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Theorem ecovidi 6549
Description: Lemma used to transfer a distributive law via an equivalence relation. (Contributed by Jim Kingdon, 17-Sep-2019.)
Hypotheses
Ref Expression
ecovidi.1 𝐷 = ((𝑆 × 𝑆) / )
ecovidi.2 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑀, 𝑁⟩] )
ecovidi.3 (((𝑥𝑆𝑦𝑆) ∧ (𝑀𝑆𝑁𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑀, 𝑁⟩] ) = [⟨𝐻, 𝐽⟩] )
ecovidi.4 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) = [⟨𝑊, 𝑋⟩] )
ecovidi.5 (((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] ) = [⟨𝑌, 𝑍⟩] )
ecovidi.6 (((𝑊𝑆𝑋𝑆) ∧ (𝑌𝑆𝑍𝑆)) → ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ) = [⟨𝐾, 𝐿⟩] )
ecovidi.7 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑀𝑆𝑁𝑆))
ecovidi.8 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝑊𝑆𝑋𝑆))
ecovidi.9 (((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑌𝑆𝑍𝑆))
ecovidi.10 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐻 = 𝐾)
ecovidi.11 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐽 = 𝐿)
Assertion
Ref Expression
ecovidi ((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 ecovidi
StepHypRef Expression
1 ecovidi.1 . 2 𝐷 = ((𝑆 × 𝑆) / )
2 oveq1 5789 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )))
3 oveq1 5789 . . . 4 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) = (𝐴 · [⟨𝑧, 𝑤⟩] ))
4 oveq1 5789 . . . 4 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] ) = (𝐴 · [⟨𝑣, 𝑢⟩] ))
53, 4oveq12d 5800 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )) = ((𝐴 · [⟨𝑧, 𝑤⟩] ) + (𝐴 · [⟨𝑣, 𝑢⟩] )))
62, 5eqeq12d 2155 . 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 5789 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = (𝐵 + [⟨𝑣, 𝑢⟩] ))
87oveq2d 5798 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] )))
9 oveq2 5790 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 · [⟨𝑧, 𝑤⟩] ) = (𝐴 · 𝐵))
109oveq1d 5797 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → ((𝐴 · [⟨𝑧, 𝑤⟩] ) + (𝐴 · [⟨𝑣, 𝑢⟩] )) = ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] )))
118, 10eqeq12d 2155 . 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 5790 . . . 4 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐵 + [⟨𝑣, 𝑢⟩] ) = (𝐵 + 𝐶))
1312oveq2d 5798 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] )) = (𝐴 · (𝐵 + 𝐶)))
14 oveq2 5790 . . . 4 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐴 · [⟨𝑣, 𝑢⟩] ) = (𝐴 · 𝐶))
1514oveq2d 5798 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] )) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
1613, 15eqeq12d 2155 . 2 ([⟨𝑣, 𝑢⟩] = 𝐶 → ((𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] )) = ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] )) ↔ (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))))
17 ecovidi.10 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐻 = 𝐾)
18 ecovidi.11 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → 𝐽 = 𝐿)
19 opeq12 3715 . . . . 5 ((𝐻 = 𝐾𝐽 = 𝐿) → ⟨𝐻, 𝐽⟩ = ⟨𝐾, 𝐿⟩)
2019eceq1d 6473 . . . 4 ((𝐻 = 𝐾𝐽 = 𝐿) → [⟨𝐻, 𝐽⟩] = [⟨𝐾, 𝐿⟩] )
2117, 18, 20syl2anc 409 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → [⟨𝐻, 𝐽⟩] = [⟨𝐾, 𝐿⟩] )
22 ecovidi.2 . . . . . . 7 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑀, 𝑁⟩] )
2322oveq2d 5798 . . . . . 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 ecovidi.7 . . . . . 6 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑀𝑆𝑁𝑆))
26 ecovidi.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 2173 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = [⟨𝐻, 𝐽⟩] )
29283impb 1178 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = [⟨𝐻, 𝐽⟩] )
30 ecovidi.4 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) = [⟨𝑊, 𝑋⟩] )
31 ecovidi.5 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] ) = [⟨𝑌, 𝑍⟩] )
3230, 31oveqan12d 5801 . . . . 5 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆))) → (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )) = ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ))
33 ecovidi.8 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) → (𝑊𝑆𝑋𝑆))
34 ecovidi.9 . . . . . 6 (((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (𝑌𝑆𝑍𝑆))
35 ecovidi.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 2173 . . . 4 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆))) → (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )) = [⟨𝐾, 𝐿⟩] )
38373impdi 1272 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )) = [⟨𝐾, 𝐿⟩] )
3921, 29, 383eqtr4d 2183 . 2 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )))
401, 6, 11, 16, 393ecoptocl 6526 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 963   = wceq 1332  wcel 1481  cop 3535   × cxp 4545  (class class class)co 5782  [cec 6435   / cqs 6436
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fv 5139  df-ov 5785  df-ec 6439  df-qs 6443
This theorem is referenced by:  distrnqg  7219  distrsrg  7591  axdistr  7706
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