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Theorem caovdir2d 5704
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 5703 . 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 5681 . . 3 (𝜑 → (𝐴𝐹𝐵) ∈ 𝑆)
96, 8, 2caovcomd 5684 . 2 (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = (𝐶𝐺(𝐴𝐹𝐵)))
106, 3, 2caovcomd 5684 . . 3 (𝜑 → (𝐴𝐺𝐶) = (𝐶𝐺𝐴))
116, 4, 2caovcomd 5684 . . 3 (𝜑 → (𝐵𝐺𝐶) = (𝐶𝐺𝐵))
1210, 11oveq12d 5557 . 2 (𝜑 → ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)))
135, 9, 123eqtr4d 2098 1 (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896   = wceq 1259  wcel 1409  (class class class)co 5539
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 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-iota 4894  df-fv 4937  df-ov 5542
This theorem is referenced by:  addcmpblnq  6522  ltanqg  6555  addcmpblnq0  6598  mulasssrg  6900  mulgt0sr  6919  mulextsr1lem  6921
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