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Theorem dvcnvrelem2 25979

Description: Lemma for dvcnvre 25980. (Contributed by Mario Carneiro, 19-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

**Hypotheses**
Ref	Expression
dvcnvre.f	⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℝ))
dvcnvre.d	⊢ (𝜑 → dom (ℝ D 𝐹) = 𝑋)
dvcnvre.z	⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))
dvcnvre.1	⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
dvcnvre.c	⊢ (𝜑 → 𝐶 ∈ 𝑋)
dvcnvre.r	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
dvcnvre.s	⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋)
dvcnvre.t	⊢ 𝑇 = (topGen‘ran (,))
dvcnvre.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)
dvcnvre.m	⊢ 𝑀 = (𝐽 ↾_t 𝑋)
dvcnvre.n	⊢ 𝑁 = (𝐽 ↾_t 𝑌)

**Assertion**
Ref	Expression
dvcnvrelem2	⊢ (𝜑 → ((𝐹‘𝐶) ∈ ((int‘𝑇)‘𝑌) ∧ ^◡𝐹 ∈ ((𝑁 CnP 𝑀)‘(𝐹‘𝐶))))

**Proof of Theorem dvcnvrelem2**
Step	Hyp	Ref	Expression
1		dvcnvre.t	. . . . 5 ⊢ 𝑇 = (topGen‘ran (,))
2		retop 24705	. . . . 5 ⊢ (topGen‘ran (,)) ∈ Top
3	1, 2	eqeltri 2832	. . . 4 ⊢ 𝑇 ∈ Top
4		dvcnvre.1	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
5		f1ofo 6781	. . . . . 6 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
6		forn 6749	. . . . . 6 ⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌)
7	4, 5, 6	3syl 18	. . . . 5 ⊢ (𝜑 → ran 𝐹 = 𝑌)
8		dvcnvre.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℝ))
9		cncff 24842	. . . . . 6 ⊢ (𝐹 ∈ (𝑋–cn→ℝ) → 𝐹:𝑋⟶ℝ)
10		frn 6669	. . . . . 6 ⊢ (𝐹:𝑋⟶ℝ → ran 𝐹 ⊆ ℝ)
11	8, 9, 10	3syl 18	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
12	7, 11	eqsstrrd 3969	. . . 4 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
13		imassrn 6030	. . . . 5 ⊢ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ran 𝐹
14	13, 7	sseqtrid 3976	. . . 4 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ 𝑌)
15		uniretop 24706	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
16	1	unieqi 4875	. . . . . 6 ⊢ ∪ 𝑇 = ∪ (topGen‘ran (,))
17	15, 16	eqtr4i 2762	. . . . 5 ⊢ ℝ = ∪ 𝑇
18	17	ntrss 22999	. . . 4 ⊢ ((𝑇 ∈ Top ∧ 𝑌 ⊆ ℝ ∧ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ 𝑌) → ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ ((int‘𝑇)‘𝑌))
19	3, 12, 14, 18	mp3an2i 1468	. . 3 ⊢ (𝜑 → ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ ((int‘𝑇)‘𝑌))
20		dvcnvre.d	. . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐹) = 𝑋)
21		dvcnvre.z	. . . . 5 ⊢ (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐹))
22		dvcnvre.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
23		dvcnvre.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
24		dvcnvre.s	. . . . 5 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋)
25	8, 20, 21, 4, 22, 23, 24	dvcnvrelem1 25978	. . . 4 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
26	1	fveq2i 6837	. . . . 5 ⊢ (int‘𝑇) = (int‘(topGen‘ran (,)))
27	26	fveq1i 6835	. . . 4 ⊢ ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
28	25, 27	eleqtrrdi 2847	. . 3 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
29	19, 28	sseldd 3934	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑇)‘𝑌))
30		f1ocnv 6786	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
31		f1of 6774	. . . . . . 7 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹:𝑌⟶𝑋)
32	4, 30, 31	3syl 18	. . . . . 6 ⊢ (𝜑 → ^◡𝐹:𝑌⟶𝑋)
33		ffun 6665	. . . . . 6 ⊢ (^◡𝐹:𝑌⟶𝑋 → Fun ^◡𝐹)
34		funcnvres 6570	. . . . . 6 ⊢ (Fun ^◡𝐹 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
35	32, 33, 34	3syl 18	. . . . 5 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
36		dvbsss 25859	. . . . . . . . . . 11 ⊢ dom (ℝ D 𝐹) ⊆ ℝ
37	20, 36	eqsstrrdi 3979	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ℝ)
38		ax-resscn 11083	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
39	37, 38	sstrdi 3946	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
40		cncfss 24848	. . . . . . . . 9 ⊢ ((((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋 ∧ 𝑋 ⊆ ℂ) → ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→𝑋))
41	24, 39, 40	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→𝑋))
42		f1of1 6773	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–1-1→𝑌)
43	4, 42	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑋–1-1→𝑌)
44		f1ores 6788	. . . . . . . . . 10 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋) → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))–1-1-onto→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
45	43, 24, 44	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))–1-1-onto→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
46		dvcnvre.j	. . . . . . . . . . . . . . 15 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
47	46	tgioo2 24747	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)
