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Theorem caofdi 7448
 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)
43ffvelrnda 6847 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝐾)
5 caofdi.3 . . . . 5 (𝜑𝐺:𝐴𝑆)
65ffvelrnda 6847 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
7 caofdi.4 . . . . 5 (𝜑𝐻:𝐴𝑆)
87ffvelrnda 6847 . . . 4 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
92, 4, 6, 8caovdid 7364 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑇((𝐺𝑤)𝑅(𝐻𝑤))) = (((𝐹𝑤)𝑇(𝐺𝑤))𝑂((𝐹𝑤)𝑇(𝐻𝑤))))
109mpteq2dva 5130 . 2 (𝜑 → (𝑤𝐴 ↦ ((𝐹𝑤)𝑇((𝐺𝑤)𝑅(𝐻𝑤)))) = (𝑤𝐴 ↦ (((𝐹𝑤)𝑇(𝐺𝑤))𝑂((𝐹𝑤)𝑇(𝐻𝑤)))))
11 caofdi.1 . . 3 (𝜑𝐴𝑉)
12 ovexd 7190 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐻𝑤)) ∈ V)
133feqmptd 6725 . . 3 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
145feqmptd 6725 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
157feqmptd 6725 . . . 4 (𝜑𝐻 = (𝑤𝐴 ↦ (𝐻𝑤)))
1611, 6, 8, 14, 15offval2 7429 . . 3 (𝜑 → (𝐺f 𝑅𝐻) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐻𝑤))))
1711, 4, 12, 13, 16offval2 7429 . 2 (𝜑 → (𝐹f 𝑇(𝐺f 𝑅𝐻)) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑇((𝐺𝑤)𝑅(𝐻𝑤)))))
18 ovexd 7190 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑇(𝐺𝑤)) ∈ V)
19 ovexd 7190 . . 3 ((𝜑𝑤𝐴) → ((𝐹𝑤)𝑇(𝐻𝑤)) ∈ V)
2011, 4, 6, 13, 14offval2 7429 . . 3 (𝜑 → (𝐹f 𝑇𝐺) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑇(𝐺𝑤))))
2111, 4, 8, 13, 15offval2 7429 . . 3 (𝜑 → (𝐹f 𝑇𝐻) = (𝑤𝐴 ↦ ((𝐹𝑤)𝑇(𝐻𝑤))))
2211, 18, 19, 20, 21offval2 7429 . 2 (𝜑 → ((𝐹f 𝑇𝐺) ∘f 𝑂(𝐹f 𝑇𝐻)) = (𝑤𝐴 ↦ (((𝐹𝑤)𝑇(𝐺𝑤))𝑂((𝐹𝑤)𝑇(𝐻𝑤)))))
2310, 17, 223eqtr4d 2803 1 (𝜑 → (𝐹f 𝑇(𝐺f 𝑅𝐻)) = ((𝐹f 𝑇𝐺) ∘f 𝑂(𝐹f 𝑇𝐻)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ↦ cmpt 5115  ⟶wf 6335  ‘cfv 6339  (class class class)co 7155   ∘f cof 7408 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7410 This theorem is referenced by:  psrlmod  20734  plydivlem4  24996  plydiveu  24998  quotcan  25009  basellem9  25778  lflvsdi2  36681  mendlmod  40538
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