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Theorem fucass 17686

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 2738	. . . . 5 ⊢ (Base‘𝐷) = (Base‘𝐷)
2		eqid 2738	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
3		eqid 2738	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
4		fucass.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
5		fucass.n	. . . . . . . . . . 11 ⊢ 𝑁 = (𝐶 Nat 𝐷)
6	5	natrcl 17666	. . . . . . . . . 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 17578	. . . . . . . 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 2738	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
14		relfunc 17577	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
15		1st2ndbr 7883	. . . . . . . 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 17581	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
18	17	ffvelrnda 6961	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
19	7	simprd 496	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
20		1st2ndbr 7883	. . . . . . . 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 17581	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
23	22	ffvelrnda 6961	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
24		fucass.t	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ (𝐻𝑁𝐾))
25	5	natrcl 17666	. . . . . . . . . 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 7883	. . . . . . . 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 17581	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
31	30	ffvelrnda 6961	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐻)‘𝑥) ∈ (Base‘𝐷))
32	5, 4	nat1st2nd 17667	. . . . . . 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 17669	. . . . 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 17667	. . . . . . 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 17669	. . . . 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 7883	. . . . . . . 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 17581	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐾):(Base‘𝐶)⟶(Base‘𝐷))
44	43	ffvelrnda 6961	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐾)‘𝑥) ∈ (Base‘𝐷))
45	5, 24	nat1st2nd 17667	. . . . . . 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 17669	. . . . 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 17395	. . . 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 17681	. . . . 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 7290	. . . 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 17681	. . . . 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 7291	. . . 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 2788	. . 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 5174	. 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 17682	. . 3 ⊢ (𝜑 → (𝑇(⟨𝐺, 𝐻⟩ ∙ 𝐾)𝑆) ∈ (𝐺𝑁𝐾))
61	49, 5, 13, 3, 50, 4, 60	fucco 17680	. 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 17682	. . 3 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) ∈ (𝐹𝑁𝐻))
63	49, 5, 13, 3, 50, 62, 24	fucco 17680	. 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 2788	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 1539 ∈ wcel 2106 ⟨cop 4567 class class class wbr 5074 ↦ cmpt 5157 Rel wrel 5594 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 Basecbs 16912 Hom chom 16973 compcco 16974 Catccat 17373 Func cfunc 17569 Nat cnat 17657 FuncCat cfuc 17658

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-hom 16986 df-cco 16987 df-cat 17377 df-func 17573 df-nat 17659 df-fuc 17660

This theorem is referenced by: fuccatid 17687

W3C validator