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Theorem caofdi 7724
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 714 . . . 4 (((𝜑𝑤𝐴) ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝑇(𝑦𝑅𝑧)) = ((𝑥𝑇𝑦)𝑂(𝑥𝑇𝑧)))
3 caofdi.2 . . . . 5 (𝜑𝐹:𝐴𝐾)
43ffvelcdmda 7094 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝐾)
5 caofdi.3 . . . . 5 (𝜑𝐺:𝐴𝑆)
65ffvelcdmda 7094 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
7 caofdi.4 . . . . 5 (𝜑𝐻:𝐴𝑆)
87ffvelcdmda 7094 . . . 4 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
92, 4, 6, 8caovdid 7636 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑇((𝐺𝑤)𝑅(𝐻𝑤))) = (((𝐹𝑤)𝑇(𝐺𝑤))𝑂((𝐹𝑤)𝑇(𝐻𝑤))))
109mpteq2dva 5248 . 2 (𝜑 → (𝑤𝐴 ↦ ((𝐹𝑤)𝑇((𝐺𝑤)𝑅(𝐻𝑤)))) = (𝑤𝐴 ↦ (((𝐹𝑤)𝑇(𝐺𝑤))𝑂((𝐹𝑤)𝑇(𝐻𝑤)))))
11 caofdi.1 . . 3 (𝜑𝐴𝑉)
12 ovexd 7455 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐻𝑤)) ∈ V)
133feqmptd 6967 . . 3 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
145feqmptd 6967 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
157feqmptd 6967 . . . 4 (𝜑𝐻 = (𝑤𝐴 ↦ (𝐻𝑤)))
1611, 6, 8, 14, 15offval2 7705 . . 3 (𝜑 → (𝐺f 𝑅𝐻) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐻𝑤))))
1711, 4, 12, 13, 16offval2 7705 . 2 (𝜑 → (𝐹f 𝑇(𝐺f 𝑅𝐻)) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑇((𝐺𝑤)𝑅(𝐻𝑤)))))
18 ovexd 7455 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑇(𝐺𝑤)) ∈ V)
19 ovexd 7455 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑇(𝐻𝑤)) ∈ V)
2011, 4, 6, 13, 14offval2 7705 . . 3 (𝜑 → (𝐹f 𝑇𝐺) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑇(𝐺𝑤))))
2111, 4, 8, 13, 15offval2 7705 . . 3 (𝜑 → (𝐹f 𝑇𝐻) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑇(𝐻𝑤))))
2211, 18, 19, 20, 21offval2 7705 . 2 (𝜑 → ((𝐹f 𝑇𝐺) ∘f 𝑂(𝐹f 𝑇𝐻)) = (𝑤𝐴 ↦ (((𝐹𝑤)𝑇(𝐺𝑤))𝑂((𝐹𝑤)𝑇(𝐻𝑤)))))
2310, 17, 223eqtr4d 2778 1 (𝜑 → (𝐹f 𝑇(𝐺f 𝑅𝐻)) = ((𝐹f 𝑇𝐺) ∘f 𝑂(𝐹f 𝑇𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  Vcvv 3471  cmpt 5231  wf 6544  cfv 6548  (class class class)co 7420  f cof 7683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7685
This theorem is referenced by:  psrlmod  21903  plydivlem4  26244  plydiveu  26246  quotcan  26257  basellem9  27034  lflvsdi2  38551  mendlmod  42617
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