Theorem caofass 7716

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 3191	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
3	2	adantr 480	. . . 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 7079	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
6		caofcom.3	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
7	6	ffvelcdmda 7079	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
8		caofass.4	. . . . . 6 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
9	8	ffvelcdmda 7079	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
10		oveq1 7417	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦) = ((𝐹‘𝑤)𝑅𝑦))
11	10	oveq1d 7425	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦)𝑇𝑧) = (((𝐹‘𝑤)𝑅𝑦)𝑇𝑧))
12		oveq1 7417	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑂(𝑦𝑃𝑧)) = ((𝐹‘𝑤)𝑂(𝑦𝑃𝑧)))
13	11, 12	eqeq12d 2752	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)) ↔ (((𝐹‘𝑤)𝑅𝑦)𝑇𝑧) = ((𝐹‘𝑤)𝑂(𝑦𝑃𝑧))))
14		oveq2 7418	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑅𝑦) = ((𝐹‘𝑤)𝑅(𝐺‘𝑤)))
15	14	oveq1d 7425	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑅𝑦)𝑇𝑧) = (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇𝑧))
16		oveq1 7417	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → (𝑦𝑃𝑧) = ((𝐺‘𝑤)𝑃𝑧))
17	16	oveq2d 7426	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑂(𝑦𝑃𝑧)) = ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃𝑧)))
18	15, 17	eqeq12d 2752	. . . . . 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 7418	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑤) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇𝑧) = (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤)))
20		oveq2 7418	. . . . . . . 8 ⊢ (𝑧 = (𝐻‘𝑤) → ((𝐺‘𝑤)𝑃𝑧) = ((𝐺‘𝑤)𝑃(𝐻‘𝑤)))
21	20	oveq2d 7426	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑤) → ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃𝑧)) = ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃(𝐻‘𝑤))))
22	19, 21	eqeq12d 2752	. . . . . 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 3622	. . . . 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 1373	. . . 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 15	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤)) = ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃(𝐻‘𝑤))))
26	25	mpteq2dva 5219	. 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 7445	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) ∈ V)
29	4	feqmptd 6952	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
30	6	feqmptd 6952	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
31	27, 5, 7, 29, 30	offval2 7696	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))))
32	8	feqmptd 6952	. . 3 ⊢ (𝜑 → 𝐻 = (𝑤 ∈ 𝐴 ↦ (𝐻‘𝑤)))
33	27, 28, 9, 31, 32	offval2 7696	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘f 𝑇𝐻) = (𝑤 ∈ 𝐴 ↦ (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤))))
34		ovexd 7445	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑃(𝐻‘𝑤)) ∈ V)
35	27, 7, 9, 30, 32	offval2 7696	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑃𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑃(𝐻‘𝑤))))
36	27, 5, 34, 29, 35	offval2 7696	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑂(𝐺 ∘f 𝑃𝐻)) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃(𝐻‘𝑤)))))
37	26, 33, 36	3eqtr4d 2781	1 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘f 𝑇𝐻) = (𝐹 ∘f 𝑂(𝐺 ∘f 𝑃𝐻)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464   ↦ cmpt 5206  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∘_f cof 7674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7676

This theorem is referenced by:  mndvass  18781  psrgrpOLD  21922  psrlmod  21925  psdmul  22109  itg2mulc  25705  plydivlem4  26261  dchrabl  27222  lfladdass  39096  lflvsass  39104  expgrowth  44326