Theorem caofdir 7707

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 727	. . . 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 7069	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
5		caofdi.4	. . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
6	5	ffvelcdmda 7069	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
7		caofdi.2	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐾)
8	7	ffvelcdmda 7069	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝐾)
9	2, 4, 6, 8	caovdird 7618	. . 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 5197	. 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 7435	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐻‘𝑤)) ∈ V)
13	3	feqmptd 6939	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
14	5	feqmptd 6939	. . . 4 ⊢ (𝜑 → 𝐻 = (𝑤 ∈ 𝐴 ↦ (𝐻‘𝑤)))
15	11, 4, 6, 13, 14	offval2 7684	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐻‘𝑤))))
16	7	feqmptd 6939	. . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
17	11, 12, 8, 15, 16	offval2 7684	. 2 ⊢ (𝜑 → ((𝐺 ∘f 𝑅𝐻) ∘f 𝑇𝐹) = (𝑤 ∈ 𝐴 ↦ (((𝐺‘𝑤)𝑅(𝐻‘𝑤))𝑇(𝐹‘𝑤))))
18		ovexd 7435	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑇(𝐹‘𝑤)) ∈ V)
19		ovexd 7435	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐻‘𝑤)𝑇(𝐹‘𝑤)) ∈ V)
20	11, 4, 8, 13, 16	offval2 7684	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑇𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑇(𝐹‘𝑤))))
21	11, 6, 8, 14, 16	offval2 7684	. . 3 ⊢ (𝜑 → (𝐻 ∘f 𝑇𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐻‘𝑤)𝑇(𝐹‘𝑤))))
22	11, 18, 19, 20, 21	offval2 7684	. 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 2810	1 ⊢ (𝜑 → ((𝐺 ∘f 𝑅𝐻) ∘f 𝑇𝐹) = ((𝐺 ∘f 𝑇𝐹) ∘f 𝑂(𝐻 ∘f 𝑇𝐹)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ↦ cmpt 5185  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ∘_f cof 7662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664

This theorem is referenced by:  psrlmod  22066  lflvsdi1  39709  mendlmod  43773  expgrowth  44904