Theorem lanup 50215

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 17977	. . 3 ⊢ (𝐷 Func 𝐸) = (Base‘(𝐷 FuncCat 𝐸))
3		lanup.s	. . . 4 ⊢ 𝑆 = (𝐶 FuncCat 𝐸)
4	3	fucbas 17977	. . 3 ⊢ (𝐶 Func 𝐸) = (Base‘𝑆)
5		lanup.m	. . . 4 ⊢ 𝑀 = (𝐷 Nat 𝐸)
6	1, 5	fuchom 17978	. . 3 ⊢ 𝑀 = (Hom ‘(𝐷 FuncCat 𝐸))
7		lanup.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐸)
8	3, 7	fuchom 17978	. . 3 ⊢ 𝑁 = (Hom ‘𝑆)
9		lanup.x	. . 3 ⊢ ∙ = (comp‘𝑆)
10		lanup.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝑁(𝐿 ∘func 𝐹)))
11	7	natrcl 17967	. . . . 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 498	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐶 Func 𝐸))
14	13	func1st2nd 49650	. . . . . 6 ⊢ (𝜑 → (1st ‘𝑋)(𝐶 Func 𝐸)(2nd ‘𝑋))
15	14	funcrcl3 49654	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Cat)
16		lanup.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
17	1, 15, 3, 16	prcoffunca 49960	. . . 4 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ −∘F 𝐹) ∈ ((𝐷 FuncCat 𝐸) Func 𝑆))
18	17	func1st2nd 49650	. . 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 49962	. . . . 5 ⊢ (𝜑 → ((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝐿) = (𝐿 ∘func 𝐹))
22	21	oveq2d 7406	. . . 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 49754	. 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 50193	. . . 4 ⊢ (𝜑 → (𝐹(⟨𝐶, 𝐷⟩ Lan 𝐸)𝑋) = ((⟨𝐷, 𝐸⟩ −∘F 𝐹)((𝐷 FuncCat 𝐸) UP 𝑆)𝑋))
27	26	breqd 5110	. . 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 49761	. . 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 281	. 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 488	. . . . . . 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 49962	. . . . . 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 7406	. . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) → (𝑋𝑁(𝑙 ∘func 𝐹)) = (𝑋𝑁((1st ‘(⟨𝐷, 𝐸⟩ −∘F 𝐹))‘𝑙)))
35	21	ad3antrrr 740	. . . . . . . . . 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 4837	. . . . . . . . 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 736	. . . . . . . . 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 7408	. . . . . . . 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 488	. . . . . . . . 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 740	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑙 ∈ (𝐷 Func 𝐸)) ∧ 𝑎 ∈ (𝑋𝑁(𝑙 ∘func 𝐹))) ∧ 𝑏 ∈ (𝐿𝑀𝑙)) → 𝐹 ∈ (𝐶 Func 𝐷))
42	5, 39, 40, 41	prcof21a 49965	. . . . . . . 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 7411	. . . . . . 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 3325	. . 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 3182	. 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 313	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)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃!wreu 3364  ⟨cop 4587   class class class wbr 5099   ∘ ccom 5649  ‘cfv 6515  (class class class)co 7390  1^st c1st 7962  2^nd c2nd 7963  compcco 17279   Func cfunc 17868   ∘_func ccofu 17870   Nat cnat 17958   FuncCat cfuc 17959   UP cup 49747   −∘_F cprcof 49947   Lan clan 50179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7712  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-er 8671  df-map 8803  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-pnf 11213  df-mnf 11214  df-xr 11215  df-ltxr 11216  df-le 11217  df-sub 11411  df-neg 11412  df-nn 12206  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12477  df-z 12564  df-dec 12684  df-uz 12835  df-fz 13508  df-struct 17164  df-slot 17199  df-ndx 17211  df-base 17227  df-hom 17291  df-cco 17292  df-cat 17681  df-cid 17682  df-func 17872  df-cofu 17874  df-nat 17960  df-fuc 17961  df-xpc 18185  df-curf 18227  df-up 49748  df-swapf 49834  df-fuco 49891  df-prcof 49948  df-lan 50181

This theorem is referenced by: (None)