Theorem lanup 49376

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⟩Lan𝐸)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 2734	. . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2	1	fucbas 17963	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
3		lanup.s	. . . 4 ⊢ 𝑆 = (𝐶 FuncCat 𝐸)
4	3	fucbas 17963	. . 3 ⊢ (𝐶 Func 𝐸) = (Base‘𝑆)
5		lanup.m	. . . 4 ⊢ 𝑀 = (𝐷 Nat 𝐸)
6	1, 5	fuchom 17964	. . 3 ⊢ 𝑀 = (Hom ‘(𝐷 FuncCat 𝐸))
7		lanup.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐸)
8	3, 7	fuchom 17964	. . 3 ⊢ 𝑁 = (Hom ‘𝑆)
9		lanup.x	. . 3 ⊢ ∙ = (comp‘𝑆)
10		lanup.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝑁(𝐿 ∘func 𝐹)))
11	7	natrcl 17953	. . . . 5 ⊢ (𝐴 ∈ (𝑋𝑁(𝐿 ∘func 𝐹)) → (𝑋 ∈ (𝐶 Func 𝐸) ∧ (𝐿 ∘func 𝐹) ∈ (𝐶 Func 𝐸)))
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝐶 Func 𝐸) ∧ (𝐿 ∘func 𝐹) ∈ (𝐶 Func 𝐸)))
13	12	simpld 494	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
14	13	func1st2nd 48936	. . . . . 6 ⊢ (𝜑 → (1st ‘𝑋)(𝐶 Func 𝐸)(2nd ‘𝑋))
15	14	funcrcl3 48938	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
16		lanup.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
17	1, 15, 3, 16	prcoffunca 49159	. . . 4 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ ((𝐷 FuncCat 𝐸) Func 𝑆))
18	17	func1st2nd 48936	. . 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 2735	. . . . . 6 ⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
21	19, 20	prcof1 49161	. . . . 5 ⊢ (𝜑 → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿) = (𝐿 ∘func 𝐹))
22	21	oveq2d 7416	. . . 4 ⊢ (𝜑 → (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)) = (𝑋𝑁(𝐿 ∘func 𝐹)))
23	10, 22	eleqtrrd 2836	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)))
24	2, 4, 6, 8, 9, 13, 18, 19, 23	isup 48981	. 2 ⊢ (𝜑 → (𝐿(⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩((𝐷 FuncCat 𝐸)UP𝑆)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 2735	. . . . 5 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = (⟨𝐷, 𝐸⟩ −∘F 𝐹))
26	1, 3, 16, 13, 25	lanval 49355	. . . 4 ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩Lan𝐸)𝑋) = ((⟨𝐷, 𝐸⟩ −∘F 𝐹)((𝐷 FuncCat 𝐸)UP𝑆)𝑋))
27	26	breqd 5128	. . 3 ⊢ (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩Lan𝐸)𝑋)𝐴 ↔ 𝐿((⟨𝐷, 𝐸⟩ −∘F 𝐹)((𝐷 FuncCat 𝐸)UP𝑆)𝑋)𝐴))
28	17	up1st2ndb 48987	. . 3 ⊢ (𝜑 → (𝐿((⟨𝐷, 𝐸⟩ −∘F 𝐹)((𝐷 FuncCat 𝐸)UP𝑆)𝑋)𝐴 ↔ 𝐿(⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩((𝐷 FuncCat 𝐸)UP𝑆)𝑋)𝐴))
29	27, 28	bitrd 279	. 2 ⊢ (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩Lan𝐸)𝑋)𝐴 ↔ 𝐿(⟨(1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)), (2nd ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))⟩((𝐷 FuncCat 𝐸)UP𝑆)𝑋)𝐴))
30		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → 𝑙 ∈ (𝐷 Func 𝐸))
31		eqidd 2735	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
32	30, 31	prcof1 49161	. . . . . 6 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙) = (𝑙 ∘func 𝐹))
33	32	eqcomd 2740	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (𝑙 ∘func 𝐹) = ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))
34	33	oveq2d 7416	. . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (𝑋𝑁(𝑙 ∘func 𝐹)) = (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙)))
35	21	ad3antrrr 730	. . . . . . . . . 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 4854	. . . . . . . . 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 726	. . . . . . . . 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 7418	. . . . . . . 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 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → 𝑏 ∈ (𝐿𝑀𝑙))
40		eqidd 2735	. . . . . . . . 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 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → 𝐹 ∈ (𝐶 Func 𝐷))
42	5, 39, 40, 41	prcof21a 49164	. . . . . . . 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 2735	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → 𝐴 = 𝐴)
44	38, 42, 43	oveq123d 7421	. . . . . . 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 2740	. . . . . 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 2745	. . . . 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 3373	. . . 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 3309	. . 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 3159	. 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 311	1 ⊢ (𝜑 → (𝐿(𝐹(⟨𝐶, 𝐷⟩Lan𝐸)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)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ∃!wreu 3355  ⟨cop 4605   class class class wbr 5117   ∘ ccom 5656  ‘cfv 6528  (class class class)co 7400  1^st c1st 7981  2^nd c2nd 7982  compcco 17270   Func cfunc 17854   ∘_func ccofu 17856   Nat cnat 17944   FuncCat cfuc 17945  UPcup 48974   −∘_F cprcof 49147  Lanclan 49343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5247  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724  ax-cnex 11178  ax-resscn 11179  ax-1cn 11180  ax-icn 11181  ax-addcl 11182  ax-addrcl 11183  ax-mulcl 11184  ax-mulrcl 11185  ax-mulcom 11186  ax-addass 11187  ax-mulass 11188  ax-distr 11189  ax-i2m1 11190  ax-1ne0 11191  ax-1rid 11192  ax-rnegex 11193  ax-rrecex 11194  ax-cnre 11195  ax-pre-lttri 11196  ax-pre-lttrn 11197  ax-pre-ltadd 11198  ax-pre-mulgt0 11199

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3357  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-pss 3944  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-tp 4604  df-op 4606  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-tr 5228  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6288  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7857  df-1st 7983  df-2nd 7984  df-frecs 8275  df-wrecs 8306  df-recs 8380  df-rdg 8419  df-1o 8475  df-er 8714  df-map 8837  df-ixp 8907  df-en 8955  df-dom 8956  df-sdom 8957  df-fin 8958  df-pnf 11264  df-mnf 11265  df-xr 11266  df-ltxr 11267  df-le 11268  df-sub 11461  df-neg 11462  df-nn 12234  df-2 12296  df-3 12297  df-4 12298  df-5 12299  df-6 12300  df-7 12301  df-8 12302  df-9 12303  df-n0 12495  df-z 12582  df-dec 12702  df-uz 12846  df-fz 13515  df-struct 17153  df-slot 17188  df-ndx 17200  df-base 17216  df-hom 17282  df-cco 17283  df-cat 17667  df-cid 17668  df-func 17858  df-cofu 17860  df-nat 17946  df-fuc 17947  df-xpc 18171  df-curf 18213  df-up 48975  df-swapf 49040  df-fuco 49091  df-prcof 49148  df-lan 49345

This theorem is referenced by: (None)