Theorem caofass 7548

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 3115	. . . . 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	ffvelrnda 6943	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
6		caofcom.3	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
7	6	ffvelrnda 6943	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
8		caofass.4	. . . . . 6 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
9	8	ffvelrnda 6943	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
10		oveq1 7262	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦) = ((𝐹‘𝑤)𝑅𝑦))
11	10	oveq1d 7270	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦)𝑇𝑧) = (((𝐹‘𝑤)𝑅𝑦)𝑇𝑧))
12		oveq1 7262	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑂(𝑦𝑃𝑧)) = ((𝐹‘𝑤)𝑂(𝑦𝑃𝑧)))
13	11, 12	eqeq12d 2754	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)) ↔ (((𝐹‘𝑤)𝑅𝑦)𝑇𝑧) = ((𝐹‘𝑤)𝑂(𝑦𝑃𝑧))))
14		oveq2 7263	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑅𝑦) = ((𝐹‘𝑤)𝑅(𝐺‘𝑤)))
15	14	oveq1d 7270	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑅𝑦)𝑇𝑧) = (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇𝑧))
16		oveq1 7262	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → (𝑦𝑃𝑧) = ((𝐺‘𝑤)𝑃𝑧))
17	16	oveq2d 7271	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑂(𝑦𝑃𝑧)) = ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃𝑧)))
18	15, 17	eqeq12d 2754	. . . . . 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 7263	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑤) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇𝑧) = (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤)))
20		oveq2 7263	. . . . . . . 8 ⊢ (𝑧 = (𝐻‘𝑤) → ((𝐺‘𝑤)𝑃𝑧) = ((𝐺‘𝑤)𝑃(𝐻‘𝑤)))
21	20	oveq2d 7271	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑤) → ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃𝑧)) = ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃(𝐻‘𝑤))))
22	19, 21	eqeq12d 2754	. . . . . 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 3565	. . . . 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 1369	. . . 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 5170	. 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 7290	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) ∈ V)
29	4	feqmptd 6819	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
30	6	feqmptd 6819	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
31	27, 5, 7, 29, 30	offval2 7531	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))))
32	8	feqmptd 6819	. . 3 ⊢ (𝜑 → 𝐻 = (𝑤 ∈ 𝐴 ↦ (𝐻‘𝑤)))
33	27, 28, 9, 31, 32	offval2 7531	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘f 𝑇𝐻) = (𝑤 ∈ 𝐴 ↦ (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤))))
34		ovexd 7290	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑃(𝐻‘𝑤)) ∈ V)
35	27, 7, 9, 30, 32	offval2 7531	. . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑃𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑃(𝐻‘𝑤))))
36	27, 5, 34, 29, 35	offval2 7531	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑂(𝐺 ∘f 𝑃𝐻)) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃(𝐻‘𝑤)))))
37	26, 33, 36	3eqtr4d 2788	1 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘f 𝑇𝐻) = (𝐹 ∘f 𝑂(𝐺 ∘f 𝑃𝐻)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ↦ cmpt 5153  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∘_f cof 7509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511

This theorem is referenced by:  psrgrp  21077  psrlmod  21080  mndvass  21451  itg2mulc  24817  plydivlem4  25361  dchrabl  26307  lfladdass  37014  lflvsass  37022  expgrowth  41842