fucolid - Mathbox for Zhi Wang

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Theorem fucolid 50172

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 2765	. . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸)
4		fucolid.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
5	1, 2, 3, 4	fucid 18049	. . 3 ⊢ (𝜑 → (𝐼‘𝐹) = ((Id‘𝐸) ∘ (1^st ‘𝐹)))
6	5	oveq1d 7431	. 2 ⊢ (𝜑 → ((𝐼‘𝐹)(⟨𝐹, 𝐺⟩𝑃⟨𝐹, 𝐻⟩)𝐴) = (((Id‘𝐸) ∘ (1^st ‘𝐹))(⟨𝐹, 𝐺⟩𝑃⟨𝐹, 𝐻⟩)𝐴))
7		eqid 2765	. . . . . . . 8 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
8		fucolid.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐶 Nat 𝐷)𝐻))
9	7, 8	nat1st2nd 18029	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
10	7, 9	natrcl2 50035	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
11	10	funcrcl2 49890	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
12	10	funcrcl3 49891	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
13	4	func1st2nd 49887	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
14	13	funcrcl3 49891	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
15		eqidd 2766	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = (⟨𝐶, 𝐷⟩ ∘_F 𝐸))
16	11, 12, 14, 15	fucoelvv 50131	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) ∈ (V × V))
17		1st2nd2 8027	. . . . 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 4847	. . . 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 2800	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), 𝑃⟩)
22		eqidd 2766	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = ⟨𝐹, 𝐺⟩)
23		eqidd 2766	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐻⟩ = ⟨𝐹, 𝐻⟩)
24		eqid 2765	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
25	1, 24, 3, 4	fucidcl 18043	. . 3 ⊢ (𝜑 → ((Id‘𝐸) ∘ (1^st ‘𝐹)) ∈ (𝐹(𝐷 Nat 𝐸)𝐹))
26	21, 22, 23, 8, 25	fuco22a 50161	. 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 2765	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
28		eqid 2765	. . . . . . 7 ⊢ (Base‘𝐸) = (Base‘𝐸)
29	13	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
30	27, 28, 29	funcf1 17941	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐹):(Base‘𝐷)⟶(Base‘𝐸))
31		eqid 2765	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
32	7, 9	natrcl3 50036	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
33	31, 27, 32	funcf1 17941	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
34	33	ffvelcdmda 7083	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐻)‘𝑥) ∈ (Base‘𝐷))
35	30, 34	fvco3d 6986	. . . . 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 7431	. . . 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 2765	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
38	14	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
39	31, 27, 10	funcf1 17941	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
40	39	ffvelcdmda 7083	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
41	30, 40	ffvelcdmd 7084	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)) ∈ (Base‘𝐸))
42		eqid 2765	. . . . 5 ⊢ (comp‘𝐸) = (comp‘𝐸)
43	30, 34	ffvelcdmd 7084	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥)) ∈ (Base‘𝐸))
44		eqid 2765	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
45	27, 44, 37, 29, 40, 34	funcf2 17943	. . . . . 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 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐴 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
47		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
48	7, 46, 31, 44, 47	natcl 18031	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐴‘𝑥) ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)))
49	45, 48	ffvelcdmd 7084	. . . . 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 17757	. . . 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 2800	. . 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 5206	. 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 2804	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 Vcvv 3457 ⟨cop 4597 class class class wbr 5111 ↦ cmpt 5194 × cxp 5661 ∘ ccom 5667 ‘cfv 6540 (class class class)co 7416 1^st c1st 7986 2^nd c2nd 7987 Basecbs 17287 Hom chom 17339 compcco 17340 Catccat 17738 Idccid 17739 Func cfunc 17929 Nat cnat 18019 FuncCat cfuc 18020 ∘_F cfuco 50127

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-z 12603 df-dec 12724 df-uz 12875 df-fz 13548 df-struct 17225 df-slot 17260 df-ndx 17272 df-base 17288 df-hom 17352 df-cco 17353 df-cat 17742 df-cid 17743 df-func 17933 df-cofu 17935 df-nat 18021 df-fuc 18022 df-fuco 50128

This theorem is used by: postcofval 50175

W3C validator