fucolid - Mathbox for Zhi Wang

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Theorem fucolid 49366

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 2729	. . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸)
4		fucolid.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
5	1, 2, 3, 4	fucid 17900	. . 3 ⊢ (𝜑 → (𝐼‘𝐹) = ((Id‘𝐸) ∘ (1^st ‘𝐹)))
6	5	oveq1d 7368	. 2 ⊢ (𝜑 → ((𝐼‘𝐹)(⟨𝐹, 𝐺⟩𝑃⟨𝐹, 𝐻⟩)𝐴) = (((Id‘𝐸) ∘ (1^st ‘𝐹))(⟨𝐹, 𝐺⟩𝑃⟨𝐹, 𝐻⟩)𝐴))
7		eqid 2729	. . . . . . . 8 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
8		fucolid.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐶 Nat 𝐷)𝐻))
9	7, 8	nat1st2nd 17880	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
10	7, 9	natrcl2 49229	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
11	10	funcrcl2 49084	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
12	10	funcrcl3 49085	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
13	4	func1st2nd 49081	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
14	13	funcrcl3 49085	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
15		eqidd 2730	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = (⟨𝐶, 𝐷⟩ ∘_F 𝐸))
16	11, 12, 14, 15	fucoelvv 49325	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) ∈ (V × V))
17		1st2nd2 7970	. . . . 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 4834	. . . 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 2764	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), 𝑃⟩)
22		eqidd 2730	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = ⟨𝐹, 𝐺⟩)
23		eqidd 2730	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐻⟩ = ⟨𝐹, 𝐻⟩)
24		eqid 2729	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
25	1, 24, 3, 4	fucidcl 17894	. . 3 ⊢ (𝜑 → ((Id‘𝐸) ∘ (1^st ‘𝐹)) ∈ (𝐹(𝐷 Nat 𝐸)𝐹))
26	21, 22, 23, 8, 25	fuco22a 49355	. 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 2729	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
28		eqid 2729	. . . . . . 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 17792	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐹):(Base‘𝐷)⟶(Base‘𝐸))
31		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
32	7, 9	natrcl3 49230	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
33	31, 27, 32	funcf1 17792	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
34	33	ffvelcdmda 7022	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐻)‘𝑥) ∈ (Base‘𝐷))
35	30, 34	fvco3d 6927	. . . . 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 7368	. . . 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 2729	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
38	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
39	31, 27, 10	funcf1 17792	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
40	39	ffvelcdmda 7022	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
41	30, 40	ffvelcdmd 7023	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)) ∈ (Base‘𝐸))
42		eqid 2729	. . . . 5 ⊢ (comp‘𝐸) = (comp‘𝐸)
43	30, 34	ffvelcdmd 7023	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥)) ∈ (Base‘𝐸))
44		eqid 2729	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
45	27, 44, 37, 29, 40, 34	funcf2 17794	. . . . . 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 17882	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐴‘𝑥) ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)))
49	45, 48	ffvelcdmd 7023	. . . . 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 17608	. . . 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 2764	. . 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 2768	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 1540 ∈ wcel 2109 Vcvv 3438 ⟨cop 4585 class class class wbr 5095 ↦ cmpt 5176 × cxp 5621 ∘ ccom 5627 ‘cfv 6486 (class class class)co 7353 1^st c1st 7929 2^nd c2nd 7930 Basecbs 17139 Hom chom 17191 compcco 17192 Catccat 17589 Idccid 17590 Func cfunc 17780 Nat cnat 17870 FuncCat cfuc 17871 ∘_F cfuco 49321

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-map 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12611 df-uz 12755 df-fz 13430 df-struct 17077 df-slot 17112 df-ndx 17124 df-base 17140 df-hom 17204 df-cco 17205 df-cat 17593 df-cid 17594 df-func 17784 df-cofu 17786 df-nat 17872 df-fuc 17873 df-fuco 49322

This theorem is referenced by: postcofval 49369

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