Theorem cofuass 17151

Description: Functor composition is associative. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
cofuass.g	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
cofuass.h	⊢ (𝜑 → 𝐻 ∈ (𝐷 Func 𝐸))
cofuass.k	⊢ (𝜑 → 𝐾 ∈ (𝐸 Func 𝐹))

Assertion

Ref	Expression
cofuass	⊢ (𝜑 → ((𝐾 ∘func 𝐻) ∘func 𝐺) = (𝐾 ∘func (𝐻 ∘func 𝐺)))

Proof of Theorem cofuass

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		coass 6085	. . . 4 ⊢ (((1st ‘𝐾) ∘ (1st ‘𝐻)) ∘ (1st ‘𝐺)) = ((1st ‘𝐾) ∘ ((1st ‘𝐻) ∘ (1st ‘𝐺)))
2		eqid 2798	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		cofuass.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (𝐷 Func 𝐸))
4		cofuass.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝐸 Func 𝐹))
5	2, 3, 4	cofu1st 17145	. . . . 5 ⊢ (𝜑 → (1st ‘(𝐾 ∘func 𝐻)) = ((1st ‘𝐾) ∘ (1st ‘𝐻)))
6	5	coeq1d 5696	. . . 4 ⊢ (𝜑 → ((1st ‘(𝐾 ∘func 𝐻)) ∘ (1st ‘𝐺)) = (((1st ‘𝐾) ∘ (1st ‘𝐻)) ∘ (1st ‘𝐺)))
7		eqid 2798	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		cofuass.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
9	7, 8, 3	cofu1st 17145	. . . . 5 ⊢ (𝜑 → (1st ‘(𝐻 ∘func 𝐺)) = ((1st ‘𝐻) ∘ (1st ‘𝐺)))
10	9	coeq2d 5697	. . . 4 ⊢ (𝜑 → ((1st ‘𝐾) ∘ (1st ‘(𝐻 ∘func 𝐺))) = ((1st ‘𝐾) ∘ ((1st ‘𝐻) ∘ (1st ‘𝐺))))
11	1, 6, 10	3eqtr4a 2859	. . 3 ⊢ (𝜑 → ((1st ‘(𝐾 ∘func 𝐻)) ∘ (1st ‘𝐺)) = ((1st ‘𝐾) ∘ (1st ‘(𝐻 ∘func 𝐺))))
12		coass 6085	. . . . 5 ⊢ (((((1st ‘𝐻)‘((1st ‘𝐺)‘𝑥))(2nd ‘𝐾)((1st ‘𝐻)‘((1st ‘𝐺)‘𝑦))) ∘ (((1st ‘𝐺)‘𝑥)(2nd ‘𝐻)((1st ‘𝐺)‘𝑦))) ∘ (𝑥(2nd ‘𝐺)𝑦)) = ((((1st ‘𝐻)‘((1st ‘𝐺)‘𝑥))(2nd ‘𝐾)((1st ‘𝐻)‘((1st ‘𝐺)‘𝑦))) ∘ ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐻)((1st ‘𝐺)‘𝑦)) ∘ (𝑥(2nd ‘𝐺)𝑦)))
13	3	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐻 ∈ (𝐷 Func 𝐸))
14	4	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐾 ∈ (𝐸 Func 𝐹))
15		relfunc 17124	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
16		1st2ndbr 7723	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
17	15, 8, 16	sylancr 590	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
18	17	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
19	7, 2, 18	funcf1 17128	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
20		simp2 1134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
21	19, 20	ffvelrnd 6829	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
22		simp3 1135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
23	19, 22	ffvelrnd 6829	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑦) ∈ (Base‘𝐷))
24	2, 13, 14, 21, 23	cofu2nd 17147	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (((1st ‘𝐺)‘𝑥)(2nd ‘(𝐾 ∘func 𝐻))((1st ‘𝐺)‘𝑦)) = ((((1st ‘𝐻)‘((1st ‘𝐺)‘𝑥))(2nd ‘𝐾)((1st ‘𝐻)‘((1st ‘𝐺)‘𝑦))) ∘ (((1st ‘𝐺)‘𝑥)(2nd ‘𝐻)((1st ‘𝐺)‘𝑦))))
25	24	coeq1d 5696	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st ‘𝐺)‘𝑥)(2nd ‘(𝐾 ∘func 𝐻))((1st ‘𝐺)‘𝑦)) ∘ (𝑥(2nd ‘𝐺)𝑦)) = (((((1st ‘𝐻)‘((1st ‘𝐺)‘𝑥))(2nd ‘𝐾)((1st ‘𝐻)‘((1st ‘𝐺)‘𝑦))) ∘ (((1st ‘𝐺)‘𝑥)(2nd ‘𝐻)((1st ‘𝐺)‘𝑦))) ∘ (𝑥(2nd ‘𝐺)𝑦)))
