Theorem cofuass 17953

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 6296	. . . 4 ⊢ (((1st ‘𝐾) ∘ (1st ‘𝐻)) ∘ (1st ‘𝐺)) = ((1st ‘𝐾) ∘ ((1st ‘𝐻) ∘ (1st ‘𝐺)))
2		eqid 2740	. . . . . 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 17947	. . . . 5 ⊢ (𝜑 → (1st ‘(𝐾 ∘func 𝐻)) = ((1st ‘𝐾) ∘ (1st ‘𝐻)))
6	5	coeq1d 5886	. . . 4 ⊢ (𝜑 → ((1st ‘(𝐾 ∘func 𝐻)) ∘ (1st ‘𝐺)) = (((1st ‘𝐾) ∘ (1st ‘𝐻)) ∘ (1st ‘𝐺)))
7		eqid 2740	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		cofuass.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
9	7, 8, 3	cofu1st 17947	. . . . 5 ⊢ (𝜑 → (1st ‘(𝐻 ∘func 𝐺)) = ((1st ‘𝐻) ∘ (1st ‘𝐺)))
10	9	coeq2d 5887	. . . 4 ⊢ (𝜑 → ((1st ‘𝐾) ∘ (1st ‘(𝐻 ∘func 𝐺))) = ((1st ‘𝐾) ∘ ((1st ‘𝐻) ∘ (1st ‘𝐺))))
11	1, 6, 10	3eqtr4a 2806	. . 3 ⊢ (𝜑 → ((1st ‘(𝐾 ∘func 𝐻)) ∘ (1st ‘𝐺)) = ((1st ‘𝐾) ∘ (1st ‘(𝐻 ∘func 𝐺))))
12		coass 6296	. . . . 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 17926	. . . . . . . . . . 11 ⊢ Rel (𝐶 Func 𝐷)
16		1st2ndbr 8083	. . . . . . . . . . 11 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺))
17	15, 8, 16	sylancr 586	. . . . . . . . . 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 17930	. . . . . . . 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 7119	. . . . . . 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 7119	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘𝐺)‘𝑦) ∈ (Base‘𝐷))
24	2, 13, 14, 21, 23	cofu2nd 17949	. . . . . 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 5886	. . . . 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 17948	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘(𝐻 ∘func 𝐺))‘𝑥) = ((1st ‘𝐻)‘((1st ‘𝐺)‘𝑥)))
28	7, 26, 13, 22	cofu1 17948	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → ((1st ‘(𝐻 ∘func 𝐺))‘𝑦) = ((1st ‘𝐻)‘((1st ‘𝐺)‘𝑦)))
29	27, 28	oveq12d 7466	. . . . . 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 17949	. . . . . 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 5889	. . . . 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 2806	. . . 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 7527	. . 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 4905	. 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 17952	. . 3 ⊢ (𝜑 → (𝐾 ∘func 𝐻) ∈ (𝐷 Func 𝐹))
36	7, 8, 35	cofuval 17946	. 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 17952	. . 3 ⊢ (𝜑 → (𝐻 ∘func 𝐺) ∈ (𝐶 Func 𝐸))
38	7, 37, 4	cofuval 17946	. 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 2790	1 ⊢ (𝜑 → ((𝐾 ∘func 𝐻) ∘func 𝐺) = (𝐾 ∘func (𝐻 ∘func 𝐺)))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ⟨cop 4654   class class class wbr 5166   ∘ ccom 5704  Rel wrel 5705  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1^st c1st 8028  2^nd c2nd 8029  Basecbs 17258   Func cfunc 17918   ∘_func ccofu 17920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-map 8886  df-ixp 8956  df-cat 17726  df-cid 17727  df-func 17922  df-cofu 17924

This theorem is referenced by:  catccatid  18173