Theorem lanup 50478

Description: The universal property of the left Kan extension; expressed explicitly. (Contributed by Zhi Wang, 4-Nov-2025.)

Hypotheses

Ref	Expression
lanup.s	⊢ 𝑆 = (𝐶 FuncCat 𝐸)
lanup.m	⊢ 𝑀 = (𝐷 Nat 𝐸)
lanup.n	⊢ 𝑁 = (𝐶 Nat 𝐸)
lanup.x	⊢ ∙ = (comp‘𝑆)
lanup.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
lanup.l	⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))
lanup.a	⊢ (𝜑 → 𝐴 ∈ (𝑋𝑁(𝐿 ∘func 𝐹)))

Assertion

Ref	Expression
lanup	⊢ (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴 ↔ ∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴)))

Distinct variable groups:   ∙ ,𝑎,𝑏,𝑙   𝐴,𝑎,𝑏,𝑙   𝐶,𝑎,𝑏,𝑙   𝐷,𝑎,𝑏,𝑙   𝐸,𝑎,𝑏,𝑙   𝐹,𝑎,𝑏,𝑙   𝐿,𝑎,𝑏,𝑙   𝑀,𝑎,𝑏,𝑙   𝑁,𝑎,𝑏,𝑙   𝑆,𝑎,𝑏,𝑙   𝑋,𝑎,𝑏,𝑙   𝜑,𝑎,𝑏,𝑙

