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Theorem caofdi 7098
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 753 . . . 4 (((𝜑𝑤𝐴) ∧ (𝑥𝐾𝑦𝑆𝑧𝑆)) → (𝑥𝑇(𝑦𝑅𝑧)) = ((𝑥𝑇𝑦)𝑂(𝑥𝑇𝑧)))
3 caofdi.2 . . . . 5 (𝜑𝐹:𝐴𝐾)
43ffvelrnda 6522 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝐾)
5 caofdi.3 . . . . 5 (𝜑𝐺:𝐴𝑆)
65ffvelrnda 6522 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
7 caofdi.4 . . . . 5 (𝜑𝐻:𝐴𝑆)
87ffvelrnda 6522 . . . 4 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
92, 4, 6, 8caovdid 7014 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑇((𝐺𝑤)𝑅(𝐻𝑤))) = (((𝐹𝑤)𝑇(𝐺𝑤))𝑂((𝐹𝑤)𝑇(𝐻𝑤))))
109mpteq2dva 4896 . 2 (𝜑 → (𝑤𝐴 ↦ ((𝐹𝑤)𝑇((𝐺𝑤)𝑅(𝐻𝑤)))) = (𝑤𝐴 ↦ (((𝐹𝑤)𝑇(𝐺𝑤))𝑂((𝐹𝑤)𝑇(𝐻𝑤)))))
11 caofdi.1 . . 3 (𝜑𝐴𝑉)
12 ovexd 6843 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐻𝑤)) ∈ V)
133feqmptd 6411 . . 3 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
145feqmptd 6411 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
157feqmptd 6411 . . . 4 (𝜑𝐻 = (𝑤𝐴 ↦ (𝐻𝑤)))
1611, 6, 8, 14, 15offval2 7079 . . 3 (𝜑 → (𝐺𝑓 𝑅𝐻) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐻𝑤))))
1711, 4, 12, 13, 16offval2 7079 . 2 (𝜑 → (𝐹𝑓 𝑇(𝐺𝑓 𝑅𝐻)) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑇((𝐺𝑤)𝑅(𝐻𝑤)))))
18 ovexd 6843 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑇(𝐺𝑤)) ∈ V)
19 ovexd 6843 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑇(𝐻𝑤)) ∈ V)
2011, 4, 6, 13, 14offval2 7079 . . 3 (𝜑 → (𝐹𝑓 𝑇𝐺) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑇(𝐺𝑤))))
2111, 4, 8, 13, 15offval2 7079 . . 3 (𝜑 → (𝐹𝑓 𝑇𝐻) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑇(𝐻𝑤))))
2211, 18, 19, 20, 21offval2 7079 . 2 (𝜑 → ((𝐹𝑓 𝑇𝐺) ∘𝑓 𝑂(𝐹𝑓 𝑇𝐻)) = (𝑤𝐴 ↦ (((𝐹𝑤)𝑇(𝐺𝑤))𝑂((𝐹𝑤)𝑇(𝐻𝑤)))))
2310, 17, 223eqtr4d 2804 1 (𝜑 → (𝐹𝑓 𝑇(𝐺𝑓 𝑅𝐻)) = ((𝐹𝑓 𝑇𝐺) ∘𝑓 𝑂(𝐹𝑓 𝑇𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  Vcvv 3340  cmpt 4881  wf 6045  cfv 6049  (class class class)co 6813  𝑓 cof 7060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062
This theorem is referenced by:  psrlmod  19603  plydivlem4  24250  plydiveu  24252  quotcan  24263  basellem9  25014  lflvsdi2  34869  mendlmod  38265
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