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Theorem caofdir 7465
Description: Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.)
Hypotheses
Ref Expression
caofdi.1 (𝜑𝐴𝑉)
caofdi.2 (𝜑𝐹:𝐴𝐾)
caofdi.3 (𝜑𝐺:𝐴𝑆)
caofdi.4 (𝜑𝐻:𝐴𝑆)
caofdir.5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧)))
Assertion
Ref Expression
caofdir (𝜑 → ((𝐺f 𝑅𝐻) ∘f 𝑇𝐹) = ((𝐺f 𝑇𝐹) ∘f 𝑂(𝐻f 𝑇𝐹)))
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 caofdir
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 caofdir.5 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧)))
21adantlr 715 . . . 4 (((𝜑𝑤𝐴) ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧)))
3 caofdi.3 . . . . 5 (𝜑𝐺:𝐴𝑆)
43ffvelrnda 6862 . . . 4 ((𝜑𝑤𝐴) → (𝐺𝑤) ∈ 𝑆)
5 caofdi.4 . . . . 5 (𝜑𝐻:𝐴𝑆)
65ffvelrnda 6862 . . . 4 ((𝜑𝑤𝐴) → (𝐻𝑤) ∈ 𝑆)
7 caofdi.2 . . . . 5 (𝜑𝐹:𝐴𝐾)
87ffvelrnda 6862 . . . 4 ((𝜑𝑤𝐴) → (𝐹𝑤) ∈ 𝐾)
92, 4, 6, 8caovdird 7383 . . 3 ((𝜑𝑤𝐴) → (((𝐺𝑤)𝑅(𝐻𝑤))𝑇(𝐹𝑤)) = (((𝐺𝑤)𝑇(𝐹𝑤))𝑂((𝐻𝑤)𝑇(𝐹𝑤))))
109mpteq2dva 5126 . 2 (𝜑 → (𝑤𝐴 ↦ (((𝐺𝑤)𝑅(𝐻𝑤))𝑇(𝐹𝑤))) = (𝑤𝐴 ↦ (((𝐺𝑤)𝑇(𝐹𝑤))𝑂((𝐻𝑤)𝑇(𝐹𝑤)))))
11 caofdi.1 . . 3 (𝜑𝐴𝑉)
12 ovexd 7206 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑅(𝐻𝑤)) ∈ V)
133feqmptd 6738 . . . 4 (𝜑𝐺 = (𝑤𝐴 ↦ (𝐺𝑤)))
145feqmptd 6738 . . . 4 (𝜑𝐻 = (𝑤𝐴 ↦ (𝐻𝑤)))
1511, 4, 6, 13, 14offval2 7445 . . 3 (𝜑 → (𝐺f 𝑅𝐻) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑅(𝐻𝑤))))
167feqmptd 6738 . . 3 (𝜑𝐹 = (𝑤𝐴 ↦ (𝐹𝑤)))
1711, 12, 8, 15, 16offval2 7445 . 2 (𝜑 → ((𝐺f 𝑅𝐻) ∘f 𝑇𝐹) = (𝑤𝐴 ↦ (((𝐺𝑤)𝑅(𝐻𝑤))𝑇(𝐹𝑤))))
18 ovexd 7206 . . 3 ((𝜑𝑤𝐴) → ((𝐺𝑤)𝑇(𝐹𝑤)) ∈ V)
19 ovexd 7206 . . 3 ((𝜑𝑤𝐴) → ((𝐻𝑤)𝑇(𝐹𝑤)) ∈ V)
2011, 4, 8, 13, 16offval2 7445 . . 3 (𝜑 → (𝐺f 𝑇𝐹) = (𝑤𝐴 ↦ ((𝐺𝑤)𝑇(𝐹𝑤))))
2111, 6, 8, 14, 16offval2 7445 . . 3 (𝜑 → (𝐻f 𝑇𝐹) = (𝑤𝐴 ↦ ((𝐻𝑤)𝑇(𝐹𝑤))))
2211, 18, 19, 20, 21offval2 7445 . 2 (𝜑 → ((𝐺f 𝑇𝐹) ∘f 𝑂(𝐻f 𝑇𝐹)) = (𝑤𝐴 ↦ (((𝐺𝑤)𝑇(𝐹𝑤))𝑂((𝐻𝑤)𝑇(𝐹𝑤)))))
2310, 17, 223eqtr4d 2783 1 (𝜑 → ((𝐺f 𝑅𝐻) ∘f 𝑇𝐹) = ((𝐺f 𝑇𝐹) ∘f 𝑂(𝐻f 𝑇𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2113  Vcvv 3398  cmpt 5111  wf 6336  cfv 6340  (class class class)co 7171  f cof 7424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3683  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7174  df-oprab 7175  df-mpo 7176  df-of 7426
This theorem is referenced by:  psrlmod  20781  lflvsdi1  36712  mendlmod  40582  expgrowth  41483
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