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Theorem fucass 17356

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 2739	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2739	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
3		eqid 2739	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
4		fucass.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
5		fucass.n	. . . . . . . . . . 11 ⊢ 𝑁 = (𝐶 Nat 𝐷)
6	5	natrcl 17338	. . . . . . . . . 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 498	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
9		funcrcl 17251	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
11	10	simprd 499	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
12	11	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
13		eqid 2739	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
14		relfunc 17250	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
15		1st2ndbr 7779	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
16	14, 8, 15	sylancr 590	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
17	13, 1, 16	funcf1 17254	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
18	17	ffvelrnda 6874	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
19	7	simprd 499	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
20		1st2ndbr 7779	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
21	14, 19, 20	sylancr 590	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
22	13, 1, 21	funcf1 17254	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
23	22	ffvelrnda 6874	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
24		fucass.t	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ (𝐻𝑁𝐾))
25	5	natrcl 17338	. . . . . . . . . 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 498	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
28		1st2ndbr 7779	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
29	14, 27, 28	sylancr 590	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
30	13, 1, 29	funcf1 17254	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
31	30	ffvelrnda 6874	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐻)‘𝑥) ∈ (Base‘𝐷))
32	5, 4	nat1st2nd 17339	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
33	32	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
34		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
35	5, 33, 13, 2, 34	natcl 17341	. . . . 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 17339	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩𝑁⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
38	37	adantr 484	. . . . . 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 17341	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑆‘𝑥) ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)))
40	26	simprd 499	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (𝐶 Func 𝐷))
41		1st2ndbr 7779	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐾 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐾)(𝐶 Func 𝐷)(2^nd ‘𝐾))
42	14, 40, 41	sylancr 590	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐾)(𝐶 Func 𝐷)(2^nd ‘𝐾))
43	13, 1, 42	funcf1 17254	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐾):(Base‘𝐶)⟶(Base‘𝐷))
44	43	ffvelrnda 6874	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐾)‘𝑥) ∈ (Base‘𝐷))
45	5, 24	nat1st2nd 17339	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩𝑁⟨(1^st ‘𝐾), (2^nd ‘𝐾)⟩))
46	45	adantr 484	. . . . . 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 17341	. . . . 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 17073	. . . 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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (𝐺𝑁𝐻))
52	24	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑇 ∈ (𝐻𝑁𝐾))
53	49, 5, 13, 3, 50, 51, 52, 34	fuccoval 17351	. . . . 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 7198	. . . 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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (𝐹𝑁𝐺))
56	49, 5, 13, 3, 50, 55, 51, 34	fuccoval 17351	. . . . 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 7199	. . . 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 2784	. . 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 5135	. 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 17352	. . 3 ⊢ (𝜑 → (𝑇(⟨𝐺, 𝐻⟩ ∙ 𝐾)𝑆) ∈ (𝐺𝑁𝐾))
61	49, 5, 13, 3, 50, 4, 60	fucco 17350	. 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 17352	. . 3 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ (𝐹𝑁𝐻))
63	49, 5, 13, 3, 50, 62, 24	fucco 17350	. 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 2784	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 399 = wceq 1542 ∈ wcel 2114 ⟨cop 4532 class class class wbr 5040 ↦ cmpt 5120 Rel wrel 5540 ‘cfv 6350 (class class class)co 7183 1^st c1st 7725 2^nd c2nd 7726 Basecbs 16599 Hom chom 16692 compcco 16693 Catccat 17051 Func cfunc 17242 Nat cnat 17329 FuncCat cfuc 17330

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-er 8333 df-map 8452 df-ixp 8521 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-5 11795 df-6 11796 df-7 11797 df-8 11798 df-9 11799 df-n0 11990 df-z 12076 df-dec 12193 df-uz 12338 df-fz 12995 df-struct 16601 df-ndx 16602 df-slot 16603 df-base 16605 df-hom 16705 df-cco 16706 df-cat 17055 df-func 17246 df-nat 17331 df-fuc 17332

This theorem is referenced by: fuccatid 17357

W3C validator