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Theorem fucass 17238

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 2821	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2821	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
3		eqid 2821	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
4		fucass.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
5		fucass.n	. . . . . . . . . . 11 ⊢ 𝑁 = (𝐶 Nat 𝐷)
6	5	natrcl 17220	. . . . . . . . . 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 497	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
9		funcrcl 17133	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
11	10	simprd 498	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
12	11	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
13		eqid 2821	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
14		relfunc 17132	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
15		1st2ndbr 7741	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
16	14, 8, 15	sylancr 589	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
17	13, 1, 16	funcf1 17136	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
18	17	ffvelrnda 6851	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
19	7	simprd 498	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
20		1st2ndbr 7741	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
21	14, 19, 20	sylancr 589	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
22	13, 1, 21	funcf1 17136	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
23	22	ffvelrnda 6851	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
24		fucass.t	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ (𝐻𝑁𝐾))
25	5	natrcl 17220	. . . . . . . . . 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 497	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
28		1st2ndbr 7741	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
29	14, 27, 28	sylancr 589	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
30	13, 1, 29	funcf1 17136	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
31	30	ffvelrnda 6851	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐻)‘𝑥) ∈ (Base‘𝐷))
32	5, 4	nat1st2nd 17221	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
33	32	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
34		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
35	5, 33, 13, 2, 34	natcl 17223	. . . . 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 17221	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩𝑁⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
38	37	adantr 483	. . . . . 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 17223	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑆‘𝑥) ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)))
40	26	simprd 498	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (𝐶 Func 𝐷))
41		1st2ndbr 7741	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐾 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐾)(𝐶 Func 𝐷)(2^nd ‘𝐾))
42	14, 40, 41	sylancr 589	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐾)(𝐶 Func 𝐷)(2^nd ‘𝐾))
43	13, 1, 42	funcf1 17136	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐾):(Base‘𝐶)⟶(Base‘𝐷))
44	43	ffvelrnda 6851	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐾)‘𝑥) ∈ (Base‘𝐷))
45	5, 24	nat1st2nd 17221	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩𝑁⟨(1^st ‘𝐾), (2^nd ‘𝐾)⟩))
46	45	adantr 483	. . . . . 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 17223	. . . . 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 16957	. . . 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 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑆 ∈ (𝐺𝑁𝐻))
52	24	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑇 ∈ (𝐻𝑁𝐾))
53	49, 5, 13, 3, 50, 51, 52, 34	fuccoval 17233	. . . . 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 7171	. . . 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 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (𝐹𝑁𝐺))
56	49, 5, 13, 3, 50, 55, 51, 34	fuccoval 17233	. . . . 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 7172	. . . 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 2866	. . 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 5161	. 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 17234	. . 3 ⊢ (𝜑 → (𝑇(⟨𝐺, 𝐻⟩ ∙ 𝐾)𝑆) ∈ (𝐺𝑁𝐾))
61	49, 5, 13, 3, 50, 4, 60	fucco 17232	. 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 17234	. . 3 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ (𝐹𝑁𝐻))
63	49, 5, 13, 3, 50, 62, 24	fucco 17232	. 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 2866	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 398 = wceq 1537 ∈ wcel 2114 ⟨cop 4573 class class class wbr 5066 ↦ cmpt 5146 Rel wrel 5560 ‘cfv 6355 (class class class)co 7156 1^st c1st 7687 2^nd c2nd 7688 Basecbs 16483 Hom chom 16576 compcco 16577 Catccat 16935 Func cfunc 17124 Nat cnat 17211 FuncCat cfuc 17212

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-hom 16589 df-cco 16590 df-cat 16939 df-func 17128 df-nat 17213 df-fuc 17214

This theorem is referenced by: fuccatid 17239

W3C validator