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Theorem caovdir2d 5879
Description: Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovdir2d.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)))
caovdir2d.2 (𝜑𝐴𝑆)
caovdir2d.3 (𝜑𝐵𝑆)
caovdir2d.4 (𝜑𝐶𝑆)
caovdir2d.cl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
caovdir2d.com ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
Assertion
Ref Expression
caovdir2d (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(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,𝑆,𝑦,𝑧

Proof of Theorem caovdir2d
StepHypRef Expression
1 caovdir2d.1 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)))
2 caovdir2d.4 . . 3 (𝜑𝐶𝑆)
3 caovdir2d.2 . . 3 (𝜑𝐴𝑆)
4 caovdir2d.3 . . 3 (𝜑𝐵𝑆)
51, 2, 3, 4caovdid 5878 . 2 (𝜑 → (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)))
6 caovdir2d.com . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
7 caovdir2d.cl . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
87, 3, 4caovcld 5856 . . 3 (𝜑 → (𝐴𝐹𝐵) ∈ 𝑆)
96, 8, 2caovcomd 5859 . 2 (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = (𝐶𝐺(𝐴𝐹𝐵)))
106, 3, 2caovcomd 5859 . . 3 (𝜑 → (𝐴𝐺𝐶) = (𝐶𝐺𝐴))
116, 4, 2caovcomd 5859 . . 3 (𝜑 → (𝐵𝐺𝐶) = (𝐶𝐺𝐵))
1210, 11oveq12d 5724 . 2 (𝜑 → ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)))
135, 9, 123eqtr4d 2142 1 (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 930   = wceq 1299  wcel 1448  (class class class)co 5706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-iota 5024  df-fv 5067  df-ov 5709
This theorem is referenced by:  addcmpblnq  7076  ltanqg  7109  addcmpblnq0  7152  mulasssrg  7454  mulgt0sr  7473  mulextsr1lem  7475
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