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Theorem fucass 17072

Description: Associativity of natural transformation composition. Remark 6.14(b) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)

**Hypotheses**
Ref	Expression
fucass.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fucass.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
fucass.x	⊢ ∙ = (comp‘𝑄)
fucass.r	⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
fucass.s	⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻))
fucass.t	⊢ (𝜑 → 𝑇 ∈ (𝐻𝑁𝐾))

**Assertion**
Ref	Expression
fucass	⊢ (𝜑 → ((𝑇(⟨𝐺, 𝐻⟩ ∙ 𝐾)𝑆)(⟨𝐹, 𝐺⟩ ∙ 𝐾)𝑅) = (𝑇(⟨𝐹, 𝐻⟩ ∙ 𝐾)(𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)))

Proof of Theorem fucass Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2795	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2795	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
3		eqid 2795	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
4		fucass.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
5		fucass.n	. . . . . . . . . . 11 ⊢ 𝑁 = (𝐶 Nat 𝐷)
6	5	natrcl 17054	. . . . . . . . . 10 ⊢ (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
7	4, 6	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
8	7	simpld 495	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
9		funcrcl 16967	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
11	10	simprd 496	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
12	11	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
13		eqid 2795	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
14		relfunc 16966	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
15		1st2ndbr 7602	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
16	14, 8, 15	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
17	13, 1, 16	funcf1 16970	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
18	17	ffvelrnda 6721	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
19	7	simprd 496	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
20		1st2ndbr 7602	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
21	14, 19, 20	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
22	13, 1, 21	funcf1 16970	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
23	22	ffvelrnda 6721	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
24		fucass.t	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ (𝐻𝑁𝐾))
25	5	natrcl 17054	. . . . . . . . . 10 ⊢ (𝑇 ∈ (𝐻𝑁𝐾) → (𝐻 ∈ (𝐶 Func 𝐷) ∧ 𝐾 ∈ (𝐶 Func 𝐷)))
26	24, 25	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐻 ∈ (𝐶 Func 𝐷) ∧ 𝐾 ∈ (𝐶 Func 𝐷)))
27	26	simpld 495	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
28		1st2ndbr 7602	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
29	14, 27, 28	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
30	13, 1, 29	funcf1 16970	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
31	30	ffvelrnda 6721	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐻)‘𝑥) ∈ (Base‘𝐷))
32	5, 4	nat1st2nd 17055	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
33	32	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
34		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
35	5, 33, 13, 2, 34	natcl 17057	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑅‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑥)))
36		fucass.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻))
37	5, 36	nat1st2nd 17055	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩𝑁⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
38	37	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩𝑁⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
39	5, 38, 13, 2, 34	natcl 17057	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑆‘𝑥) ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)))
40	26	simprd 496	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (𝐶 Func 𝐷))
41		1st2ndbr 7602	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐾 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐾)(𝐶 Func 𝐷)(2^nd ‘𝐾))
42	14, 40, 41	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐾)(𝐶 Func 𝐷)(2^nd ‘𝐾))
43	13, 1, 42	funcf1 16970	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐾):(Base‘𝐶)⟶(Base‘𝐷))
