Theorem caofdi 7718

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 727	. . . 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 7081	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝐾)
5		caofdi.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
6	5	ffvelcdmda 7081	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
7		caofdi.4	. . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
8	7	ffvelcdmda 7081	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
9	2, 4, 6, 8	caovdid 7627	. . 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 5205	. 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 7447	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐻‘𝑤)) ∈ V)
13	3	feqmptd 6951	. . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
14	5	feqmptd 6951	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
15	7	feqmptd 6951	. . . 4 ⊢ (𝜑 → 𝐻 = (𝑤 ∈ 𝐴 ↦ (𝐻‘𝑤)))
16	11, 6, 8, 14, 15	offval2 7696	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐻‘𝑤))))
17	11, 4, 12, 13, 16	offval2 7696	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑇(𝐺 ∘f 𝑅𝐻)) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇((𝐺‘𝑤)𝑅(𝐻‘𝑤)))))
18		ovexd 7447	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑇(𝐺‘𝑤)) ∈ V)
19		ovexd 7447	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑇(𝐻‘𝑤)) ∈ V)
20	11, 4, 6, 13, 14	offval2 7696	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑇𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇(𝐺‘𝑤))))
21	11, 4, 8, 13, 15	offval2 7696	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑇𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇(𝐻‘𝑤))))
22	11, 18, 19, 20, 21	offval2 7696	. 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 2808	1 ⊢ (𝜑 → (𝐹 ∘f 𝑇(𝐺 ∘f 𝑅𝐻)) = ((𝐹 ∘f 𝑇𝐺) ∘f 𝑂(𝐹 ∘f 𝑇𝐻)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ↦ cmpt 5193  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412   ∘_f cof 7674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676

This theorem is referenced by:  psrlmod  22090  plydivlem4  26438  plydiveu  26440  quotcan  26451  basellem9  27234  lflvsdi2  39834  mendlmod  43899