fucolid - Mathbox for Zhi Wang

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Theorem fucolid 49522

Description: Post-compose a natural transformation with an identity natural transformation. (Contributed by Zhi Wang, 11-Oct-2025.)

**Hypotheses**
Ref	Expression
fucolid.p	⊢ (𝜑 → (2^nd ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)) = 𝑃)
fucolid.i	⊢ 𝐼 = (Id‘𝑄)
fucolid.q	⊢ 𝑄 = (𝐷 FuncCat 𝐸)
fucolid.a	⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐶 Nat 𝐷)𝐻))
fucolid.f	⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))

**Assertion**
Ref	Expression
fucolid	⊢ (𝜑 → ((𝐼‘𝐹)(⟨𝐹, 𝐺⟩𝑃⟨𝐹, 𝐻⟩)𝐴) = (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐶 𝑥,𝐷 𝑥,𝐸 𝑥,𝐹 𝑥,𝐺 𝑥,𝐻 𝜑,𝑥 Allowed substitution hints: 𝑃(𝑥) 𝑄(𝑥) 𝐼(𝑥)

**Proof of Theorem fucolid**
Step	Hyp	Ref	Expression
1		fucolid.q	. . . 4 ⊢ 𝑄 = (𝐷 FuncCat 𝐸)
2		fucolid.i	. . . 4 ⊢ 𝐼 = (Id‘𝑄)
3		eqid 2733	. . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸)
4		fucolid.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
5	1, 2, 3, 4	fucid 17889	. . 3 ⊢ (𝜑 → (𝐼‘𝐹) = ((Id‘𝐸) ∘ (1^st ‘𝐹)))
6	5	oveq1d 7370	. 2 ⊢ (𝜑 → ((𝐼‘𝐹)(⟨𝐹, 𝐺⟩𝑃⟨𝐹, 𝐻⟩)𝐴) = (((Id‘𝐸) ∘ (1^st ‘𝐹))(⟨𝐹, 𝐺⟩𝑃⟨𝐹, 𝐻⟩)𝐴))
7		eqid 2733	. . . . . . . 8 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
8		fucolid.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐶 Nat 𝐷)𝐻))
9	7, 8	nat1st2nd 17869	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
10	7, 9	natrcl2 49385	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
11	10	funcrcl2 49240	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
12	10	funcrcl3 49241	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
13	4	func1st2nd 49237	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
14	13	funcrcl3 49241	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
15		eqidd 2734	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = (⟨𝐶, 𝐷⟩ ∘_F 𝐸))
16	11, 12, 14, 15	fucoelvv 49481	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) ∈ (V × V))
17		1st2nd2 7969	. . . . 5 ⊢ ((⟨𝐶, 𝐷⟩ ∘_F 𝐸) ∈ (V × V) → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), (2^nd ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸))⟩)
18	16, 17	syl 17	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), (2^nd ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸))⟩)
19		fucolid.p	. . . . 5 ⊢ (𝜑 → (2^nd ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)) = 𝑃)
20	19	opeq2d 4833	. . . 4 ⊢ (𝜑 → ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), (2^nd ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸))⟩ = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), 𝑃⟩)
21	18, 20	eqtrd 2768	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), 𝑃⟩)
22		eqidd 2734	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = ⟨𝐹, 𝐺⟩)
23		eqidd 2734	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐻⟩ = ⟨𝐹, 𝐻⟩)
24		eqid 2733	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
25	1, 24, 3, 4	fucidcl 17883	. . 3 ⊢ (𝜑 → ((Id‘𝐸) ∘ (1^st ‘𝐹)) ∈ (𝐹(𝐷 Nat 𝐸)𝐹))
26	21, 22, 23, 8, 25	fuco22a 49511	. 2 ⊢ (𝜑 → (((Id‘𝐸) ∘ (1^st ‘𝐹))(⟨𝐹, 𝐺⟩𝑃⟨𝐹, 𝐻⟩)𝐴) = (𝑥 ∈ (Base‘𝐶) ↦ ((((Id‘𝐸) ∘ (1^st ‘𝐹))‘((1^st ‘𝐻)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥)))))
27		eqid 2733	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
28		eqid 2733	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
29	13	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
30	27, 28, 29	funcf1 17781	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐹):(Base‘𝐷)⟶(Base‘𝐸))
31		eqid 2733	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
32	7, 9	natrcl3 49386	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
33	31, 27, 32	funcf1 17781	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
34	33	ffvelcdmda 7026	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐻)‘𝑥) ∈ (Base‘𝐷))
35	30, 34	fvco3d 6931	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((Id‘𝐸) ∘ (1^st ‘𝐹))‘((1^st ‘𝐻)‘𝑥)) = ((Id‘𝐸)‘((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥))))
36	35	oveq1d 7370	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((((Id‘𝐸) ∘ (1^st ‘𝐹))‘((1^st ‘𝐻)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))) = (((Id‘𝐸)‘((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))))
37		eqid 2733	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
38	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
39	31, 27, 10	funcf1 17781	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
40	39	ffvelcdmda 7026	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
41	30, 40	ffvelcdmd 7027	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)) ∈ (Base‘𝐸))
42		eqid 2733	. . . . 5 ⊢ (comp‘𝐸) = (comp‘𝐸)
43	30, 34	ffvelcdmd 7027	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥)) ∈ (Base‘𝐸))
44		eqid 2733	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
45	27, 44, 37, 29, 40, 34	funcf2 17783	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥)):(((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥))⟶(((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥))(Hom ‘𝐸)((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥))))
46	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐴 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
47		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
48	7, 46, 31, 44, 47	natcl 17871	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐴‘𝑥) ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)))
49	45, 48	ffvelcdmd 7027	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥)) ∈ (((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥))(Hom ‘𝐸)((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥))))
50	28, 37, 3, 38, 41, 42, 43, 49	catlid 17597	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (((Id‘𝐸)‘((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥)))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))) = ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥)))
51	36, 50	eqtrd 2768	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((((Id‘𝐸) ∘ (1^st ‘𝐹))‘((1^st ‘𝐻)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))) = ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥)))
52	51	mpteq2dva 5188	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((((Id‘𝐸) ∘ (1^st ‘𝐹))‘((1^st ‘𝐻)‘𝑥))(⟨((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)), ((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥))⟩(comp‘𝐸)((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥)))((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥)))) = (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))))
53	6, 26, 52	3eqtrd 2772	1 ⊢ (𝜑 → ((𝐼‘𝐹)(⟨𝐹, 𝐺⟩𝑃⟨𝐹, 𝐻⟩)𝐴) = (𝑥 ∈ (Base‘𝐶) ↦ ((((1^st ‘𝐺)‘𝑥)(2^nd ‘𝐹)((1^st ‘𝐻)‘𝑥))‘(𝐴‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Vcvv 3437 ⟨cop 4583 class class class wbr 5095 ↦ cmpt 5176 × cxp 5619 ∘ ccom 5625 ‘cfv 6489 (class class class)co 7355 1^st c1st 7928 2^nd c2nd 7929 Basecbs 17127 Hom chom 17179 compcco 17180 Catccat 17578 Idccid 17579 Func cfunc 17769 Nat cnat 17859 FuncCat cfuc 17860 ∘_F cfuco 49477

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-er 8631 df-map 8761 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-dec 12599 df-uz 12743 df-fz 13415 df-struct 17065 df-slot 17100 df-ndx 17112 df-base 17128 df-hom 17192 df-cco 17193 df-cat 17582 df-cid 17583 df-func 17773 df-cofu 17775 df-nat 17861 df-fuc 17862 df-fuco 49478

This theorem is referenced by: postcofval 49525

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