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Theorem caovdig 5776
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 2452 . 2 (𝜑 → ∀𝑥𝐾𝑦𝑆𝑧𝑆 (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)))
3 oveq1 5620 . . . 4 (𝑥 = 𝐴 → (𝑥𝐺(𝑦𝐹𝑧)) = (𝐴𝐺(𝑦𝐹𝑧)))
4 oveq1 5620 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
5 oveq1 5620 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐺𝑧) = (𝐴𝐺𝑧))
64, 5oveq12d 5631 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)) = ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧)))
73, 6eqeq12d 2099 . . 3 (𝑥 = 𝐴 → ((𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)) ↔ (𝐴𝐺(𝑦𝐹𝑧)) = ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧))))
8 oveq1 5620 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐹𝑧) = (𝐵𝐹𝑧))
98oveq2d 5629 . . . 4 (𝑦 = 𝐵 → (𝐴𝐺(𝑦𝐹𝑧)) = (𝐴𝐺(𝐵𝐹𝑧)))
10 oveq2 5621 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
1110oveq1d 5628 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧)))
129, 11eqeq12d 2099 . . 3 (𝑦 = 𝐵 → ((𝐴𝐺(𝑦𝐹𝑧)) = ((𝐴𝐺𝑦)𝐻(𝐴𝐺𝑧)) ↔ (𝐴𝐺(𝐵𝐹𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧))))
13 oveq2 5621 . . . . 5 (𝑧 = 𝐶 → (𝐵𝐹𝑧) = (𝐵𝐹𝐶))
1413oveq2d 5629 . . . 4 (𝑧 = 𝐶 → (𝐴𝐺(𝐵𝐹𝑧)) = (𝐴𝐺(𝐵𝐹𝐶)))
15 oveq2 5621 . . . . 5 (𝑧 = 𝐶 → (𝐴𝐺𝑧) = (𝐴𝐺𝐶))
1615oveq2d 5629 . . . 4 (𝑧 = 𝐶 → ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))
1714, 16eqeq12d 2099 . . 3 (𝑧 = 𝐶 → ((𝐴𝐺(𝐵𝐹𝑧)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝑧)) ↔ (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶))))
187, 12, 17rspc3v 2729 . 2 ((𝐴𝐾𝐵𝑆𝐶𝑆) → (∀𝑥𝐾𝑦𝑆𝑧𝑆 (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐻(𝑥𝐺𝑧)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶))))
192, 18mpan9 275 1 ((𝜑 ∧ (𝐴𝐾𝐵𝑆𝐶𝑆)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐻(𝐴𝐺𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 922   = wceq 1287  wcel 1436  wral 2355  (class class class)co 5613
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-iota 4946  df-fv 4989  df-ov 5616
This theorem is referenced by:  caovdid  5777  caovdi  5781
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