Theorem caofass 6931

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 2972	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)))
3	2	adantr 481	. . . 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 6359	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆)
6		caofcom.3	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆)
7	6	ffvelrnda 6359	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆)
8		caofass.4	. . . . . 6 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆)
9	8	ffvelrnda 6359	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆)
10		oveq1 6657	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑦) = ((𝐹‘𝑤)𝑅𝑦))
11	10	oveq1d 6665	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → ((𝑥𝑅𝑦)𝑇𝑧) = (((𝐹‘𝑤)𝑅𝑦)𝑇𝑧))
12		oveq1 6657	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑂(𝑦𝑃𝑧)) = ((𝐹‘𝑤)𝑂(𝑦𝑃𝑧)))
13	11, 12	eqeq12d 2637	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → (((𝑥𝑅𝑦)𝑇𝑧) = (𝑥𝑂(𝑦𝑃𝑧)) ↔ (((𝐹‘𝑤)𝑅𝑦)𝑇𝑧) = ((𝐹‘𝑤)𝑂(𝑦𝑃𝑧))))
14		oveq2 6658	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑅𝑦) = ((𝐹‘𝑤)𝑅(𝐺‘𝑤)))
15	14	oveq1d 6665	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑤) → (((𝐹‘𝑤)𝑅𝑦)𝑇𝑧) = (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇𝑧))
16		oveq1 6657	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑤) → (𝑦𝑃𝑧) = ((𝐺‘𝑤)𝑃𝑧))
17	16	oveq2d 6666	. . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑤) → ((𝐹‘𝑤)𝑂(𝑦𝑃𝑧)) = ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃𝑧)))
18	15, 17	eqeq12d 2637	. . . . . 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 6658	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑤) → (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇𝑧) = (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤)))
20		oveq2 6658	. . . . . . . 8 ⊢ (𝑧 = (𝐻‘𝑤) → ((𝐺‘𝑤)𝑃𝑧) = ((𝐺‘𝑤)𝑃(𝐻‘𝑤)))
21	20	oveq2d 6666	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑤) → ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃𝑧)) = ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃(𝐻‘𝑤))))
22	19, 21	eqeq12d 2637	. . . . . 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 3325	. . . . 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 1326	. . . 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 4744	. 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 6680	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) ∈ V)
29	4	feqmptd 6249	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤)))
30	6	feqmptd 6249	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤)))
31	27, 5, 7, 29, 30	offval2 6914	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))))
32	8	feqmptd 6249	. . 3 ⊢ (𝜑 → 𝐻 = (𝑤 ∈ 𝐴 ↦ (𝐻‘𝑤)))
33	27, 28, 9, 31, 32	offval2 6914	. 2 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) ∘𝑓 𝑇𝐻) = (𝑤 ∈ 𝐴 ↦ (((𝐹‘𝑤)𝑅(𝐺‘𝑤))𝑇(𝐻‘𝑤))))
34		ovexd 6680	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑃(𝐻‘𝑤)) ∈ V)
35	27, 7, 9, 30, 32	offval2 6914	. . 3 ⊢ (𝜑 → (𝐺 ∘𝑓 𝑃𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑃(𝐻‘𝑤))))
36	27, 5, 34, 29, 35	offval2 6914	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑂(𝐺 ∘𝑓 𝑃𝐻)) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑂((𝐺‘𝑤)𝑃(𝐻‘𝑤)))))
37	26, 33, 36	3eqtr4d 2666	1 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) ∘𝑓 𝑇𝐻) = (𝐹 ∘𝑓 𝑂(𝐺 ∘𝑓 𝑃𝐻)))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897

This theorem is referenced by:  psrgrp  19398  psrlmod  19401  mndvass  20198  itg2mulc  23514  plydivlem4  24051  dchrabl  24979  lfladdass  34360  lflvsass  34368  expgrowth  38534