Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ecovidi GIF version

Theorem ecovidi 6307
 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 5572 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )))
3 oveq1 5572 . . . 4 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) = (𝐴 · [⟨𝑧, 𝑤⟩] ))
4 oveq1 5572 . . . 4 ([⟨𝑥, 𝑦⟩] = 𝐴 → ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] ) = (𝐴 · [⟨𝑣, 𝑢⟩] ))
53, 4oveq12d 5583 . . 3 ([⟨𝑥, 𝑦⟩] = 𝐴 → (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )) = ((𝐴 · [⟨𝑧, 𝑤⟩] ) + (𝐴 · [⟨𝑣, 𝑢⟩] )))
62, 5eqeq12d 2097 . 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 5572 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = (𝐵 + [⟨𝑣, 𝑢⟩] ))
87oveq2d 5581 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] )))
9 oveq2 5573 . . . 4 ([⟨𝑧, 𝑤⟩] = 𝐵 → (𝐴 · [⟨𝑧, 𝑤⟩] ) = (𝐴 · 𝐵))
109oveq1d 5580 . . 3 ([⟨𝑧, 𝑤⟩] = 𝐵 → ((𝐴 · [⟨𝑧, 𝑤⟩] ) + (𝐴 · [⟨𝑣, 𝑢⟩] )) = ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] )))
118, 10eqeq12d 2097 . 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 5573 . . . 4 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐵 + [⟨𝑣, 𝑢⟩] ) = (𝐵 + 𝐶))
1312oveq2d 5581 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐴 · (𝐵 + [⟨𝑣, 𝑢⟩] )) = (𝐴 · (𝐵 + 𝐶)))
14 oveq2 5573 . . . 4 ([⟨𝑣, 𝑢⟩] = 𝐶 → (𝐴 · [⟨𝑣, 𝑢⟩] ) = (𝐴 · 𝐶))
1514oveq2d 5581 . . 3 ([⟨𝑣, 𝑢⟩] = 𝐶 → ((𝐴 · 𝐵) + (𝐴 · [⟨𝑣, 𝑢⟩] )) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
1613, 15eqeq12d 2097 . 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 3593 . . . . 5 ((𝐻 = 𝐾𝐽 = 𝐿) → ⟨𝐻, 𝐽⟩ = ⟨𝐾, 𝐿⟩)
2019eceq1d 6231 . . . 4 ((𝐻 = 𝐾𝐽 = 𝐿) → [⟨𝐻, 𝐽⟩] = [⟨𝐾, 𝐿⟩] )
2117, 18, 20syl2anc 403 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → [⟨𝐻, 𝐽⟩] = [⟨𝐾, 𝐿⟩] )
22 ecovidi.2 . . . . . . 7 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] ) = [⟨𝑀, 𝑁⟩] )
2322oveq2d 5581 . . . . . 6 (((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = ([⟨𝑥, 𝑦⟩] · [⟨𝑀, 𝑁⟩] ))
2423adantl 271 . . . . 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 280 . . . . 5 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] · [⟨𝑀, 𝑁⟩] ) = [⟨𝐻, 𝐽⟩] )
2824, 27eqtrd 2115 . . . 4 (((𝑥𝑆𝑦𝑆) ∧ ((𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = [⟨𝐻, 𝐽⟩] )
29283impb 1135 . . 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 5584 . . . . 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 283 . . . . 5 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆))) → ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ) = [⟨𝐾, 𝐿⟩] )
3732, 36eqtrd 2115 . . . 4 ((((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆)) ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑣𝑆𝑢𝑆))) → (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )) = [⟨𝐾, 𝐿⟩] )
38373impdi 1225 . . 3 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )) = [⟨𝐾, 𝐿⟩] )
3921, 29, 383eqtr4d 2125 . 2 (((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆) ∧ (𝑣𝑆𝑢𝑆)) → ([⟨𝑥, 𝑦⟩] · ([⟨𝑧, 𝑤⟩] + [⟨𝑣, 𝑢⟩] )) = (([⟨𝑥, 𝑦⟩] · [⟨𝑧, 𝑤⟩] ) + ([⟨𝑥, 𝑦⟩] · [⟨𝑣, 𝑢⟩] )))
401, 6, 11, 16, 393ecoptocl 6284 1 ((𝐴𝐷𝐵𝐷𝐶𝐷) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 920   = wceq 1285   ∈ wcel 1434  ⟨cop 3420   × cxp 4390  (class class class)co 5565  [cec 6193   / cqs 6194 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-xp 4398  df-cnv 4400  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fv 4961  df-ov 5568  df-ec 6197  df-qs 6201 This theorem is referenced by:  distrnqg  6716  distrsrg  7075  axdistr  7179
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