟩] ) = (𝐴 + 𝐵))
6 oveq1 5675 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → ([⟨𝑧, 𝑤⟩] + 𝐴) = (𝐵 + 𝐴))
75, 6eqeq12d 2103 . 2 ([⟨𝑧, 𝑤⟩] = 𝐵 → ((𝐴 + [⟨𝑧, 𝑤⟩] ) = ([⟨𝑧, 𝑤⟩] + 𝐴) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴)))
8 ecovicom.4 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → 𝐷 = 𝐻)
9 ecovicom.5 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → 𝐺 = 𝐽)
10 opeq12 3632 . . . . 5 ((𝐷 = 𝐻𝐺 = 𝐽) → ⟨𝐷, 𝐺⟩ = ⟨𝐻, 𝐽⟩)
1110eceq1d 6344 . . . 4 ((𝐷 = 𝐻𝐺 = 𝐽) → [⟨𝐷, 𝐺⟩] = [⟨𝐻, 𝐽⟩] )
128, 9, 11syl2anc 404 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → [⟨𝐷, 𝐺⟩] = [⟨𝐻, 𝐽⟩] )
13 ecovicom.2 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐷, 𝐺⟩] )
14 ecovicom.3 . . . 4 (((𝑧𝑆𝑤𝑆) ∧ (𝑥𝑆𝑦𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = [⟨𝐻, 𝐽⟩] )
1514ancoms 265 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ) = [⟨𝐻, 𝐽⟩] )
1612, 13, 153eqtr4d 2131 . 2 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = ([⟨𝑧, 𝑤⟩] + [⟨𝑥, 𝑦⟩] ))
171, 4, 7, 162ecoptocl 6396 1 ((𝐴𝐶𝐵𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1290  wcel 1439  cop 3455   × cxp 4452  (class class class)co 5668  [cec 6306   / cqs 6307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-xp 4460  df-cnv 4462  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fv 5038  df-ov 5671  df-ec 6310  df-qs 6314
This theorem is referenced by:  addcomnqg  7003  mulcomnqg  7005  addcomsrg  7364  mulcomsrg  7366  axmulcom  7469
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