Theorem cofuass 17984

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 6266	. . . 4 ⊢ (((1st ‘𝐾) ∘ (1st ‘𝐻)) ∘ (1st ‘𝐺)) = ((1st ‘𝐾) ∘ ((1st ‘𝐻) ∘ (1st ‘𝐺)))
2		eqid 2762	. . . . . 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 17978	. . . . 5 ⊢ (𝜑 → (1st ‘(𝐾 ∘func 𝐻)) = ((1st ‘𝐾) ∘ (1st ‘𝐻)))
6	5	coeq1d 5845	. . . 4 ⊢ (𝜑 → ((1st ‘(𝐾 ∘func 𝐻)) ∘ (1st ‘𝐺)) = (((1st ‘𝐾) ∘ (1st ‘𝐻)) ∘ (1st ‘𝐺)))
7		eqid 2762	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		cofuass.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
9	7, 8, 3	cofu1st 17978	. . . . 5 ⊢ (𝜑 → (1st ‘(𝐻 ∘func 𝐺)) = ((1st ‘𝐻) ∘ (1st ‘𝐺)))
10	9	coeq2d 5846	. . . 4 ⊢ (𝜑 → ((1st ‘𝐾) ∘ (1st ‘(𝐻 ∘func 𝐺))) = ((1st ‘𝐾) ∘ ((1st ‘𝐻) ∘ (1st ‘𝐺))))
11	1, 6, 10	3eqtr4a 2823	. . 3 ⊢ (𝜑 → ((1st ‘(𝐾 ∘func 𝐻)) ∘ (1st ‘𝐺)) = ((1st ‘𝐾) ∘ (1st ‘(𝐻 ∘func 𝐺))))
12		coass 6266	. . . . 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 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐻 ∈ (𝐷 Func 𝐸))
14	4	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐾 ∈ (𝐸 Func 𝐹))
15		relfunc 17957	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
16		1st2ndbr 8043	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
17	15, 8, 16	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
18	17	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
19	7, 2, 18	funcf1 17961	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (1st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
20		simp2 1155	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
21	19, 20	ffvelcdmd 7082	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
22		simp3 1156	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝑦 ∈ (Base‘𝐶))
23	19, 22	ffvelcdmd 7082	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑦) ∈ (Base‘𝐷))
24	2, 13, 14, 21, 23	cofu2nd 17980	. . . . . 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 5845	. . . . 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 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → 𝐺 ∈ (𝐶 Func 𝐷))
27	7, 26, 13, 20	cofu1 17979	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘(𝐻 ∘func 𝐺))‘𝑥) = ((1st ‘𝐻)‘((1st ‘𝐺)‘𝑥)))
28	7, 26, 13, 22	cofu1 17979	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘(𝐻 ∘func 𝐺))‘𝑦) = ((1st ‘𝐻)‘((1st ‘𝐺)‘𝑦)))
29	27, 28	oveq12d 7435	. . . . . 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 17980	. . . . . 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 5848	. . . . 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 2823	. . . 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 7494	. . 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 4844	. 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 17983	. . 3 ⊢ (𝜑 → (𝐾 ∘func 𝐻) ∈ (𝐷 Func 𝐹))
36	7, 8, 35	cofuval 17977	. 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 17983	. . 3 ⊢ (𝜑 → (𝐻 ∘func 𝐺) ∈ (𝐶 Func 𝐸))
38	7, 37, 4	cofuval 17977	. 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 2807	1 ⊢ (𝜑 → ((𝐾 ∘func 𝐻) ∘func 𝐺) = (𝐾 ∘func (𝐻 ∘func 𝐺)))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ⟨cop 4593   class class class wbr 5107   ∘ ccom 5663  Rel wrel 5664  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  1^st c1st 7988  2^nd c2nd 7989  Basecbs 17307   Func cfunc 17949   ∘_func ccofu 17951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-map 8832  df-ixp 8909  df-cat 17762  df-cid 17763  df-func 17953  df-cofu 17955

This theorem is used by:  catccatid  18201  uobeqw  50153