26	8	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐺 ∈ (𝐶 Func 𝐷))
27	7, 26, 13, 20	cofu1 17146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘(𝐻 ∘func 𝐺))‘𝑥) = ((1st ‘𝐻)‘((1st ‘𝐺)‘𝑥)))
28	7, 26, 13, 22	cofu1 17146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘(𝐻 ∘func 𝐺))‘𝑦) = ((1st ‘𝐻)‘((1st ‘𝐺)‘𝑦)))
29	27, 28	oveq12d 7153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (((1st ‘(𝐻 ∘func 𝐺))‘𝑥)(2nd ‘𝐾)((1st ‘(𝐻 ∘func 𝐺))‘𝑦)) = (((1st ‘𝐻)‘((1st ‘𝐺)‘𝑥))(2nd ‘𝐾)((1st ‘𝐻)‘((1st ‘𝐺)‘𝑦))))
30	7, 26, 13, 20, 22	cofu2nd 17147	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(2nd ‘(𝐻 ∘func 𝐺))𝑦) = ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐻)((1st ‘𝐺)‘𝑦)) ∘ (𝑥(2nd ‘𝐺)𝑦)))
31	29, 30	coeq12d 5699	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st ‘(𝐻 ∘func 𝐺))‘𝑥)(2nd ‘𝐾)((1st ‘(𝐻 ∘func 𝐺))‘𝑦)) ∘ (𝑥(2nd ‘(𝐻 ∘func 𝐺))𝑦)) = ((((1st ‘𝐻)‘((1st ‘𝐺)‘𝑥))(2nd ‘𝐾)((1st ‘𝐻)‘((1st ‘𝐺)‘𝑦))) ∘ ((((1st ‘𝐺)‘𝑥)(2nd ‘𝐻)((1st ‘𝐺)‘𝑦)) ∘ (𝑥(2nd ‘𝐺)𝑦))))
32	12, 25, 31	3eqtr4a 2859	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((((1st ‘𝐺)‘𝑥)(2nd ‘(𝐾 ∘func 𝐻))((1st ‘𝐺)‘𝑦)) ∘ (𝑥(2nd ‘𝐺)𝑦)) = ((((1st ‘(𝐻 ∘func 𝐺))‘𝑥)(2nd ‘𝐾)((1st ‘(𝐻 ∘func 𝐺))‘𝑦)) ∘ (𝑥(2nd ‘(𝐻 ∘func 𝐺))𝑦)))
33	32	mpoeq3dva 7210	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐺)‘𝑥)(2nd ‘(𝐾 ∘func 𝐻))((1st ‘𝐺)‘𝑦)) ∘ (𝑥(2nd ‘𝐺)𝑦))) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘(𝐻 ∘func 𝐺))‘𝑥)(2nd ‘𝐾)((1st ‘(𝐻 ∘func 𝐺))‘𝑦)) ∘ (𝑥(2nd ‘(𝐻 ∘func 𝐺))𝑦))))
34	11, 33	opeq12d 4773	. 2 ⊢ (𝜑 → ⟨((1st ‘(𝐾 ∘func 𝐻)) ∘ (1st ‘𝐺)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐺)‘𝑥)(2nd ‘(𝐾 ∘func 𝐻))((1st ‘𝐺)‘𝑦)) ∘ (𝑥(2nd ‘𝐺)𝑦)))⟩ = ⟨((1st ‘𝐾) ∘ (1st ‘(𝐻 ∘func 𝐺))), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘(𝐻 ∘func 𝐺))‘𝑥)(2nd ‘𝐾)((1st ‘(𝐻 ∘func 𝐺))‘𝑦)) ∘ (𝑥(2nd ‘(𝐻 ∘func 𝐺))𝑦)))⟩)
35	3, 4	cofucl 17150	. . 3 ⊢ (𝜑 → (𝐾 ∘func 𝐻) ∈ (𝐷 Func 𝐹))
36	7, 8, 35	cofuval 17144	. 2 ⊢ (𝜑 → ((𝐾 ∘func 𝐻) ∘func 𝐺) = ⟨((1st ‘(𝐾 ∘func 𝐻)) ∘ (1st ‘𝐺)), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘𝐺)‘𝑥)(2nd ‘(𝐾 ∘func 𝐻))((1st ‘𝐺)‘𝑦)) ∘ (𝑥(2nd ‘𝐺)𝑦)))⟩)
37	8, 3	cofucl 17150	. . 3 ⊢ (𝜑 → (𝐻 ∘func 𝐺) ∈ (𝐶 Func 𝐸))
38	7, 37, 4	cofuval 17144	. 2 ⊢ (𝜑 → (𝐾 ∘func (𝐻 ∘func 𝐺)) = ⟨((1st ‘𝐾) ∘ (1st ‘(𝐻 ∘func 𝐺))), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ((((1st ‘(𝐻 ∘func 𝐺))‘𝑥)(2nd ‘𝐾)((1st ‘(𝐻 ∘func 𝐺))‘𝑦)) ∘ (𝑥(2nd ‘(𝐻 ∘func 𝐺))𝑦)))⟩)
39	34, 36, 38	3eqtr4d 2843	1 ⊢ (𝜑 → ((𝐾 ∘func 𝐻) ∘func 𝐺) = (𝐾 ∘func (𝐻 ∘func 𝐺)))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   class class class wbr 5030   ∘ ccom 5523  Rel wrel 5524  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  1^st c1st 7669  2^nd c2nd 7670  Basecbs 16475   Func cfunc 17116   ∘_func ccofu 17118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391  df-ixp 8445  df-cat 16931  df-cid 16932  df-func 17120  df-cofu 17122

This theorem is referenced by:  catccatid  17354