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Theorem ecovass 8764
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 7365 . . . 4 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = (𝐴 + [⟨𝑧, 𝑤⟩] ))
32oveq1d 7373 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ((𝐴 + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ))
4 oveq1 7365 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )))
53, 4eqeq12d 2753 . 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 7366 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 + [⟨𝑧, 𝑤⟩] ) = (𝐴 + 𝐵))
76oveq1d 7373 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → ((𝐴 + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ))
8 oveq1 7365 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = (𝐵 + [⟨𝑣, 𝑢⟩] ))
98oveq2d 7374 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] )))
107, 9eqeq12d 2753 . 2 ([⟨𝑧, 𝑤⟩] = 𝐵 → (((𝐴 + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = (𝐴 + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) ↔ ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] ))))
11 oveq2 7366 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ) = ((𝐴 + 𝐵) + 𝐶))
12 oveq2 7366 . . . 4 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐵 + [⟨𝑣, 𝑢⟩] ) = (𝐵 + 𝐶))
1312oveq2d 7374 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] )) = (𝐴 + (𝐵 + 𝐶)))
1411, 13eqeq12d 2753 . 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 4833 . . . . 5 ((𝐽 = 𝐿𝐾 = 𝑀) → ⟨𝐽, 𝐾⟩ = ⟨𝐿, 𝑀⟩)
1817eceq1d 8688 . . . 4 ((𝐽 = 𝐿𝐾 = 𝑀) → [⟨𝐽, 𝐾⟩] = [⟨𝐿, 𝑀⟩] )
1915, 16, 18mp2an 691 . . 3 [⟨𝐽, 𝐾⟩] = [⟨𝐿, 𝑀⟩]
20 ecovass.2 . . . . . . 7 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) = [⟨𝐺, 𝐻⟩] )
2120oveq1d 7373 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ))
2221adantr 482 . . . . 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 581 . . . . 5 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
2622, 25eqtrd 2777 . . . 4 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = [⟨𝐽, 𝐾⟩] )
27263impa 1111 . . 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 7374 . . . . . 6 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ))
3029adantl 483 . . . . 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 594 . . . . 5 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )
3430, 33eqtrd 2777 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = [⟨𝐿, 𝑀⟩] )
35343impb 1116 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = [⟨𝐿, 𝑀⟩] )
3619, 27, 353eqtr4a 2803 . 2 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] + [⟨𝑧, 𝑤⟩] ) + [⟨𝑣, 𝑢⟩] ) = ([⟨𝑥, 𝑦⟩] + ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )))
371, 5, 10, 14, 363ecoptocl 8749 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  cop 4593   × cxp 5632  (class class class)co 7358  [cec 8647   / cqs 8648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fv 6505  df-ov 7361  df-ec 8651  df-qs 8655
This theorem is referenced by:  addasssr  11025  mulasssr  11027  axaddass  11093  axmulass  11094
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