Theorem caofdig 6269

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)
caofdig.r	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦) ∈ 𝑉)
caofdig.t	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑇𝑦) ∈ 𝑊)
caofdi.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑇(𝑦𝑅𝑧)) = ((𝑥𝑇𝑦)𝑂(𝑥𝑇𝑧)))

Assertion

Ref	Expression
caofdig	⊢ (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   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦

Allowed substitution hints:   𝑉(𝑧)   𝑊(𝑧)

Proof of Theorem caofdig

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 477	. . . 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 5782	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝐾)
5		caofdi.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
6	5	ffvelcdmda 5782	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
7		caofdi.4	. . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
8	7	ffvelcdmda 5782	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
9	2, 4, 6, 8	caovdid 6198	. . 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 4179	. 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		oveq2 6026	. . . . 5 ⊢ (𝑦 = (𝐻‘𝑤) → ((𝐺‘𝑤)𝑅𝑦) = ((𝐺‘𝑤)𝑅(𝐻‘𝑤)))
13	12	eleq1d 2300	. . . 4 ⊢ (𝑦 = (𝐻‘𝑤) → (((𝐺‘𝑤)𝑅𝑦) ∈ 𝑉 ↔ ((𝐺‘𝑤)𝑅(𝐻‘𝑤)) ∈ 𝑉))
14		oveq1 6025	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑤) → (𝑥𝑅𝑦) = ((𝐺‘𝑤)𝑅𝑦))
15	14	eleq1d 2300	. . . . . 6 ⊢ (𝑥 = (𝐺‘𝑤) → ((𝑥𝑅𝑦) ∈ 𝑉 ↔ ((𝐺‘𝑤)𝑅𝑦) ∈ 𝑉))
16	15	ralbidv 2532	. . . . 5 ⊢ (𝑥 = (𝐺‘𝑤) → (∀𝑦 ∈ 𝑆 (𝑥𝑅𝑦) ∈ 𝑉 ↔ ∀𝑦 ∈ 𝑆 ((𝐺‘𝑤)𝑅𝑦) ∈ 𝑉))
17		caofdig.r	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦) ∈ 𝑉)
18	17	ralrimivva 2614	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑅𝑦) ∈ 𝑉)
19	18	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥𝑅𝑦) ∈ 𝑉)
20	16, 19, 6	rspcdva 2915	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑦 ∈ 𝑆 ((𝐺‘𝑤)𝑅𝑦) ∈ 𝑉)
21	13, 20, 8	rspcdva 2915	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐻‘𝑤)) ∈ 𝑉)
22	3	feqmptd 5699	. . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
23	5	feqmptd 5699	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
24	7	feqmptd 5699	. . . 4 ⊢ (𝜑 → 𝐻 = (𝑤 ∈ 𝐴 ↦ (𝐻‘𝑤)))
25	11, 6, 8, 23, 24	offval2 6251	. . 3 ⊢ (𝜑 → (𝐺 ∘𝑓 𝑅𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐻‘𝑤))))
26	11, 4, 21, 22, 25	offval2 6251	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑇(𝐺 ∘𝑓 𝑅𝐻)) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇((𝐺‘𝑤)𝑅(𝐻‘𝑤)))))
27		oveq2 6026	. . . . 5 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑇𝑦) = ((𝐹‘𝑤)𝑇(𝐺‘𝑤)))
28	27	eleq1d 2300	. . . 4 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑇𝑦) ∈ 𝑊 ↔ ((𝐹‘𝑤)𝑇(𝐺‘𝑤)) ∈ 𝑊))
29		oveq1 6025	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑇𝑦) = ((𝐹‘𝑤)𝑇𝑦))
30	29	eleq1d 2300	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑇𝑦) ∈ 𝑊 ↔ ((𝐹‘𝑤)𝑇𝑦) ∈ 𝑊))
31	30	ralbidv 2532	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → (∀𝑦 ∈ 𝑆 (𝑥𝑇𝑦) ∈ 𝑊 ↔ ∀𝑦 ∈ 𝑆 ((𝐹‘𝑤)𝑇𝑦) ∈ 𝑊))
32		caofdig.t	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑇𝑦) ∈ 𝑊)
33	32	ralrimivva 2614	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑆 (𝑥𝑇𝑦) ∈ 𝑊)
34	33	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑆 (𝑥𝑇𝑦) ∈ 𝑊)
35	31, 34, 4	rspcdva 2915	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑦 ∈ 𝑆 ((𝐹‘𝑤)𝑇𝑦) ∈ 𝑊)
36	28, 35, 6	rspcdva 2915	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑇(𝐺‘𝑤)) ∈ 𝑊)
37		oveq2 6026	. . . . 5 ⊢ (𝑦 = (𝐻‘𝑤) → ((𝐹‘𝑤)𝑇𝑦) = ((𝐹‘𝑤)𝑇(𝐻‘𝑤)))
38	37	eleq1d 2300	. . . 4 ⊢ (𝑦 = (𝐻‘𝑤) → (((𝐹‘𝑤)𝑇𝑦) ∈ 𝑊 ↔ ((𝐹‘𝑤)𝑇(𝐻‘𝑤)) ∈ 𝑊))
39	38, 35, 8	rspcdva 2915	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑇(𝐻‘𝑤)) ∈ 𝑊)
40	11, 4, 6, 22, 23	offval2 6251	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑇𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇(𝐺‘𝑤))))
41	11, 4, 8, 22, 24	offval2 6251	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑇𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇(𝐻‘𝑤))))
42	11, 36, 39, 40, 41	offval2 6251	. 2 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑇𝐺) ∘𝑓 𝑂(𝐹 ∘𝑓 𝑇𝐻)) = (𝑤 ∈ 𝐴 ↦ (((𝐹‘𝑤)𝑇(𝐺‘𝑤))𝑂((𝐹‘𝑤)𝑇(𝐻‘𝑤)))))
43	10, 26, 42	3eqtr4d 2274	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑇(𝐺 ∘𝑓 𝑅𝐻)) = ((𝐹 ∘𝑓 𝑇𝐺) ∘𝑓 𝑂(𝐹 ∘𝑓 𝑇𝐻)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510   ↦ cmpt 4150  ⟶wf 5322  ‘cfv 5326  (class class class)co 6018   ∘_𝑓 cof 6233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-of 6235

This theorem is referenced by: (None)