Theorem caofass 7738

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 3204	. . . . 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 7103	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
6		caofcom.3	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
7	6	ffvelcdmda 7103	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
8		caofass.4	. . . . . 6 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
9	8	ffvelcdmda 7103	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
10		oveq1 7439	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦) = ((𝐹‘𝑤)𝑅𝑦))
11	10	oveq1d 7447	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦)𝑇𝑧) = (((𝐹‘𝑤)𝑅𝑦)𝑇𝑧))
12		oveq1 7439	. . . . . . 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 7440	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑅𝑦) = ((𝐹‘𝑤)𝑅(𝐺‘𝑤)))
15	14	oveq1d 7447	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑅𝑦)𝑇𝑧) = (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇𝑧))
16		oveq1 7439	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → (𝑦𝑃𝑧) = ((𝐺‘𝑤)𝑃𝑧))
17	16	oveq2d 7448	. . . . . . 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 7440	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑤) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇𝑧) = (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤)))
20		oveq2 7440	. . . . . . . 8 ⊢ (𝑧 = (𝐻‘𝑤) → ((𝐺‘𝑤)𝑃𝑧) = ((𝐺‘𝑤)𝑃(𝐻‘𝑤)))
21	20	oveq2d 7448	. . . . . . 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 3637	. . . . 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 1372	. . . 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 5241	. 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 7467	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) ∈ V)
29	4	feqmptd 6976	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
30	6	feqmptd 6976	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
31	27, 5, 7, 29, 30	offval2 7718	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))))
32	8	feqmptd 6976	. . 3 ⊢ (𝜑 → 𝐻 = (𝑤 ∈ 𝐴 ↦ (𝐻‘𝑤)))
33	27, 28, 9, 31, 32	offval2 7718	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘f 𝑇𝐻) = (𝑤 ∈ 𝐴 ↦ (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤))))
34		ovexd 7467	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑃(𝐻‘𝑤)) ∈ V)
35	27, 7, 9, 30, 32	offval2 7718	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑃𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑃(𝐻‘𝑤))))
36	27, 5, 34, 29, 35	offval2 7718	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑂(𝐺 ∘f 𝑃𝐻)) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃(𝐻‘𝑤)))))
37	26, 33, 36	3eqtr4d 2786	1 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘f 𝑇𝐻) = (𝐹 ∘f 𝑂(𝐺 ∘f 𝑃𝐻)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3060  Vcvv 3479   ↦ cmpt 5224  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ∘_f cof 7696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698

This theorem is referenced by:  mndvass  18812  psrgrpOLD  21978  psrlmod  21981  psdmul  22171  itg2mulc  25783  plydivlem4  26339  dchrabl  27299  lfladdass  39075  lflvsass  39083  expgrowth  44359