Theorem caofass 7717

Description: Transfer an associative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)

Hypotheses

Ref	Expression
caofref.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
caofref.2	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
caofcom.3	⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
caofass.4	⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
caofass.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))

Assertion

Ref	Expression
caofass	⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘f 𝑇𝐻) = (𝐹 ∘f 𝑂(𝐺 ∘f 𝑃𝐻)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐹   𝑥,𝐺,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝑂,𝑦,𝑧   𝑥,𝑃,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥, 𝑦, 𝑧)   𝑉(𝑥, 𝑦, 𝑧)

Proof of Theorem caofass

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofass.5	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
2	1	ralrimivvva 3208	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
3	2	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
4		caofref.2	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
5	4	ffvelcdmda 7073	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
6		caofcom.3	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
7	6	ffvelcdmda 7073	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
8		caofass.4	. . . . . 6 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
9	8	ffvelcdmda 7073	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
10		oveq1 7416	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦) = ((𝐹‘𝑤)𝑅𝑦))
11	10	oveq1d 7424	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦)𝑇𝑧) = (((𝐹‘𝑤)𝑅𝑦)𝑇𝑧))
12		oveq1 7416	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑂(𝑦𝑃𝑧)) = ((𝐹‘𝑤)𝑂(𝑦𝑃𝑧)))
13	11, 12	eqeq12d 2776	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)) ↔ (((𝐹‘𝑤)𝑅𝑦)𝑇𝑧) = ((𝐹‘𝑤)𝑂(𝑦𝑃𝑧))))
14		oveq2 7417	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑅𝑦) = ((𝐹‘𝑤)𝑅(𝐺‘𝑤)))
15	14	oveq1d 7424	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑅𝑦)𝑇𝑧) = (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇𝑧))
16		oveq1 7416	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → (𝑦𝑃𝑧) = ((𝐺‘𝑤)𝑃𝑧))
17	16	oveq2d 7425	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑂(𝑦𝑃𝑧)) = ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃𝑧)))
18	15, 17	eqeq12d 2776	. . . . . 6 ⊢ (𝑦 = (𝐺‘𝑤) → ((((𝐹‘𝑤)𝑅𝑦)𝑇𝑧) = ((𝐹‘𝑤)𝑂(𝑦𝑃𝑧)) ↔ (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇𝑧) = ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃𝑧))))
19		oveq2 7417	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑤) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇𝑧) = (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤)))
20		oveq2 7417	. . . . . . . 8 ⊢ (𝑧 = (𝐻‘𝑤) → ((𝐺‘𝑤)𝑃𝑧) = ((𝐺‘𝑤)𝑃(𝐻‘𝑤)))
21	20	oveq2d 7425	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑤) → ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃𝑧)) = ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃(𝐻‘𝑤))))
22	19, 21	eqeq12d 2776	. . . . . 6 ⊢ (𝑧 = (𝐻‘𝑤) → ((((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇𝑧) = ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃𝑧)) ↔ (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤)) = ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃(𝐻‘𝑤)))))
23	13, 18, 22	rspc3v 3592	. . . . 5 ⊢ (((𝐹‘𝑤) ∈ 𝑆 ∧ (𝐺‘𝑤) ∈ 𝑆 ∧ (𝐻‘𝑤) ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤)) = ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃(𝐻‘𝑤)))))
24	5, 7, 9, 23	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤)) = ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃(𝐻‘𝑤)))))
25	3, 24	mpd 16	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤)) = ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃(𝐻‘𝑤))))
26	25	mpteq2dva 5198	. 2 ⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤))) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃(𝐻‘𝑤)))))
27		caofref.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
28		ovexd 7444	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) ∈ V)
29	4	feqmptd 6942	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
30	6	feqmptd 6942	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
31	27, 5, 7, 29, 30	offval2 7697	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))))
32	8	feqmptd 6942	. . 3 ⊢ (𝜑 → 𝐻 = (𝑤 ∈ 𝐴 ↦ (𝐻‘𝑤)))
33	27, 28, 9, 31, 32	offval2 7697	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘f 𝑇𝐻) = (𝑤 ∈ 𝐴 ↦ (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤))))
34		ovexd 7444	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑃(𝐻‘𝑤)) ∈ V)
35	27, 7, 9, 30, 32	offval2 7697	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑃𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑃(𝐻‘𝑤))))
36	27, 5, 34, 29, 35	offval2 7697	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑂(𝐺 ∘f 𝑃𝐻)) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃(𝐻‘𝑤)))))
37	26, 33, 36	3eqtr4d 2805	1 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘f 𝑇𝐻) = (𝐹 ∘f 𝑂(𝐺 ∘f 𝑃𝐻)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ↦ cmpt 5186  ⟶wf 6524  ‘cfv 6528  (class class class)co 7409   ∘_f cof 7675

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7677

This theorem is used by:  mndvass  18940  psrlmod  22214  psdmul  22434  itg2mulc  26015  plydivlem4  26566  dchrabl  27530  lfladdass  40044  lflvsass  40052  expgrowth  45257