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Theorem caofdi 7705
Description: Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypotheses
Ref Expression
caofdi.1 (𝜑𝐴𝑉)
caofdi.2 (𝜑𝐹:𝐴𝐾)
caofdi.3 (𝜑𝐺:𝐴𝑆)
caofdi.4 (𝜑𝐻:𝐴𝑆)
caofdi.5 ((𝜑 ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝑇(𝑦𝑅𝑧)) = ((𝑥𝑇𝑦)𝑂(𝑥𝑇𝑧)))
Assertion
Ref Expression
caofdi (𝜑 → (𝐹f 𝑇(𝐺f 𝑅𝐻)) = ((𝐹f 𝑇𝐺) ∘f 𝑂(𝐹f 𝑇𝐻)))
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 caofdi
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofdi.5 . . . . 5 ((𝜑 ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝑇(𝑦𝑅𝑧)) = ((𝑥𝑇𝑦)𝑂(𝑥𝑇𝑧)))
21adantlr 713 . . . 4 (((𝜑𝑤𝐴) ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝑇(𝑦𝑅𝑧)) = ((𝑥𝑇𝑦)𝑂(𝑥𝑇𝑧)))
3 caofdi.2 . . . . 5 (𝜑𝐹:𝐴𝐾)
43ffvelcdmda 7083 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝐾)
5 caofdi.3 . . . . 5 (𝜑𝐺:𝐴𝑆)
65ffvelcdmda 7083 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
7 caofdi.4 . . . . 5 (𝜑𝐻:𝐴𝑆)
87ffvelcdmda 7083 . . . 4 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
92, 4, 6, 8caovdid 7618 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑇((𝐺𝑤)𝑅(𝐻𝑤))) = (((𝐹𝑤)𝑇(𝐺𝑤))𝑂((𝐹𝑤)𝑇(𝐻𝑤))))
109mpteq2dva 5247 . 2 (𝜑 → (𝑤𝐴 ↦ ((𝐹𝑤)𝑇((𝐺𝑤)𝑅(𝐻𝑤)))) = (𝑤𝐴 ↦ (((𝐹𝑤)𝑇(𝐺𝑤))𝑂((𝐹𝑤)𝑇(𝐻𝑤)))))
11 caofdi.1 . . 3 (𝜑𝐴𝑉)
12 ovexd 7440 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐻𝑤)) ∈ V)
133feqmptd 6957 . . 3 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
145feqmptd 6957 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
157feqmptd 6957 . . . 4 (𝜑𝐻 = (𝑤𝐴 ↦ (𝐻𝑤)))
1611, 6, 8, 14, 15offval2 7686 . . 3 (𝜑 → (𝐺f 𝑅𝐻) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐻𝑤))))
1711, 4, 12, 13, 16offval2 7686 . 2 (𝜑 → (𝐹f 𝑇(𝐺f 𝑅𝐻)) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑇((𝐺𝑤)𝑅(𝐻𝑤)))))
18 ovexd 7440 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑇(𝐺𝑤)) ∈ V)
19 ovexd 7440 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑇(𝐻𝑤)) ∈ V)
2011, 4, 6, 13, 14offval2 7686 . . 3 (𝜑 → (𝐹f 𝑇𝐺) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑇(𝐺𝑤))))
2111, 4, 8, 13, 15offval2 7686 . . 3 (𝜑 → (𝐹f 𝑇𝐻) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑇(𝐻𝑤))))
2211, 18, 19, 20, 21offval2 7686 . 2 (𝜑 → ((𝐹f 𝑇𝐺) ∘f 𝑂(𝐹f 𝑇𝐻)) = (𝑤𝐴 ↦ (((𝐹𝑤)𝑇(𝐺𝑤))𝑂((𝐹𝑤)𝑇(𝐻𝑤)))))
2310, 17, 223eqtr4d 2782 1 (𝜑 → (𝐹f 𝑇(𝐺f 𝑅𝐻)) = ((𝐹f 𝑇𝐺) ∘f 𝑂(𝐹f 𝑇𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3474  cmpt 5230  wf 6536  cfv 6540  (class class class)co 7405  f cof 7664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666
This theorem is referenced by:  psrlmod  21512  plydivlem4  25800  plydiveu  25802  quotcan  25813  basellem9  26582  lflvsdi2  37937  mendlmod  41920
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