Proof of Theorem lanup

Step	Hyp	Ref	Expression
1		eqid 2765	. . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2	1	fucbas 18044	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
3		lanup.s	. . . 4 ⊢ 𝑆 = (𝐶 FuncCat 𝐸)
4	3	fucbas 18044	. . 3 ⊢ (𝐶 Func 𝐸) = (Base‘𝑆)
5		lanup.m	. . . 4 ⊢ 𝑀 = (𝐷 Nat 𝐸)
6	1, 5	fuchom 18045	. . 3 ⊢ 𝑀 = (Hom ‘(𝐷 FuncCat 𝐸))
7		lanup.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐸)
8	3, 7	fuchom 18045	. . 3 ⊢ 𝑁 = (Hom ‘𝑆)
9		lanup.x	. . 3 ⊢ ∙ = (comp‘𝑆)
10		lanup.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝑁(𝐿 ∘func 𝐹)))
11	7	natrcl 18034	. . . . 5 ⊢ (𝐴 ∈ (𝑋𝑁(𝐿 ∘func 𝐹)) → (𝑋 ∈ (𝐶 Func 𝐸) ∧ (𝐿 ∘func 𝐹) ∈ (𝐶 Func 𝐸)))
12	10, 11	syl 18	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝐶 Func 𝐸) ∧ (𝐿 ∘func 𝐹) ∈ (𝐶 Func 𝐸)))
13	12	simpld 500	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
14	13	func1st2nd 49913	. . . . . 6 ⊢ (𝜑 → (1st ‘𝑋)(𝐶 Func 𝐸)(2nd ‘𝑋))
15	14	funcrcl3 49917	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
16		lanup.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
17	1, 15, 3, 16	prcoffunca 50223	. . . 4 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ ((𝐷 FuncCat 𝐸) Func 𝑆))
18	17	func1st2nd 49913	. . 3 ⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))((𝐷 FuncCat 𝐸) Func 𝑆)(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
19		lanup.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ (𝐷 Func 𝐸))
20		eqidd 2766	. . . . . 6 ⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
21	19, 20	prcof1 50225	. . . . 5 ⊢ (𝜑 → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿) = (𝐿 ∘func 𝐹))
22	21	oveq2d 7435	. . . 4 ⊢ (𝜑 → (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)) = (𝑋𝑁(𝐿 ∘func 𝐹)))
23	10, 22	eleqtrrd 2868	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)))
24	2, 4, 6, 8, 9, 13, 18, 19, 23	isup 50017	. 2 ⊢ (𝜑 → (𝐿(⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩((𝐷 FuncCat 𝐸) UP 𝑆)𝑋)𝐴 ↔ ∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = (((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏)(⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))𝐴)))
25		eqidd 2766	. . . . 5 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = (⟨𝐷, 𝐸⟩ −∘F 𝐹))
26	1, 3, 16, 13, 25	lanval 50456	. . . 4 ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = ((⟨𝐷, 𝐸⟩ −∘F 𝐹)((𝐷 FuncCat 𝐸) UP 𝑆)𝑋))
27	26	breqd 5122	. . 3 ⊢ (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴 ↔ 𝐿((⟨𝐷, 𝐸⟩ −∘F 𝐹)((𝐷 FuncCat 𝐸) UP 𝑆)𝑋)𝐴))
28	17	up1st2ndb 50024	. . 3 ⊢ (𝜑 → (𝐿((⟨𝐷, 𝐸⟩ −∘F 𝐹)((𝐷 FuncCat 𝐸) UP 𝑆)𝑋)𝐴 ↔ 𝐿(⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩((𝐷 FuncCat 𝐸) UP 𝑆)𝑋)𝐴))
29	27, 28	bitrd 282	. 2 ⊢ (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴 ↔ 𝐿(⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩((𝐷 FuncCat 𝐸) UP 𝑆)𝑋)𝐴))
30		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → 𝑙 ∈ (𝐷 Func 𝐸))
31		eqidd 2766	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
32	30, 31	prcof1 50225	. . . . . 6 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙) = (𝑙 ∘func 𝐹))
33	32	eqcomd 2771	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (𝑙 ∘func 𝐹) = ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))
34	33	oveq2d 7435	. . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (𝑋𝑁(𝑙 ∘func 𝐹)) = (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙)))
35	21	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿) = (𝐿 ∘func 𝐹))
36	35	opeq2d 4847	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → ⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ = ⟨𝑋, (𝐿 ∘func 𝐹)⟩)
37	32	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙) = (𝑙 ∘func 𝐹))
38	36, 37	oveq12d 7437	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → (⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙)) = (⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹)))
39		simpr 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → 𝑏 ∈ (𝐿𝑀𝑙))
40		eqidd 2766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
41	16	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → 𝐹 ∈ (𝐶 Func 𝐷))
42	5, 39, 40, 41	prcof21a 50228	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → ((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏) = (𝑏 ∘ (1st ‘𝐹)))
43		eqidd 2766	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → 𝐴 = 𝐴)
44	38, 42, 43	oveq123d 7440	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → (((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏)(⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))𝐴) = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴))
45	44	eqcomd 2771	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴) = (((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏)(⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))𝐴))
46	45	eqeq2d 2776	. . . . 5 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → (𝑎 = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴) ↔ 𝑎 = (((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏)(⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))𝐴)))
47	46	reubidva 3385	. . . 4 ⊢ (((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) → (∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴) ↔ ∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = (((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏)(⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))𝐴)))
48	34, 47	raleqbidva 3331	. . 3 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (∀𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴) ↔ ∀𝑎 ∈ (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = (((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏)(⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))𝐴)))
49	48	ralbidva 3188	. 2 ⊢ (𝜑 → (∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴) ↔ ∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = (((𝐿(2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))𝑙)‘𝑏)(⟨𝑋, ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)⟩ ∙ ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))𝐴)))
50	24, 29, 49	3bitr4d 314	1 ⊢ (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋)𝐴 ↔ ∀𝑙 ∈ (𝐷 Func 𝐸)∀𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))∃!𝑏 ∈ (𝐿𝑀𝑙)𝑎 = ((𝑏 ∘ (1st ‘𝐹))(⟨𝑋, (𝐿 ∘func 𝐹)⟩ ∙ (𝑙 ∘func 𝐹))𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃!wreu 3369  ⟨cop 4597   class class class wbr 5111   ∘ ccom 5667  ‘cfv 6540  (class class class)co 7419  1^st c1st 7990  2^nd c2nd 7991  compcco 17346   Func cfunc 17935   ∘_func ccofu 17937   Nat cnat 18025   FuncCat cfuc 18026   UP cup 50010   −∘_F cprcof 50210   Lan clan 50442

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 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194

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 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-nn 12251  df-2 12320  df-3 12321  df-4 12322  df-5 12323  df-6 12324  df-7 12325  df-8 12326  df-9 12327  df-n0 12522  df-z 12609  df-dec 12730  df-uz 12881  df-fz 13554  df-struct 17231  df-slot 17266  df-ndx 17278  df-base 17294  df-hom 17358  df-cco 17359  df-cat 17748  df-cid 17749  df-func 17939  df-cofu 17941  df-nat 18027  df-fuc 18028  df-xpc 18252  df-curf 18294  df-up 50011  df-swapf 50097  df-fuco 50154  df-prcof 50211  df-lan 50444

This theorem is used by: (None)