Theorem cofuass 17793

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 6213	. . . 4 ⊢ (((1st ‘𝐾) ∘ (1st ‘𝐻)) ∘ (1st ‘𝐺)) = ((1st ‘𝐾) ∘ ((1st ‘𝐻) ∘ (1st ‘𝐺)))
2		eqid 2731	. . . . . 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 17787	. . . . 5 ⊢ (𝜑 → (1st ‘(𝐾 ∘func 𝐻)) = ((1st ‘𝐾) ∘ (1st ‘𝐻)))
6	5	coeq1d 5801	. . . 4 ⊢ (𝜑 → ((1st ‘(𝐾 ∘func 𝐻)) ∘ (1st ‘𝐺)) = (((1st ‘𝐾) ∘ (1st ‘𝐻)) ∘ (1st ‘𝐺)))
7		eqid 2731	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		cofuass.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
9	7, 8, 3	cofu1st 17787	. . . . 5 ⊢ (𝜑 → (1st ‘(𝐻 ∘func 𝐺)) = ((1st ‘𝐻) ∘ (1st ‘𝐺)))
10	9	coeq2d 5802	. . . 4 ⊢ (𝜑 → ((1st ‘𝐾) ∘ (1st ‘(𝐻 ∘func 𝐺))) = ((1st ‘𝐾) ∘ ((1st ‘𝐻) ∘ (1st ‘𝐺))))
11	1, 6, 10	3eqtr4a 2792	. . 3 ⊢ (𝜑 → ((1st ‘(𝐾 ∘func 𝐻)) ∘ (1st ‘𝐺)) = ((1st ‘𝐾) ∘ (1st ‘(𝐻 ∘func 𝐺))))
12		coass 6213	. . . . 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 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐻 ∈ (𝐷 Func 𝐸))
14	4	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐾 ∈ (𝐸 Func 𝐹))
15		relfunc 17766	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
16		1st2ndbr 7974	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
17	15, 8, 16	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
18	17	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
19	7, 2, 18	funcf1 17770	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
20		simp2 1137	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
21	19, 20	ffvelcdmd 7018	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
22		simp3 1138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
23	19, 22	ffvelcdmd 7018	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑦) ∈ (Base‘𝐷))
24	2, 13, 14, 21, 23	cofu2nd 17789	. . . . . 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 5801	. . . . 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 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐺 ∈ (𝐶 Func 𝐷))
27	7, 26, 13, 20	cofu1 17788	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘(𝐻 ∘func 𝐺))‘𝑥) = ((1st ‘𝐻)‘((1st ‘𝐺)‘𝑥)))
28	7, 26, 13, 22	cofu1 17788	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘(𝐻 ∘func 𝐺))‘𝑦) = ((1st ‘𝐻)‘((1st ‘𝐺)‘𝑦)))
29	27, 28	oveq12d 7364	. . . . . 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 17789	. . . . . 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 5804	. . . . 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 2792	. . . 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 7423	. . 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 4833	. 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 17792	. . 3 ⊢ (𝜑 → (𝐾 ∘func 𝐻) ∈ (𝐷 Func 𝐹))
36	7, 8, 35	cofuval 17786	. 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 17792	. . 3 ⊢ (𝜑 → (𝐻 ∘func 𝐺) ∈ (𝐶 Func 𝐸))
38	7, 37, 4	cofuval 17786	. 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 2776	1 ⊢ (𝜑 → ((𝐾 ∘func 𝐻) ∘func 𝐺) = (𝐾 ∘func (𝐻 ∘func 𝐺)))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ⟨cop 4582   class class class wbr 5091   ∘ ccom 5620  Rel wrel 5621  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  1^st c1st 7919  2^nd c2nd 7920  Basecbs 17117   Func cfunc 17758   ∘_func ccofu 17760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752  df-ixp 8822  df-cat 17571  df-cid 17572  df-func 17762  df-cofu 17764

This theorem is referenced by:  catccatid  18010  uobeqw  49250