Theorem lanup 50748

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 2761	. . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸)
2	1	fucbas 18138	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
3		lanup.s	. . . 4 ⊢ 𝑆 = (𝐶 FuncCat 𝐸)
4	3	fucbas 18138	. . 3 ⊢ (𝐶 Func 𝐸) = (Base‘𝑆)
5		lanup.m	. . . 4 ⊢ 𝑀 = (𝐷 Nat 𝐸)
6	1, 5	fuchom 18139	. . 3 ⊢ 𝑀 = (Hom ‘(𝐷 FuncCat 𝐸))
7		lanup.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐸)
8	3, 7	fuchom 18139	. . 3 ⊢ 𝑁 = (Hom ‘𝑆)
9		lanup.x	. . 3 ⊢ ∙ = (comp‘𝑆)
10		lanup.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝑁(𝐿 ∘func 𝐹)))
11	7	natrcl 18128	. . . . 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 50183	. . . . . 6 ⊢ (𝜑 → (1st ‘𝑋)(𝐶 Func 𝐸)(2nd ‘𝑋))
15	14	funcrcl3 50187	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
16		lanup.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
17	1, 15, 3, 16	prcoffunca 50493	. . . 4 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ ((𝐷 FuncCat 𝐸) Func 𝑆))
18	17	func1st2nd 50183	. . 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 2762	. . . . . 6 ⊢ (𝜑 → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
21	19, 20	prcof1 50495	. . . . 5 ⊢ (𝜑 → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿) = (𝐿 ∘func 𝐹))
22	21	oveq2d 7436	. . . 4 ⊢ (𝜑 → (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)) = (𝑋𝑁(𝐿 ∘func 𝐹)))
23	10, 22	eleqtrrd 2864	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿)))
24	2, 4, 6, 8, 9, 13, 18, 19, 23	isup 50287	. 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 2762	. . . . 5 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) = (⟨𝐷, 𝐸⟩ −∘F 𝐹))
26	1, 3, 16, 13, 25	lanval 50726	. . . 4 ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = ((⟨𝐷, 𝐸⟩ −∘F 𝐹)((𝐷 FuncCat 𝐸) UP 𝑆)𝑋))
27	26	breqd 5114	. . 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 50294	. . 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 2762	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)) = (1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹)))
32	30, 31	prcof1 50495	. . . . . 6 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙) = (𝑙 ∘func 𝐹))
33	32	eqcomd 2767	. . . . 5 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (𝑙 ∘func 𝐹) = ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙))
34	33	oveq2d 7436	. . . 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 4840	. . . . . . . . 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 7438	. . . . . . . 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 2762	. . . . . . . . 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 50498	. . . . . . . 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 2762	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → 𝐴 = 𝐴)
44	38, 42, 43	oveq123d 7441	. . . . . . 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 2767	. . . . . 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 2772	. . . . 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 3380	. . . 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 3326	. . 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 3184	. 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 2145  ∀wral 3077  ∃!wreu 3364  ⟨cop 4590   class class class wbr 5103   ∘ ccom 5655  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  compcco 17440   Func cfunc 18029   ∘_func ccofu 18031   Nat cnat 18119   FuncCat cfuc 18120   UP cup 50280   −∘_F cprcof 50480   Lan clan 50712

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-slot 17360  df-ndx 17372  df-base 17388  df-hom 17452  df-cco 17453  df-cat 17842  df-cid 17843  df-func 18033  df-cofu 18035  df-nat 18121  df-fuc 18122  df-xpc 18346  df-curf 18388  df-up 50281  df-swapf 50367  df-fuco 50424  df-prcof 50481  df-lan 50714

This theorem is used by: (None)