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Theorem caovdig 5700
Description: Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
caovdig.1 ((𝜑 ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)))
Assertion
Ref Expression
caovdig ((𝜑 ∧ (𝐴𝐾𝐵𝑆𝐶𝑆)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caovdig
StepHypRef Expression
1 caovdig.1 . . 3 ((𝜑 ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)))
21ralrimivvva 2417 . 2 (𝜑 → ∀𝑥𝐾𝑦𝑆𝑧𝑆 (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)))
3 oveq1 5544 . . . 4 (𝑥 = 𝐴 → (𝑥𝐺(𝑦𝐹𝑧)) = (𝐴𝐺(𝑦𝐹𝑧)))
4 oveq1 5544 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
5 oveq1 5544 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐺𝑧) = (𝐴𝐺𝑧))
64, 5oveq12d 5555 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)) = ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧)))
73, 6eqeq12d 2068 . . 3 (𝑥 = 𝐴 → ((𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)) ↔ (𝐴𝐺(𝑦𝐹𝑧)) = ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧))))
8 oveq1 5544 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐹𝑧) = (𝐵𝐹𝑧))
98oveq2d 5553 . . . 4 (𝑦 = 𝐵 → (𝐴𝐺(𝑦𝐹𝑧)) = (𝐴𝐺(𝐵𝐹𝑧)))
10 oveq2 5545 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
1110oveq1d 5552 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧)))
129, 11eqeq12d 2068 . . 3 (𝑦 = 𝐵 → ((𝐴𝐺(𝑦𝐹𝑧)) = ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧)) ↔ (𝐴𝐺(𝐵𝐹𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧))))
13 oveq2 5545 . . . . 5 (𝑧 = 𝐶 → (𝐵𝐹𝑧) = (𝐵𝐹𝐶))
1413oveq2d 5553 . . . 4 (𝑧 = 𝐶 → (𝐴𝐺(𝐵𝐹𝑧)) = (𝐴𝐺(𝐵𝐹𝐶)))
15 oveq2 5545 . . . . 5 (𝑧 = 𝐶 → (𝐴𝐺𝑧) = (𝐴𝐺𝐶))
1615oveq2d 5553 . . . 4 (𝑧 = 𝐶 → ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))
1714, 16eqeq12d 2068 . . 3 (𝑧 = 𝐶 → ((𝐴𝐺(𝐵𝐹𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧)) ↔ (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶))))
187, 12, 17rspc3v 2685 . 2 ((𝐴𝐾𝐵𝑆𝐶𝑆) → (∀𝑥𝐾𝑦𝑆𝑧𝑆 (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶))))
192, 18mpan9 269 1 ((𝜑 ∧ (𝐴𝐾𝐵𝑆𝐶𝑆)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 894   = wceq 1257  wcel 1407  wral 2321  (class class class)co 5537
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-un 2947  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-br 3790  df-iota 4892  df-fv 4935  df-ov 5540
This theorem is referenced by:  caovdid  5701  caovdi  5705
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