Theorem caofdig 6336

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 481	. . . 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 5843	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝐾)
5		caofdi.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
6	5	ffvelcdmda 5843	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
7		caofdi.4	. . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
8	7	ffvelcdmda 5843	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
9	2, 4, 6, 8	caovdid 6265	. . 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 4221	. 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 6093	. . . . 5 ⊢ (𝑦 = (𝐻‘𝑤) → ((𝐺‘𝑤)𝑅𝑦) = ((𝐺‘𝑤)𝑅(𝐻‘𝑤)))
13	12	eleq1d 2307	. . . 4 ⊢ (𝑦 = (𝐻‘𝑤) → (((𝐺‘𝑤)𝑅𝑦) ∈ 𝑉 ↔ ((𝐺‘𝑤)𝑅(𝐻‘𝑤)) ∈ 𝑉))
14		oveq1 6092	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑤) → (𝑥𝑅𝑦) = ((𝐺‘𝑤)𝑅𝑦))
15	14	eleq1d 2307	. . . . . 6 ⊢ (𝑥 = (𝐺‘𝑤) → ((𝑥𝑅𝑦) ∈ 𝑉 ↔ ((𝐺‘𝑤)𝑅𝑦) ∈ 𝑉))
16	15	ralbidv 2550	. . . . 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 2632	. . . . . 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 2934	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑦 ∈ 𝑆 ((𝐺‘𝑤)𝑅𝑦) ∈ 𝑉)
21	13, 20, 8	rspcdva 2934	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐻‘𝑤)) ∈ 𝑉)
22	3	feqmptd 5756	. . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
23	5	feqmptd 5756	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
24	7	feqmptd 5756	. . . 4 ⊢ (𝜑 → 𝐻 = (𝑤 ∈ 𝐴 ↦ (𝐻‘𝑤)))
25	11, 6, 8, 23, 24	offval2 6318	. . 3 ⊢ (𝜑 → (𝐺 ∘𝑓 𝑅𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐻‘𝑤))))
26	11, 4, 21, 22, 25	offval2 6318	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑇(𝐺 ∘𝑓 𝑅𝐻)) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇((𝐺‘𝑤)𝑅(𝐻‘𝑤)))))
27		oveq2 6093	. . . . 5 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑇𝑦) = ((𝐹‘𝑤)𝑇(𝐺‘𝑤)))
28	27	eleq1d 2307	. . . 4 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑇𝑦) ∈ 𝑊 ↔ ((𝐹‘𝑤)𝑇(𝐺‘𝑤)) ∈ 𝑊))
29		oveq1 6092	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑇𝑦) = ((𝐹‘𝑤)𝑇𝑦))
30	29	eleq1d 2307	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑇𝑦) ∈ 𝑊 ↔ ((𝐹‘𝑤)𝑇𝑦) ∈ 𝑊))
31	30	ralbidv 2550	. . . . 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 2632	. . . . . 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 2934	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑦 ∈ 𝑆 ((𝐹‘𝑤)𝑇𝑦) ∈ 𝑊)
36	28, 35, 6	rspcdva 2934	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑇(𝐺‘𝑤)) ∈ 𝑊)
37		oveq2 6093	. . . . 5 ⊢ (𝑦 = (𝐻‘𝑤) → ((𝐹‘𝑤)𝑇𝑦) = ((𝐹‘𝑤)𝑇(𝐻‘𝑤)))
38	37	eleq1d 2307	. . . 4 ⊢ (𝑦 = (𝐻‘𝑤) → (((𝐹‘𝑤)𝑇𝑦) ∈ 𝑊 ↔ ((𝐹‘𝑤)𝑇(𝐻‘𝑤)) ∈ 𝑊))
39	38, 35, 8	rspcdva 2934	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑇(𝐻‘𝑤)) ∈ 𝑊)
40	11, 4, 6, 22, 23	offval2 6318	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑇𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇(𝐺‘𝑤))))
41	11, 4, 8, 22, 24	offval2 6318	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑇𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑇(𝐻‘𝑤))))
42	11, 36, 39, 40, 41	offval2 6318	. 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 2281	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑇(𝐺 ∘𝑓 𝑅𝐻)) = ((𝐹 ∘𝑓 𝑇𝐺) ∘𝑓 𝑂(𝐹 ∘𝑓 𝑇𝐻)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ↦ cmpt 4192  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085   ∘_𝑓 cof 6300

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302

This theorem is used by: (None)