Theorem caofdi 7448

Description: Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-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)
caofdi.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑇(𝑦𝑅𝑧)) = ((𝑥𝑇𝑦)𝑂(𝑥𝑇𝑧)))

Assertion

Ref	Expression
caofdi	⊢ (𝜑 → (𝐹 ∘f 𝑇(𝐺 ∘f 𝑅𝐻)) = ((𝐹 ∘f 𝑇𝐺) ∘f 𝑂(𝐹 ∘f 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.

Step	Hyp	Ref	Expression
1		caofdi.5	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑇(𝑦𝑅𝑧)) = ((𝑥𝑇𝑦)𝑂(𝑥𝑇𝑧)))
2	1	adantlr 714	. . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑇(𝑦𝑅𝑧)) = ((𝑥𝑇𝑦)𝑂(𝑥𝑇𝑧)))
3		caofdi.2	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐾)
4	3	ffvelrnda 6847	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝐾)
5		caofdi.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
6	5	ffvelrnda 6847	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
7		caofdi.4	. . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
8	7	ffvelrnda 6847	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
9	2, 4, 6, 8	caovdid 7364	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑇((𝐺‘𝑤)𝑅(𝐻‘𝑤))) = (((𝐹‘𝑤)𝑇(𝐺‘𝑤))𝑂((𝐹‘𝑤)𝑇(𝐻‘𝑤))))
10	9	mpteq2dva 5130	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇((𝐺‘𝑤)𝑅(𝐻‘𝑤)))) = (𝑤 ∈ 𝐴 ↦ (((𝐹‘𝑤)𝑇(𝐺‘𝑤))𝑂((𝐹‘𝑤)𝑇(𝐻‘𝑤)))))
11		caofdi.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
12		ovexd 7190	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐻‘𝑤)) ∈ V)
13	3	feqmptd 6725	. . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
14	5	feqmptd 6725	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
15	7	feqmptd 6725	. . . 4 ⊢ (𝜑 → 𝐻 = (𝑤 ∈ 𝐴 ↦ (𝐻‘𝑤)))
16	11, 6, 8, 14, 15	offval2 7429	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐻‘𝑤))))
17	11, 4, 12, 13, 16	offval2 7429	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑇(𝐺 ∘f 𝑅𝐻)) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇((𝐺‘𝑤)𝑅(𝐻‘𝑤)))))
18		ovexd 7190	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑇(𝐺‘𝑤)) ∈ V)
19		ovexd 7190	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑇(𝐻‘𝑤)) ∈ V)
20	11, 4, 6, 13, 14	offval2 7429	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑇𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇(𝐺‘𝑤))))
21	11, 4, 8, 13, 15	offval2 7429	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑇𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇(𝐻‘𝑤))))
22	11, 18, 19, 20, 21	offval2 7429	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑇𝐺) ∘f 𝑂(𝐹 ∘f 𝑇𝐻)) = (𝑤 ∈ 𝐴 ↦ (((𝐹‘𝑤)𝑇(𝐺‘𝑤))𝑂((𝐹‘𝑤)𝑇(𝐻‘𝑤)))))
23	10, 17, 22	3eqtr4d 2803	1 ⊢ (𝜑 → (𝐹 ∘f 𝑇(𝐺 ∘f 𝑅𝐻)) = ((𝐹 ∘f 𝑇𝐺) ∘f 𝑂(𝐹 ∘f 𝑇𝐻)))

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