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Theorem ecovass 7800
Description: Lemma used to transfer an associative law via an equivalence relation. (Contributed by NM, 31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)
Hypotheses
Ref Expression
ecovass.1 𝐷 = ((𝑆 × 𝑆) / )
ecovass.2 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐺, 𝐻⟩] )
ecovass.3 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑁, 𝑄⟩] )
ecovass.4 (((𝐺𝑆𝐻𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
ecovass.5 (((𝑥𝑆𝑦𝑆) ∧ (𝑁𝑆𝑄𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )
ecovass.6 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝐺𝑆𝐻𝑆))
ecovass.7 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑁𝑆𝑄𝑆))
ecovass.8 𝐽 = 𝐿
ecovass.9 𝐾 = 𝑀
Assertion
Ref Expression
ecovass ((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 ecovass
StepHypRef Expression
1 ecovass.1 . 2 𝐷 = ((𝑆 × 𝑆) / )
2 oveq1 6611 . . . 4 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = (𝐴 + [⟨𝑧, 𝑤⟩] ))
32oveq1d 6619 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ((𝐴 + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ))
4 oveq1 6611 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )))
53, 4eqeq12d 2636 . 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 6612 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 + [⟨𝑧, 𝑤⟩] ) = (𝐴 + 𝐵))
76oveq1d 6619 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → ((𝐴 + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ))
8 oveq1 6611 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = (𝐵 + [⟨𝑣, 𝑢⟩] ))
98oveq2d 6620 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] )))
107, 9eqeq12d 2636 . 2 ([⟨𝑧, 𝑤⟩] = 𝐵 → (((𝐴 + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = (𝐴 + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) ↔ ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] ))))
11 oveq2 6612 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ) = ((𝐴 + 𝐵) + 𝐶))
12 oveq2 6612 . . . 4 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐵 + [⟨𝑣, 𝑢⟩] ) = (𝐵 + 𝐶))
1312oveq2d 6620 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] )) = (𝐴 + (𝐵 + 𝐶)))
1411, 13eqeq12d 2636 . 2 ([⟨𝑣, 𝑢⟩] = 𝐶 → (((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] )) ↔ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))))
15 ecovass.8 . . . 4 𝐽 = 𝐿
16 ecovass.9 . . . 4 𝐾 = 𝑀
17 opeq12 4372 . . . . 5 ((𝐽 = 𝐿𝐾 = 𝑀) → ⟨𝐽, 𝐾⟩ = ⟨𝐿, 𝑀⟩)
1817eceq1d 7728 . . . 4 ((𝐽 = 𝐿𝐾 = 𝑀) → [⟨𝐽, 𝐾⟩] = [⟨𝐿, 𝑀⟩] )
1915, 16, 18mp2an 707 . . 3 [⟨𝐽, 𝐾⟩] = [⟨𝐿, 𝑀⟩]
20 ecovass.2 . . . . . . 7 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐺, 𝐻⟩] )
2120oveq1d 6619 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ))
2221adantr 481 . . . . 5 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ))
23 ecovass.6 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝐺𝑆𝐻𝑆))
24 ecovass.4 . . . . . 6 (((𝐺𝑆𝐻𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
2523, 24sylan 488 . . . . 5 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
2622, 25eqtrd 2655 . . . 4 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
27263impa 1256 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
28 ecovass.3 . . . . . . 7 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑁, 𝑄⟩] )
2928oveq2d 6620 . . . . . 6 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ))
3029adantl 482 . . . . 5 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ))
31 ecovass.7 . . . . . 6 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑁𝑆𝑄𝑆))
32 ecovass.5 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑁𝑆𝑄𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )
3331, 32sylan2 491 . . . . 5 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )
3430, 33eqtrd 2655 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = [⟨𝐿, 𝑀⟩] )
35343impb 1257 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = [⟨𝐿, 𝑀⟩] )
3619, 27, 353eqtr4a 2681 . 2 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )))
371, 5, 10, 14, 363ecoptocl 7784 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  cop 4154   × cxp 5072  (class class class)co 6604  [cec 7685   / cqs 7686
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 4741  ax-nul 4749  ax-pr 4867
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 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fv 5855  df-ov 6607  df-ec 7689  df-qs 7693
This theorem is referenced by:  addasssr  9853  mulasssr  9855  axaddass  9921  axmulass  9922
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