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Theorem ecovdi 6508
Description: Lemma used to transfer a distributive law via an equivalence relation. Most likely ecovidi 6509 will be more helpful. (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 5749 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )))
3 oveq1 5749 . . . 4 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) = (𝐴 · [⟨𝑧, 𝑤⟩] ))
4 oveq1 5749 . . . 4 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] ) = (𝐴 · [⟨𝑣, 𝑢⟩] ))
53, 4oveq12d 5760 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )) = ((𝐴 · [⟨𝑧, 𝑤⟩] ) + (𝐴 · [⟨𝑣, 𝑢⟩] )))
62, 5eqeq12d 2132 . 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 5749 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = (𝐵 + [⟨𝑣, 𝑢⟩] ))
87oveq2d 5758 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] )))
9 oveq2 5750 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 · [⟨𝑧, 𝑤⟩] ) = (𝐴 · 𝐵))
109oveq1d 5757 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → ((𝐴 · [⟨𝑧, 𝑤⟩] ) + (𝐴 · [⟨𝑣, 𝑢⟩] )) = ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] )))
118, 10eqeq12d 2132 . 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 5750 . . . 4 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐵 + [⟨𝑣, 𝑢⟩] ) = (𝐵 + 𝐶))
1312oveq2d 5758 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] )) = (𝐴 · (𝐵 + 𝐶)))
14 oveq2 5750 . . . 4 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐴 · [⟨𝑣, 𝑢⟩] ) = (𝐴 · 𝐶))
1514oveq2d 5758 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] )) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
1613, 15eqeq12d 2132 . 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 3677 . . . . 5 ((𝐻 = 𝐾𝐽 = 𝐿) → ⟨𝐻, 𝐽⟩ = ⟨𝐾, 𝐿⟩)
2019eceq1d 6433 . . . 4 ((𝐻 = 𝐾𝐽 = 𝐿) → [⟨𝐻, 𝐽⟩] = [⟨𝐾, 𝐿⟩] )
2117, 18, 20mp2an 422 . . 3 [⟨𝐻, 𝐽⟩] = [⟨𝐾, 𝐿⟩]
22 ecovdi.2 . . . . . . 7 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑀, 𝑁⟩] )
2322oveq2d 5758 . . . . . 6 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = ([⟨𝑥, 𝑦⟩] · [⟨𝑀, 𝑁⟩] ))
2423adantl 275 . . . . 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 284 . . . . 5 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] · [⟨𝑀, 𝑁⟩] ) = [⟨𝐻, 𝐽⟩] )
2824, 27eqtrd 2150 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = [⟨𝐻, 𝐽⟩] )
29283impb 1162 . . 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 5761 . . . . 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 287 . . . . 5 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ) = [⟨𝐾, 𝐿⟩] )
3732, 36eqtrd 2150 . . . 4 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆))) → (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )) = [⟨𝐾, 𝐿⟩] )
38373impdi 1256 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )) = [⟨𝐾, 𝐿⟩] )
3921, 29, 383eqtr4a 2176 . 2 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )))
401, 6, 11, 16, 393ecoptocl 6486 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 947   = wceq 1316  wcel 1465  cop 3500   × cxp 4507  (class class class)co 5742  [cec 6395   / cqs 6396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fv 5101  df-ov 5745  df-ec 6399  df-qs 6403
This theorem is referenced by: (None)
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