fucolid - Mathbox for Zhi Wang

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Theorem fucolid 50019

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 2769	. . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸)
4		fucolid.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
5	1, 2, 3, 4	fucid 18027	. . 3 ⊢ (𝜑 → (𝐼‘𝐹) = ((Id‘𝐸) ∘ (1^st ‘𝐹)))
6	5	oveq1d 7423	. 2 ⊢ (𝜑 → ((𝐼‘𝐹)(⟨𝐹, 𝐺⟩𝑃⟨𝐹, 𝐻⟩)𝐴) = (((Id‘𝐸) ∘ (1^st ‘𝐹))(⟨𝐹, 𝐺⟩𝑃⟨𝐹, 𝐻⟩)𝐴))
7		eqid 2769	. . . . . . . 8 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
8		fucolid.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐶 Nat 𝐷)𝐻))
9	7, 8	nat1st2nd 18007	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
10	7, 9	natrcl2 49882	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
11	10	funcrcl2 49737	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
12	10	funcrcl3 49738	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
13	4	func1st2nd 49734	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
14	13	funcrcl3 49738	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
15		eqidd 2770	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = (⟨𝐶, 𝐷⟩ ∘_F 𝐸))
16	11, 12, 14, 15	fucoelvv 49978	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) ∈ (V × V))
17		1st2nd2 8021	. . . . 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 18	. . . 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 4846	. . . 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 2804	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), 𝑃⟩)
22		eqidd 2770	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = ⟨𝐹, 𝐺⟩)
23		eqidd 2770	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐻⟩ = ⟨𝐹, 𝐻⟩)
24		eqid 2769	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
25	1, 24, 3, 4	fucidcl 18021	. . 3 ⊢ (𝜑 → ((Id‘𝐸) ∘ (1^st ‘𝐹)) ∈ (𝐹(𝐷 Nat 𝐸)𝐹))
26	21, 22, 23, 8, 25	fuco22a 50008	. 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 2769	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
28		eqid 2769	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
29	13	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
30	27, 28, 29	funcf1 17919	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐹):(Base‘𝐷)⟶(Base‘𝐸))
31		eqid 2769	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
32	7, 9	natrcl3 49883	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
33	31, 27, 32	funcf1 17919	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
34	33	ffvelcdmda 7077	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐻)‘𝑥) ∈ (Base‘𝐷))
35	30, 34	fvco3d 6980	. . . . 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 7423	. . . 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 2769	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
38	14	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
39	31, 27, 10	funcf1 17919	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
40	39	ffvelcdmda 7077	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
41	30, 40	ffvelcdmd 7078	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)) ∈ (Base‘𝐸))
42		eqid 2769	. . . . 5 ⊢ (comp‘𝐸) = (comp‘𝐸)
43	30, 34	ffvelcdmd 7078	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥)) ∈ (Base‘𝐸))
44		eqid 2769	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
45	27, 44, 37, 29, 40, 34	funcf2 17921	. . . . . 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 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐴 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
47		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
48	7, 46, 31, 44, 47	natcl 18009	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐴‘𝑥) ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)))
49	45, 48	ffvelcdmd 7078	. . . . 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 17735	. . . 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 2804	. . 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 5205	. 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 2808	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 400 = wceq 1567 ∈ wcel 2149 Vcvv 3463 ⟨cop 4597 class class class wbr 5110 ↦ cmpt 5193 × cxp 5657 ∘ ccom 5663 ‘cfv 6534 (class class class)co 7408 1^st c1st 7980 2^nd c2nd 7981 Basecbs 17265 Hom chom 17317 compcco 17318 Catccat 17716 Idccid 17717 Func cfunc 17907 Nat cnat 17997 FuncCat cfuc 17998 ∘_F cfuco 49974

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6300 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12501 df-z 12588 df-dec 12708 df-uz 12859 df-fz 13532 df-struct 17203 df-slot 17238 df-ndx 17250 df-base 17266 df-hom 17330 df-cco 17331 df-cat 17720 df-cid 17721 df-func 17911 df-cofu 17913 df-nat 17999 df-fuc 18000 df-fuco 49975

This theorem is referenced by: postcofval 50022

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