Theorem caofdir 7714

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 712	. . . 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 7086	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
5		caofdi.4	. . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
6	5	ffvelcdmda 7086	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
7		caofdi.2	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐾)
8	7	ffvelcdmda 7086	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝐾)
9	2, 4, 6, 8	caovdird 7629	. . 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 5248	. 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 7447	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐻‘𝑤)) ∈ V)
13	3	feqmptd 6960	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
14	5	feqmptd 6960	. . . 4 ⊢ (𝜑 → 𝐻 = (𝑤 ∈ 𝐴 ↦ (𝐻‘𝑤)))
15	11, 4, 6, 13, 14	offval2 7694	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐻‘𝑤))))
16	7	feqmptd 6960	. . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
17	11, 12, 8, 15, 16	offval2 7694	. 2 ⊢ (𝜑 → ((𝐺 ∘f 𝑅𝐻) ∘f 𝑇𝐹) = (𝑤 ∈ 𝐴 ↦ (((𝐺‘𝑤)𝑅(𝐻‘𝑤))𝑇(𝐹‘𝑤))))
18		ovexd 7447	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑇(𝐹‘𝑤)) ∈ V)
19		ovexd 7447	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐻‘𝑤)𝑇(𝐹‘𝑤)) ∈ V)
20	11, 4, 8, 13, 16	offval2 7694	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑇𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑇(𝐹‘𝑤))))
21	11, 6, 8, 14, 16	offval2 7694	. . 3 ⊢ (𝜑 → (𝐻 ∘f 𝑇𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐻‘𝑤)𝑇(𝐹‘𝑤))))
22	11, 18, 19, 20, 21	offval2 7694	. 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 2781	1 ⊢ (𝜑 → ((𝐺 ∘f 𝑅𝐻) ∘f 𝑇𝐹) = ((𝐺 ∘f 𝑇𝐹) ∘f 𝑂(𝐻 ∘f 𝑇𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  Vcvv 3473   ↦ cmpt 5231  ⟶wf 6539  ‘cfv 6543  (class class class)co 7412   ∘_f cof 7672

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674

This theorem is referenced by:  psrlmod  21833  lflvsdi1  38415  mendlmod  42401  expgrowth  43560