44	43	ffvelrnda 6721	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐾)‘𝑥) ∈ (Base‘𝐷))
45	5, 24	nat1st2nd 17055	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩𝑁⟨(1^st ‘𝐾), (2^nd ‘𝐾)⟩))
46	45	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑇 ∈ (⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩𝑁⟨(1^st ‘𝐾), (2^nd ‘𝐾)⟩))
47	5, 46, 13, 2, 34	natcl 17057	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑇‘𝑥) ∈ (((1^st ‘𝐻)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐾)‘𝑥)))
48	1, 2, 3, 12, 18, 23, 31, 35, 39, 44, 47	catass 16791	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((𝑇‘𝑥)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐾)‘𝑥))(𝑆‘𝑥))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐾)‘𝑥))(𝑅‘𝑥)) = ((𝑇‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐾)‘𝑥))((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥))))
49		fucass.q	. . . . . 6 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
50		fucass.x	. . . . . 6 ⊢ ∙ = (comp‘𝑄)
51	36	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (𝐺𝑁𝐻))
52	24	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑇 ∈ (𝐻𝑁𝐾))
53	49, 5, 13, 3, 50, 51, 52, 34	fuccoval 17067	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑇(⟨𝐺, 𝐻⟩ ∙ 𝐾)𝑆)‘𝑥) = ((𝑇‘𝑥)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐾)‘𝑥))(𝑆‘𝑥)))
54	53	oveq1d 7036	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((𝑇(⟨𝐺, 𝐻⟩ ∙ 𝐾)𝑆)‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐾)‘𝑥))(𝑅‘𝑥)) = (((𝑇‘𝑥)(⟨((1^st ‘𝐺)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐾)‘𝑥))(𝑆‘𝑥))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐾)‘𝑥))(𝑅‘𝑥)))
55	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (𝐹𝑁𝐺))
56	49, 5, 13, 3, 50, 55, 51, 34	fuccoval 17067	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑥) = ((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥)))
57	56	oveq2d 7037	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑇‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐾)‘𝑥))((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑥)) = ((𝑇‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐾)‘𝑥))((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥))))
58	48, 54, 57	3eqtr4d 2841	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((𝑇(⟨𝐺, 𝐻⟩ ∙ 𝐾)𝑆)‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐾)‘𝑥))(𝑅‘𝑥)) = ((𝑇‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐾)‘𝑥))((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑥)))
59	58	mpteq2dva 5060	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ (((𝑇(⟨𝐺, 𝐻⟩ ∙ 𝐾)𝑆)‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐾)‘𝑥))(𝑅‘𝑥))) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑇‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐾)‘𝑥))((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑥))))
60	49, 5, 50, 36, 24	fuccocl 17068	. . 3 ⊢ (𝜑 → (𝑇(⟨𝐺, 𝐻⟩ ∙ 𝐾)𝑆) ∈ (𝐺𝑁𝐾))
61	49, 5, 13, 3, 50, 4, 60	fucco 17066	. 2 ⊢ (𝜑 → ((𝑇(⟨𝐺, 𝐻⟩ ∙ 𝐾)𝑆)(⟨𝐹, 𝐺⟩ ∙ 𝐾)𝑅) = (𝑥 ∈ (Base‘𝐶) ↦ (((𝑇(⟨𝐺, 𝐻⟩ ∙ 𝐾)𝑆)‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐾)‘𝑥))(𝑅‘𝑥))))
62	49, 5, 50, 4, 36	fuccocl 17068	. . 3 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ (𝐹𝑁𝐻))
63	49, 5, 13, 3, 50, 62, 24	fucco 17066	. 2 ⊢ (𝜑 → (𝑇(⟨𝐹, 𝐻⟩ ∙ 𝐾)(𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑇‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐻)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐾)‘𝑥))((𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)‘𝑥))))
64	59, 61, 63	3eqtr4d 2841	1 ⊢ (𝜑 → ((𝑇(⟨𝐺, 𝐻⟩ ∙ 𝐾)𝑆)(⟨𝐹, 𝐺⟩ ∙ 𝐾)𝑅) = (𝑇(⟨𝐹, 𝐻⟩ ∙ 𝐾)(𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ⟨cop 4482 class class class wbr 4966 ↦ cmpt 5045 Rel wrel 5453 ‘cfv 6230 (class class class)co 7021 1^st c1st 7548 2^nd c2nd 7549 Basecbs 16317 Hom chom 16410 compcco 16411 Catccat 16769 Func cfunc 16958 Nat cnat 17045 FuncCat cfuc 17046

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-map 8263 df-ixp 8316 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-dec 11953 df-uz 12099 df-fz 12748 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-hom 16423 df-cco 16424 df-cat 16773 df-func 16962 df-nat 17047 df-fuc 17048

This theorem is referenced by: fuccatid 17073

W3C validator