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Theorem caofdir 7710
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 714 . . . 4 (((𝜑𝑤𝐴) ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧)))
3 caofdi.3 . . . . 5 (𝜑𝐺:𝐴𝑆)
43ffvelcdmda 7087 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
5 caofdi.4 . . . . 5 (𝜑𝐻:𝐴𝑆)
65ffvelcdmda 7087 . . . 4 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
7 caofdi.2 . . . . 5 (𝜑𝐹:𝐴𝐾)
87ffvelcdmda 7087 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝐾)
92, 4, 6, 8caovdird 7625 . . 3 ((𝜑𝑤𝐴) → (((𝐺𝑤)𝑅(𝐻𝑤))𝑇(𝐹𝑤)) = (((𝐺𝑤)𝑇(𝐹𝑤))𝑂((𝐻𝑤)𝑇(𝐹𝑤))))
109mpteq2dva 5249 . 2 (𝜑 → (𝑤𝐴 ↦ (((𝐺𝑤)𝑅(𝐻𝑤))𝑇(𝐹𝑤))) = (𝑤𝐴 ↦ (((𝐺𝑤)𝑇(𝐹𝑤))𝑂((𝐻𝑤)𝑇(𝐹𝑤)))))
11 caofdi.1 . . 3 (𝜑𝐴𝑉)
12 ovexd 7444 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐻𝑤)) ∈ V)
133feqmptd 6961 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
145feqmptd 6961 . . . 4 (𝜑𝐻 = (𝑤𝐴 ↦ (𝐻𝑤)))
1511, 4, 6, 13, 14offval2 7690 . . 3 (𝜑 → (𝐺f 𝑅𝐻) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐻𝑤))))
167feqmptd 6961 . . 3 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
1711, 12, 8, 15, 16offval2 7690 . 2 (𝜑 → ((𝐺f 𝑅𝐻) ∘f 𝑇𝐹) = (𝑤𝐴 ↦ (((𝐺𝑤)𝑅(𝐻𝑤))𝑇(𝐹𝑤))))
18 ovexd 7444 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑇(𝐹𝑤)) ∈ V)
19 ovexd 7444 . . 3 ((𝜑𝑤𝐴) → ((𝐻𝑤)𝑇(𝐹𝑤)) ∈ V)
2011, 4, 8, 13, 16offval2 7690 . . 3 (𝜑 → (𝐺f 𝑇𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑇(𝐹𝑤))))
2111, 6, 8, 14, 16offval2 7690 . . 3 (𝜑 → (𝐻f 𝑇𝐹) = (𝑤𝐴 ↦ ((𝐻𝑤)𝑇(𝐹𝑤))))
2211, 18, 19, 20, 21offval2 7690 . 2 (𝜑 → ((𝐺f 𝑇𝐹) ∘f 𝑂(𝐻f 𝑇𝐹)) = (𝑤𝐴 ↦ (((𝐺𝑤)𝑇(𝐹𝑤))𝑂((𝐻𝑤)𝑇(𝐹𝑤)))))
2310, 17, 223eqtr4d 2783 1 (𝜑 → ((𝐺f 𝑅𝐻) ∘f 𝑇𝐹) = ((𝐺f 𝑇𝐹) ∘f 𝑂(𝐻f 𝑇𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3475  cmpt 5232  wf 6540  cfv 6544  (class class class)co 7409  f cof 7668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670
This theorem is referenced by:  psrlmod  21521  lflvsdi1  37948  mendlmod  41935  expgrowth  43094
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