Theorem caofdi 7664

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 715	. . . 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	ffvelcdmda 7029	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝐾)
5		caofdi.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
6	5	ffvelcdmda 7029	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
7		caofdi.4	. . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
8	7	ffvelcdmda 7029	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
9	2, 4, 6, 8	caovdid 7573	. . 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 5191	. 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 7393	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐻‘𝑤)) ∈ V)
13	3	feqmptd 6902	. . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
14	5	feqmptd 6902	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
15	7	feqmptd 6902	. . . 4 ⊢ (𝜑 → 𝐻 = (𝑤 ∈ 𝐴 ↦ (𝐻‘𝑤)))
16	11, 6, 8, 14, 15	offval2 7642	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐻‘𝑤))))
17	11, 4, 12, 13, 16	offval2 7642	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑇(𝐺 ∘f 𝑅𝐻)) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇((𝐺‘𝑤)𝑅(𝐻‘𝑤)))))
18		ovexd 7393	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑇(𝐺‘𝑤)) ∈ V)
19		ovexd 7393	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑇(𝐻‘𝑤)) ∈ V)
20	11, 4, 6, 13, 14	offval2 7642	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑇𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇(𝐺‘𝑤))))
21	11, 4, 8, 13, 15	offval2 7642	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑇𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇(𝐻‘𝑤))))
22	11, 18, 19, 20, 21	offval2 7642	. 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 2781	1 ⊢ (𝜑 → (𝐹 ∘f 𝑇(𝐺 ∘f 𝑅𝐻)) = ((𝐹 ∘f 𝑇𝐺) ∘f 𝑂(𝐹 ∘f 𝑇𝐻)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ↦ cmpt 5179  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∘_f cof 7620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622

This theorem is referenced by:  psrlmod  21915  plydivlem4  26260  plydiveu  26262  quotcan  26273  basellem9  27055  lflvsdi2  39339  mendlmod  43431