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Theorem caovdirg 7083
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 3151 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝐾 ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))
3 oveq1 6883 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
43oveq1d 6891 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝐴𝐹𝑦)𝐺𝑧))
5 oveq1 6883 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐺𝑧) = (𝐴𝐺𝑧))
65oveq1d 6891 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)) = ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧)))
74, 6eqeq12d 2812 . . 3 (𝑥 = 𝐴 → (((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)) ↔ ((𝐴𝐹𝑦)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧))))
8 oveq2 6884 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
98oveq1d 6891 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐹𝑦)𝐺𝑧) = ((𝐴𝐹𝐵)𝐺𝑧))
10 oveq1 6883 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐺𝑧) = (𝐵𝐺𝑧))
1110oveq2d 6892 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧)) = ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧)))
129, 11eqeq12d 2812 . . 3 (𝑦 = 𝐵 → (((𝐴𝐹𝑦)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝑦𝐺𝑧)) ↔ ((𝐴𝐹𝐵)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧))))
13 oveq2 6884 . . . 4 (𝑧 = 𝐶 → ((𝐴𝐹𝐵)𝐺𝑧) = ((𝐴𝐹𝐵)𝐺𝐶))
14 oveq2 6884 . . . . 5 (𝑧 = 𝐶 → (𝐴𝐺𝑧) = (𝐴𝐺𝐶))
15 oveq2 6884 . . . . 5 (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶))
1614, 15oveq12d 6894 . . . 4 (𝑧 = 𝐶 → ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧)) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))
1713, 16eqeq12d 2812 . . 3 (𝑧 = 𝐶 → (((𝐴𝐹𝐵)𝐺𝑧) = ((𝐴𝐺𝑧)𝐻(𝐵𝐺𝑧)) ↔ ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶))))
187, 12, 17rspc3v 3511 . 2 ((𝐴𝑆𝐵𝑆𝐶𝐾) → (∀𝑥𝑆𝑦𝑆𝑧𝐾 ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶))))
192, 18mpan9 503 1 ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝐾)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3087  (class class class)co 6876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-iota 6062  df-fv 6107  df-ov 6879
This theorem is referenced by:  caovdird  7084  srgi  18823  ringi  18872
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