fucolid - Mathbox for Zhi Wang

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Theorem fucolid 50590

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 2760	. . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸)
4		fucolid.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐷 Func 𝐸))
5	1, 2, 3, 4	fucid 18156	. . 3 ⊢ (𝜑 → (𝐼‘𝐹) = ((Id‘𝐸) ∘ (1^st ‘𝐹)))
6	5	oveq1d 7437	. 2 ⊢ (𝜑 → ((𝐼‘𝐹)(⟨𝐹, 𝐺⟩𝑃⟨𝐹, 𝐻⟩)𝐴) = (((Id‘𝐸) ∘ (1^st ‘𝐹))(⟨𝐹, 𝐺⟩𝑃⟨𝐹, 𝐻⟩)𝐴))
7		eqid 2760	. . . . . . . 8 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
8		fucolid.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ (𝐺(𝐶 Nat 𝐷)𝐻))
9	7, 8	nat1st2nd 18136	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩(𝐶 Nat 𝐷)⟨(1^st ‘𝐻), (2^nd ‘𝐻)⟩))
10	7, 9	natrcl2 50447	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
11	10	funcrcl2 50300	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
12	10	funcrcl3 50301	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
13	4	func1st2nd 50297	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐸)(2^nd ‘𝐹))
14	13	funcrcl3 50301	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ Cat)
15		eqidd 2761	. . . . . 6 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = (⟨𝐶, 𝐷⟩ ∘_F 𝐸))
16	11, 12, 14, 15	fucoelvv 50549	. . . . 5 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) ∈ (V × V))
17		1st2nd2 8042	. . . . 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 2795	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘_F 𝐸) = ⟨(1^st ‘(⟨𝐶, 𝐷⟩ ∘_F 𝐸)), 𝑃⟩)
22		eqidd 2761	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = ⟨𝐹, 𝐺⟩)
23		eqidd 2761	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐻⟩ = ⟨𝐹, 𝐻⟩)
24		eqid 2760	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
25	1, 24, 3, 4	fucidcl 18150	. . 3 ⊢ (𝜑 → ((Id‘𝐸) ∘ (1^st ‘𝐹)) ∈ (𝐹(𝐷 Nat 𝐸)𝐹))
26	21, 22, 23, 8, 25	fuco22a 50579	. 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 2760	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
28		eqid 2760	. . . . . . 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 18048	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (1^st ‘𝐹):(Base‘𝐷)⟶(Base‘𝐸))
31		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
32	7, 9	natrcl3 50448	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐻)(𝐶 Func 𝐷)(2^nd ‘𝐻))
33	31, 27, 32	funcf1 18048	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐻):(Base‘𝐶)⟶(Base‘𝐷))
34	33	ffvelcdmda 7087	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐻)‘𝑥) ∈ (Base‘𝐷))
35	30, 34	fvco3d 6989	. . . . 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 7437	. . . 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 2760	. . . . 5 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
38	14	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
39	31, 27, 10	funcf1 18048	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
40	39	ffvelcdmda 7087	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
41	30, 40	ffvelcdmd 7088	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘((1^st ‘𝐺)‘𝑥)) ∈ (Base‘𝐸))
42		eqid 2760	. . . . 5 ⊢ (comp‘𝐸) = (comp‘𝐸)
43	30, 34	ffvelcdmd 7088	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘((1^st ‘𝐻)‘𝑥)) ∈ (Base‘𝐸))
44		eqid 2760	. . . . . . 7 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
45	27, 44, 37, 29, 40, 34	funcf2 18050	. . . . . 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 18138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝐴‘𝑥) ∈ (((1^st ‘𝐺)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐻)‘𝑥)))
49	45, 48	ffvelcdmd 7088	. . . . 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 17864	. . . 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 2795	. . 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 5197	. 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 2799	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 1569 ∈ wcel 2144 Vcvv 3450 ⟨cop 4834 class class class wbr 5102 ↦ cmpt 5185 × cxp 5648 ∘ ccom 5654 ‘cfv 6541 (class class class)co 7422 1^st c1st 8001 2^nd c2nd 8002 Basecbs 17394 Hom chom 17446 compcco 17447 Catccat 17845 Idccid 17846 Func cfunc 18036 Nat cnat 18126 FuncCat cfuc 18127 ∘_F cfuco 50545

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-om 7880 df-1st 8003 df-2nd 8004 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-1o 8481 df-er 8722 df-map 8854 df-ixp 8931 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-nn 12341 df-2 12410 df-3 12411 df-4 12412 df-5 12413 df-6 12414 df-7 12415 df-8 12416 df-9 12417 df-n0 12612 df-z 12699 df-dec 12820 df-uz 12971 df-fz 13645 df-struct 17332 df-slot 17367 df-ndx 17379 df-base 17395 df-hom 17459 df-cco 17460 df-cat 17849 df-cid 17850 df-func 18040 df-cofu 18042 df-nat 18128 df-fuc 18129 df-fuco 50546

This theorem is used by: postcofval 50603

W3C validator