Theorem caofdir 7721

Description: Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.)

Hypotheses

Ref	Expression
caofdi.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
caofdi.2	⊢ (𝜑 → 𝐹:𝐴⟶𝐾)
caofdi.3	⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
caofdi.4	⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
caofdir.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧)))

Assertion

Ref	Expression
caofdir	⊢ (𝜑 → ((𝐺 ∘f 𝑅𝐻) ∘f 𝑇𝐹) = ((𝐺 ∘f 𝑇𝐹) ∘f 𝑂(𝐻 ∘f 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.

Step	Hyp	Ref	Expression
1		caofdir.5	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧)))
2	1	adantlr 728	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧)))
3		caofdi.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
4	3	ffvelcdmda 7077	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
5		caofdi.4	. . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
6	5	ffvelcdmda 7077	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
7		caofdi.2	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐾)
8	7	ffvelcdmda 7077	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝐾)
9	2, 4, 6, 8	caovdird 7632	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐺‘𝑤)𝑅(𝐻‘𝑤))𝑇(𝐹‘𝑤)) = (((𝐺‘𝑤)𝑇(𝐹‘𝑤))𝑂((𝐻‘𝑤)𝑇(𝐹‘𝑤))))
10	9	mpteq2dva 5198	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ (((𝐺‘𝑤)𝑅(𝐻‘𝑤))𝑇(𝐹‘𝑤))) = (𝑤 ∈ 𝐴 ↦ (((𝐺‘𝑤)𝑇(𝐹‘𝑤))𝑂((𝐻‘𝑤)𝑇(𝐹‘𝑤)))))
11		caofdi.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
12		ovexd 7448	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐻‘𝑤)) ∈ V)
13	3	feqmptd 6946	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
14	5	feqmptd 6946	. . . 4 ⊢ (𝜑 → 𝐻 = (𝑤 ∈ 𝐴 ↦ (𝐻‘𝑤)))
15	11, 4, 6, 13, 14	offval2 7698	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐻‘𝑤))))
16	7	feqmptd 6946	. . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
17	11, 12, 8, 15, 16	offval2 7698	. 2 ⊢ (𝜑 → ((𝐺 ∘f 𝑅𝐻) ∘f 𝑇𝐹) = (𝑤 ∈ 𝐴 ↦ (((𝐺‘𝑤)𝑅(𝐻‘𝑤))𝑇(𝐹‘𝑤))))
18		ovexd 7448	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑇(𝐹‘𝑤)) ∈ V)
19		ovexd 7448	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐻‘𝑤)𝑇(𝐹‘𝑤)) ∈ V)
20	11, 4, 8, 13, 16	offval2 7698	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑇𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑇(𝐹‘𝑤))))
21	11, 6, 8, 14, 16	offval2 7698	. . 3 ⊢ (𝜑 → (𝐻 ∘f 𝑇𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐻‘𝑤)𝑇(𝐹‘𝑤))))
22	11, 18, 19, 20, 21	offval2 7698	. 2 ⊢ (𝜑 → ((𝐺 ∘f 𝑇𝐹) ∘f 𝑂(𝐻 ∘f 𝑇𝐹)) = (𝑤 ∈ 𝐴 ↦ (((𝐺‘𝑤)𝑇(𝐹‘𝑤))𝑂((𝐻‘𝑤)𝑇(𝐹‘𝑤)))))
23	10, 17, 22	3eqtr4d 2805	1 ⊢ (𝜑 → ((𝐺 ∘f 𝑅𝐻) ∘f 𝑇𝐹) = ((𝐺 ∘f 𝑇𝐹) ∘f 𝑂(𝐻 ∘f 𝑇𝐹)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ↦ cmpt 5186  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413   ∘_f cof 7676

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678

This theorem is used by:  psrlmod  22174  lflvsdi1  39951  mendlmod  44030  expgrowth  45159