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Theorem caofdi 7661
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 7040 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝐾)
5 caofdi.3 . . . . 5 (𝜑𝐺:𝐴𝑆)
65ffvelcdmda 7040 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
7 caofdi.4 . . . . 5 (𝜑𝐻:𝐴𝑆)
87ffvelcdmda 7040 . . . 4 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
92, 4, 6, 8caovdid 7574 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑇((𝐺𝑤)𝑅(𝐻𝑤))) = (((𝐹𝑤)𝑇(𝐺𝑤))𝑂((𝐹𝑤)𝑇(𝐻𝑤))))
109mpteq2dva 5210 . 2 (𝜑 → (𝑤𝐴 ↦ ((𝐹𝑤)𝑇((𝐺𝑤)𝑅(𝐻𝑤)))) = (𝑤𝐴 ↦ (((𝐹𝑤)𝑇(𝐺𝑤))𝑂((𝐹𝑤)𝑇(𝐻𝑤)))))
11 caofdi.1 . . 3 (𝜑𝐴𝑉)
12 ovexd 7397 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐻𝑤)) ∈ V)
133feqmptd 6915 . . 3 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
145feqmptd 6915 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
157feqmptd 6915 . . . 4 (𝜑𝐻 = (𝑤𝐴 ↦ (𝐻𝑤)))
1611, 6, 8, 14, 15offval2 7642 . . 3 (𝜑 → (𝐺f 𝑅𝐻) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐻𝑤))))
1711, 4, 12, 13, 16offval2 7642 . 2 (𝜑 → (𝐹f 𝑇(𝐺f 𝑅𝐻)) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑇((𝐺𝑤)𝑅(𝐻𝑤)))))
18 ovexd 7397 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑇(𝐺𝑤)) ∈ V)
19 ovexd 7397 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑇(𝐻𝑤)) ∈ V)
2011, 4, 6, 13, 14offval2 7642 . . 3 (𝜑 → (𝐹f 𝑇𝐺) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑇(𝐺𝑤))))
2111, 4, 8, 13, 15offval2 7642 . . 3 (𝜑 → (𝐹f 𝑇𝐻) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑇(𝐻𝑤))))
2211, 18, 19, 20, 21offval2 7642 . 2 (𝜑 → ((𝐹f 𝑇𝐺) ∘f 𝑂(𝐹f 𝑇𝐻)) = (𝑤𝐴 ↦ (((𝐹𝑤)𝑇(𝐺𝑤))𝑂((𝐹𝑤)𝑇(𝐻𝑤)))))
2310, 17, 223eqtr4d 2787 1 (𝜑 → (𝐹f 𝑇(𝐺f 𝑅𝐻)) = ((𝐹f 𝑇𝐺) ∘f 𝑂(𝐹f 𝑇𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3448  cmpt 5193  wf 6497  cfv 6501  (class class class)co 7362  f cof 7620
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622
This theorem is referenced by:  psrlmod  21386  plydivlem4  25672  plydiveu  25674  quotcan  25685  basellem9  26454  lflvsdi2  37570  mendlmod  41549
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