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Theorem caofdir 6899
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 750 . . . 4 (((𝜑𝑤𝐴) ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧)))
3 caofdi.3 . . . . 5 (𝜑𝐺:𝐴𝑆)
43ffvelrnda 6325 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
5 caofdi.4 . . . . 5 (𝜑𝐻:𝐴𝑆)
65ffvelrnda 6325 . . . 4 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
7 caofdi.2 . . . . 5 (𝜑𝐹:𝐴𝐾)
87ffvelrnda 6325 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝐾)
92, 4, 6, 8caovdird 6817 . . 3 ((𝜑𝑤𝐴) → (((𝐺𝑤)𝑅(𝐻𝑤))𝑇(𝐹𝑤)) = (((𝐺𝑤)𝑇(𝐹𝑤))𝑂((𝐻𝑤)𝑇(𝐹𝑤))))
109mpteq2dva 4714 . 2 (𝜑 → (𝑤𝐴 ↦ (((𝐺𝑤)𝑅(𝐻𝑤))𝑇(𝐹𝑤))) = (𝑤𝐴 ↦ (((𝐺𝑤)𝑇(𝐹𝑤))𝑂((𝐻𝑤)𝑇(𝐹𝑤)))))
11 caofdi.1 . . 3 (𝜑𝐴𝑉)
12 ovexd 6645 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐻𝑤)) ∈ V)
133feqmptd 6216 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
145feqmptd 6216 . . . 4 (𝜑𝐻 = (𝑤𝐴 ↦ (𝐻𝑤)))
1511, 4, 6, 13, 14offval2 6879 . . 3 (𝜑 → (𝐺𝑓 𝑅𝐻) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐻𝑤))))
167feqmptd 6216 . . 3 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
1711, 12, 8, 15, 16offval2 6879 . 2 (𝜑 → ((𝐺𝑓 𝑅𝐻) ∘𝑓 𝑇𝐹) = (𝑤𝐴 ↦ (((𝐺𝑤)𝑅(𝐻𝑤))𝑇(𝐹𝑤))))
18 ovexd 6645 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑇(𝐹𝑤)) ∈ V)
19 ovexd 6645 . . 3 ((𝜑𝑤𝐴) → ((𝐻𝑤)𝑇(𝐹𝑤)) ∈ V)
2011, 4, 8, 13, 16offval2 6879 . . 3 (𝜑 → (𝐺𝑓 𝑇𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑇(𝐹𝑤))))
2111, 6, 8, 14, 16offval2 6879 . . 3 (𝜑 → (𝐻𝑓 𝑇𝐹) = (𝑤𝐴 ↦ ((𝐻𝑤)𝑇(𝐹𝑤))))
2211, 18, 19, 20, 21offval2 6879 . 2 (𝜑 → ((𝐺𝑓 𝑇𝐹) ∘𝑓 𝑂(𝐻𝑓 𝑇𝐹)) = (𝑤𝐴 ↦ (((𝐺𝑤)𝑇(𝐹𝑤))𝑂((𝐻𝑤)𝑇(𝐹𝑤)))))
2310, 17, 223eqtr4d 2665 1 (𝜑 → ((𝐺𝑓 𝑅𝐻) ∘𝑓 𝑇𝐹) = ((𝐺𝑓 𝑇𝐹) ∘𝑓 𝑂(𝐻𝑓 𝑇𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3190  cmpt 4683  wf 5853  cfv 5857  (class class class)co 6615  𝑓 cof 6860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862
This theorem is referenced by:  psrlmod  19341  lflvsdi1  33884  mendlmod  37283  expgrowth  38055
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