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Theorem ecoviass 6354
Description: Lemma used to transfer an associative law via an equivalence relation. (Contributed by Jim Kingdon, 16-Sep-2019.)
Hypotheses
Ref Expression
ecoviass.1 𝐷 = ((𝑆 × 𝑆) / )
ecoviass.2 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐺, 𝐻⟩] )
ecoviass.3 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑁, 𝑄⟩] )
ecoviass.4 (((𝐺𝑆𝐻𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
ecoviass.5 (((𝑥𝑆𝑦𝑆) ∧ (𝑁𝑆𝑄𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )
ecoviass.6 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝐺𝑆𝐻𝑆))
ecoviass.7 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑁𝑆𝑄𝑆))
ecoviass.8 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐽 = 𝐿)
ecoviass.9 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐾 = 𝑀)
Assertion
Ref Expression
ecoviass ((𝐴𝐷𝐵𝐷𝐶𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝐴   𝑧,𝐵,𝑤,𝑣,𝑢   𝑥,𝐶,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥, + ,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥, ,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥,𝑆,𝑦,𝑧,𝑤,𝑣,𝑢   𝑧,𝐷,𝑤,𝑣,𝑢
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑄(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐺(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐻(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐽(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐾(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐿(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑀(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑁(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem ecoviass
StepHypRef Expression
1 ecoviass.1 . 2 𝐷 = ((𝑆 × 𝑆) / )
2 oveq1 5620 . . . 4 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = (𝐴 + [⟨𝑧, 𝑤⟩] ))
32oveq1d 5628 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ((𝐴 + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ))
4 oveq1 5620 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )))
53, 4eqeq12d 2099 . 2 ([⟨𝑥, 𝑦⟩] = 𝐴 → ((([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) ↔ ((𝐴 + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = (𝐴 + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ))))
6 oveq2 5621 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 + [⟨𝑧, 𝑤⟩] ) = (𝐴 + 𝐵))
76oveq1d 5628 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → ((𝐴 + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ))
8 oveq1 5620 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = (𝐵 + [⟨𝑣, 𝑢⟩] ))
98oveq2d 5629 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] )))
107, 9eqeq12d 2099 . 2 ([⟨𝑧, 𝑤⟩] = 𝐵 → (((𝐴 + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = (𝐴 + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) ↔ ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] ))))
11 oveq2 5621 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ) = ((𝐴 + 𝐵) + 𝐶))
12 oveq2 5621 . . . 4 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐵 + [⟨𝑣, 𝑢⟩] ) = (𝐵 + 𝐶))
1312oveq2d 5629 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] )) = (𝐴 + (𝐵 + 𝐶)))
1411, 13eqeq12d 2099 . 2 ([⟨𝑣, 𝑢⟩] = 𝐶 → (((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] )) ↔ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))))
15 ecoviass.8 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐽 = 𝐿)
16 ecoviass.9 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐾 = 𝑀)
17 opeq12 3607 . . . . 5 ((𝐽 = 𝐿𝐾 = 𝑀) → ⟨𝐽, 𝐾⟩ = ⟨𝐿, 𝑀⟩)
1817eceq1d 6280 . . . 4 ((𝐽 = 𝐿𝐾 = 𝑀) → [⟨𝐽, 𝐾⟩] = [⟨𝐿, 𝑀⟩] )
1915, 16, 18syl2anc 403 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → [⟨𝐽, 𝐾⟩] = [⟨𝐿, 𝑀⟩] )
20 ecoviass.2 . . . . . . 7 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐺, 𝐻⟩] )
2120oveq1d 5628 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ))
2221adantr 270 . . . . 5 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ))
23 ecoviass.6 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝐺𝑆𝐻𝑆))
24 ecoviass.4 . . . . . 6 (((𝐺𝑆𝐻𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
2523, 24sylan 277 . . . . 5 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
2622, 25eqtrd 2117 . . . 4 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
27263impa 1136 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
28 ecoviass.3 . . . . . . 7 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑁, 𝑄⟩] )
2928oveq2d 5629 . . . . . 6 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ))
3029adantl 271 . . . . 5 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ))
31 ecoviass.7 . . . . . 6 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑁𝑆𝑄𝑆))
32 ecoviass.5 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑁𝑆𝑄𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )
3331, 32sylan2 280 . . . . 5 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )
3430, 33eqtrd 2117 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = [⟨𝐿, 𝑀⟩] )
35343impb 1137 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = [⟨𝐿, 𝑀⟩] )
3619, 27, 353eqtr4d 2127 . 2 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )))
371, 5, 10, 14, 363ecoptocl 6333 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 922   = wceq 1287  wcel 1436  cop 3434   × cxp 4409  (class class class)co 5613  [cec 6242   / cqs 6243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-xp 4417  df-cnv 4419  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fv 4989  df-ov 5616  df-ec 6246  df-qs 6250
This theorem is referenced by:  addassnqg  6885  mulassnqg  6887  addasssrg  7246  mulasssrg  7248  axaddass  7351  axmulass  7352
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