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Theorem ecovdi 8397
Description: Lemma used to transfer a distributive law via an equivalence relation. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.)
Hypotheses
Ref Expression
ecovdi.1 𝐷 = ((𝑆 × 𝑆) / )
ecovdi.2 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑀, 𝑁⟩] )
ecovdi.3 (((𝑥𝑆𝑦𝑆) ∧ (𝑀𝑆𝑁𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑀, 𝑁⟩] ) = [⟨𝐻, 𝐽⟩] )
ecovdi.4 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) = [⟨𝑊, 𝑋⟩] )
ecovdi.5 (((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] ) = [⟨𝑌, 𝑍⟩] )
ecovdi.6 (((𝑊𝑆𝑋𝑆) ∧ (𝑌𝑆𝑍𝑆)) → ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ) = [⟨𝐾, 𝐿⟩] )
ecovdi.7 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑀𝑆𝑁𝑆))
ecovdi.8 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝑊𝑆𝑋𝑆))
ecovdi.9 (((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑌𝑆𝑍𝑆))
ecovdi.10 𝐻 = 𝐾
ecovdi.11 𝐽 = 𝐿
Assertion
Ref Expression
ecovdi ((𝐴𝐷𝐵𝐷𝐶𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝐴   𝑧,𝐵,𝑤,𝑣,𝑢   𝑤,𝐶,𝑣,𝑢   𝑥, + ,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥, ,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥,𝑆,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥, · ,𝑦,𝑧,𝑤,𝑣,𝑢   𝑧,𝐷,𝑤,𝑣,𝑢
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦,𝑧)   𝐷(𝑥,𝑦)   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)   𝑊(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑋(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑌(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑍(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem ecovdi
StepHypRef Expression
1 ecovdi.1 . 2 𝐷 = ((𝑆 × 𝑆) / )
2 oveq1 7155 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )))
3 oveq1 7155 . . . 4 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) = (𝐴 · [⟨𝑧, 𝑤⟩] ))
4 oveq1 7155 . . . 4 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] ) = (𝐴 · [⟨𝑣, 𝑢⟩] ))
53, 4oveq12d 7166 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )) = ((𝐴 · [⟨𝑧, 𝑤⟩] ) + (𝐴 · [⟨𝑣, 𝑢⟩] )))
62, 5eqeq12d 2835 . 2 ([⟨𝑥, 𝑦⟩] = 𝐴 → (([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )) ↔ (𝐴 · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = ((𝐴 · [⟨𝑧, 𝑤⟩] ) + (𝐴 · [⟨𝑣, 𝑢⟩] ))))
7 oveq1 7155 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = (𝐵 + [⟨𝑣, 𝑢⟩] ))
87oveq2d 7164 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] )))
9 oveq2 7156 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 · [⟨𝑧, 𝑤⟩] ) = (𝐴 · 𝐵))
109oveq1d 7163 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → ((𝐴 · [⟨𝑧, 𝑤⟩] ) + (𝐴 · [⟨𝑣, 𝑢⟩] )) = ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] )))
118, 10eqeq12d 2835 . 2 ([⟨𝑧, 𝑤⟩] = 𝐵 → ((𝐴 · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = ((𝐴 · [⟨𝑧, 𝑤⟩] ) + (𝐴 · [⟨𝑣, 𝑢⟩] )) ↔ (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] )) = ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] ))))
12 oveq2 7156 . . . 4 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐵 + [⟨𝑣, 𝑢⟩] ) = (𝐵 + 𝐶))
1312oveq2d 7164 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] )) = (𝐴 · (𝐵 + 𝐶)))
14 oveq2 7156 . . . 4 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐴 · [⟨𝑣, 𝑢⟩] ) = (𝐴 · 𝐶))
1514oveq2d 7164 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] )) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
1613, 15eqeq12d 2835 . 2 ([⟨𝑣, 𝑢⟩] = 𝐶 → ((𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] )) = ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] )) ↔ (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))))
17 ecovdi.10 . . . 4 𝐻 = 𝐾
18 ecovdi.11 . . . 4 𝐽 = 𝐿
19 opeq12 4797 . . . . 5 ((𝐻 = 𝐾𝐽 = 𝐿) → ⟨𝐻, 𝐽⟩ = ⟨𝐾, 𝐿⟩)
2019eceq1d 8320 . . . 4 ((𝐻 = 𝐾𝐽 = 𝐿) → [⟨𝐻, 𝐽⟩] = [⟨𝐾, 𝐿⟩] )
2117, 18, 20mp2an 690 . . 3 [⟨𝐻, 𝐽⟩] = [⟨𝐾, 𝐿⟩]
22 ecovdi.2 . . . . . . 7 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑀, 𝑁⟩] )
2322oveq2d 7164 . . . . . 6 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = ([⟨𝑥, 𝑦⟩] · [⟨𝑀, 𝑁⟩] ))
2423adantl 484 . . . . 5 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = ([⟨𝑥, 𝑦⟩] · [⟨𝑀, 𝑁⟩] ))
25 ecovdi.7 . . . . . 6 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑀𝑆𝑁𝑆))
26 ecovdi.3 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑀𝑆𝑁𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑀, 𝑁⟩] ) = [⟨𝐻, 𝐽⟩] )
2725, 26sylan2 594 . . . . 5 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] · [⟨𝑀, 𝑁⟩] ) = [⟨𝐻, 𝐽⟩] )
2824, 27eqtrd 2854 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = [⟨𝐻, 𝐽⟩] )
29283impb 1109 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = [⟨𝐻, 𝐽⟩] )
30 ecovdi.4 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) = [⟨𝑊, 𝑋⟩] )
31 ecovdi.5 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] ) = [⟨𝑌, 𝑍⟩] )
3230, 31oveqan12d 7167 . . . . 5 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆))) → (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )) = ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ))
33 ecovdi.8 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝑊𝑆𝑋𝑆))
34 ecovdi.9 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑌𝑆𝑍𝑆))
35 ecovdi.6 . . . . . 6 (((𝑊𝑆𝑋𝑆) ∧ (𝑌𝑆𝑍𝑆)) → ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ) = [⟨𝐾, 𝐿⟩] )
3633, 34, 35syl2an 597 . . . . 5 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ) = [⟨𝐾, 𝐿⟩] )
3732, 36eqtrd 2854 . . . 4 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆))) → (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )) = [⟨𝐾, 𝐿⟩] )
38373impdi 1344 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )) = [⟨𝐾, 𝐿⟩] )
3921, 29, 383eqtr4a 2880 . 2 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )))
401, 6, 11, 16, 393ecoptocl 8381 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1081   = wceq 1530  wcel 2107  cop 4565   × cxp 5546  (class class class)co 7148  [cec 8279   / cqs 8280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356  df-ov 7151  df-ec 8283  df-qs 8287
This theorem is referenced by:  distrsr  10505  axdistr  10572
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