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Theorem caofdir 6832
 Description: Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.)
Hypotheses
Ref Expression
caofdi.1 (𝜑𝐴𝑉)
caofdi.2 (𝜑𝐹:𝐴𝐾)
caofdi.3 (𝜑𝐺:𝐴𝑆)
caofdi.4 (𝜑𝐻:𝐴𝑆)
caofdir.5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧)))
Assertion
Ref Expression
caofdir (𝜑 → ((𝐺𝑓 𝑅𝐻) ∘𝑓 𝑇𝐹) = ((𝐺𝑓 𝑇𝐹) ∘𝑓 𝑂(𝐻𝑓 𝑇𝐹)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧   𝑥,𝑂,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem caofdir
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofdir.5 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧)))
21adantlr 747 . . . 4 (((𝜑𝑤𝐴) ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧)))
3 caofdi.3 . . . . 5 (𝜑𝐺:𝐴𝑆)
43ffvelrnda 6267 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
5 caofdi.4 . . . . 5 (𝜑𝐻:𝐴𝑆)
65ffvelrnda 6267 . . . 4 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
7 caofdi.2 . . . . 5 (𝜑𝐹:𝐴𝐾)
87ffvelrnda 6267 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝐾)
92, 4, 6, 8caovdird 6750 . . 3 ((𝜑𝑤𝐴) → (((𝐺𝑤)𝑅(𝐻𝑤))𝑇(𝐹𝑤)) = (((𝐺𝑤)𝑇(𝐹𝑤))𝑂((𝐻𝑤)𝑇(𝐹𝑤))))
109mpteq2dva 4672 . 2 (𝜑 → (𝑤𝐴 ↦ (((𝐺𝑤)𝑅(𝐻𝑤))𝑇(𝐹𝑤))) = (𝑤𝐴 ↦ (((𝐺𝑤)𝑇(𝐹𝑤))𝑂((𝐻𝑤)𝑇(𝐹𝑤)))))
11 caofdi.1 . . 3 (𝜑𝐴𝑉)
12 ovex 6577 . . . 4 ((𝐺𝑤)𝑅(𝐻𝑤)) ∈ V
1312a1i 11 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐻𝑤)) ∈ V)
143feqmptd 6159 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
155feqmptd 6159 . . . 4 (𝜑𝐻 = (𝑤𝐴 ↦ (𝐻𝑤)))
1611, 4, 6, 14, 15offval2 6812 . . 3 (𝜑 → (𝐺𝑓 𝑅𝐻) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐻𝑤))))
177feqmptd 6159 . . 3 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
1811, 13, 8, 16, 17offval2 6812 . 2 (𝜑 → ((𝐺𝑓 𝑅𝐻) ∘𝑓 𝑇𝐹) = (𝑤𝐴 ↦ (((𝐺𝑤)𝑅(𝐻𝑤))𝑇(𝐹𝑤))))
19 ovex 6577 . . . 4 ((𝐺𝑤)𝑇(𝐹𝑤)) ∈ V
2019a1i 11 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑇(𝐹𝑤)) ∈ V)
21 ovex 6577 . . . 4 ((𝐻𝑤)𝑇(𝐹𝑤)) ∈ V
2221a1i 11 . . 3 ((𝜑𝑤𝐴) → ((𝐻𝑤)𝑇(𝐹𝑤)) ∈ V)
2311, 4, 8, 14, 17offval2 6812 . . 3 (𝜑 → (𝐺𝑓 𝑇𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑇(𝐹𝑤))))
2411, 6, 8, 15, 17offval2 6812 . . 3 (𝜑 → (𝐻𝑓 𝑇𝐹) = (𝑤𝐴 ↦ ((𝐻𝑤)𝑇(𝐹𝑤))))
2511, 20, 22, 23, 24offval2 6812 . 2 (𝜑 → ((𝐺𝑓 𝑇𝐹) ∘𝑓 𝑂(𝐻𝑓 𝑇𝐹)) = (𝑤𝐴 ↦ (((𝐺𝑤)𝑇(𝐹𝑤))𝑂((𝐻𝑤)𝑇(𝐹𝑤)))))
2610, 18, 253eqtr4d 2654 1 (𝜑 → ((𝐺𝑓 𝑅𝐻) ∘𝑓 𝑇𝐹) = ((𝐺𝑓 𝑇𝐹) ∘𝑓 𝑂(𝐻𝑓 𝑇𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793 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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 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-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795 This theorem is referenced by:  psrlmod  19222  lflvsdi1  33383  mendlmod  36782  expgrowth  37556
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