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Theorem caovdirg 6749
Description: Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.)
Hypothesis
Ref Expression
caovdirg.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))
Assertion
Ref Expression
caovdirg ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝐾)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))
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 caovdirg
StepHypRef Expression
1 caovdirg.1 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))
21ralrimivvva 2955 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝐾 ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))
3 oveq1 6556 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
43oveq1d 6564 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝐴𝐹𝑦)𝐺𝑧))
5 oveq1 6556 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐺𝑧) = (𝐴𝐺𝑧))
65oveq1d 6564 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)) = ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧)))
74, 6eqeq12d 2625 . . 3 (𝑥 = 𝐴 → (((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)) ↔ ((𝐴𝐹𝑦)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧))))
8 oveq2 6557 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
98oveq1d 6564 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐹𝑦)𝐺𝑧) = ((𝐴𝐹𝐵)𝐺𝑧))
10 oveq1 6556 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐺𝑧) = (𝐵𝐺𝑧))
1110oveq2d 6565 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧)) = ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧)))
129, 11eqeq12d 2625 . . 3 (𝑦 = 𝐵 → (((𝐴𝐹𝑦)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧)) ↔ ((𝐴𝐹𝐵)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧))))
13 oveq2 6557 . . . 4 (𝑧 = 𝐶 → ((𝐴𝐹𝐵)𝐺𝑧) = ((𝐴𝐹𝐵)𝐺𝐶))
14 oveq2 6557 . . . . 5 (𝑧 = 𝐶 → (𝐴𝐺𝑧) = (𝐴𝐺𝐶))
15 oveq2 6557 . . . . 5 (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶))
1614, 15oveq12d 6567 . . . 4 (𝑧 = 𝐶 → ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧)) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))
1713, 16eqeq12d 2625 . . 3 (𝑧 = 𝐶 → (((𝐴𝐹𝐵)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧)) ↔ ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶))))
187, 12, 17rspc3v 3296 . 2 ((𝐴𝑆𝐵𝑆𝐶𝐾) → (∀𝑥𝑆𝑦𝑆𝑧𝐾 ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶))))
192, 18mpan9 485 1 ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝐾)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  (class class class)co 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552
This theorem is referenced by:  caovdird  6750  srgi  18334  ringi  18383
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