48	1, 47	eqtri 2759	. . . . . . . . . . . . 13 ⊢ 𝑇 = (𝐽 ↾_t ℝ)
49	48	oveq1i 7368	. . . . . . . . . . . 12 ⊢ (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ((𝐽 ↾_t ℝ) ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
50	46	cnfldtop 24727	. . . . . . . . . . . . 13 ⊢ 𝐽 ∈ Top
51	24, 37	sstrd 3944	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ ℝ)
52		reex 11117	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
53	52	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ∈ V)
54		restabs 23109	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ ℝ ∧ ℝ ∈ V) → ((𝐽 ↾_t ℝ) ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
55	50, 51, 53, 54	mp3an2i 1468	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐽 ↾_t ℝ) ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
56	49, 55	eqtrid 2783	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
57	37, 22	sseldd 3934	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ)
58	23	rpred 12949	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℝ)
59	57, 58	resubcld 11565	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ)
60	57, 58	readdcld 11161	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ)
61		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
62	1, 61	icccmp 24770	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝑅) ∈ ℝ ∧ (𝐶 + 𝑅) ∈ ℝ) → (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ Comp)
63	59, 60, 62	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ Comp)
64	56, 63	eqeltrrd 2837	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ Comp)
65		f1of 6774	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))–1-1-onto→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))⟶(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
66	45, 65	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))⟶(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
67	11, 38	sstrdi 3946	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
68	13, 67	sstrid 3945	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℂ)
69		rescncf 24846	. . . . . . . . . . . . 13 ⊢ (((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ 𝑋 → (𝐹 ∈ (𝑋–cn→ℝ) → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→ℝ)))
70	24, 8, 69	sylc 65	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→ℝ))
71		cncfcdm 24847	. . . . . . . . . . . 12 ⊢ (((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℂ ∧ (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→ℝ)) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ↔ (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))⟶(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
72	68, 70, 71	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ↔ (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))⟶(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
73	66, 72	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
74		eqid 2736	. . . . . . . . . . 11 ⊢ (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
75	46, 74	cncfcnvcn 24875	. . . . . . . . . 10 ⊢ (((𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ Comp ∧ (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))) → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))–1-1-onto→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ↔ ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
76	64, 73, 75	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))–1-1-onto→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ↔ ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
77	45, 76	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
78	41, 77	sseldd 3934	. . . . . . 7 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→𝑋))
79		eqid 2736	. . . . . . . . 9 ⊢ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
80		dvcnvre.m	. . . . . . . . 9 ⊢ 𝑀 = (𝐽 ↾_t 𝑋)
81	46, 79, 80	cncfcn 24859	. . . . . . . 8 ⊢ (((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℂ ∧ 𝑋 ⊆ ℂ) → ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→𝑋) = ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) Cn 𝑀))
82	68, 39, 81	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→𝑋) = ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) Cn 𝑀))
83	78, 82	eleqtrd 2838	. . . . . 6 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) Cn 𝑀))
84	57, 23	ltsubrpd 12981	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 − 𝑅) < 𝐶)
85	59, 57, 84	ltled 11281	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 − 𝑅) ≤ 𝐶)
86	57, 23	ltaddrpd 12982	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 < (𝐶 + 𝑅))
87	57, 60, 86	ltled 11281	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ≤ (𝐶 + 𝑅))
88		elicc2 13327	. . . . . . . . . 10 ⊢ (((𝐶 − 𝑅) ∈ ℝ ∧ (𝐶 + 𝑅) ∈ ℝ) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ↔ (𝐶 ∈ ℝ ∧ (𝐶 − 𝑅) ≤ 𝐶 ∧ 𝐶 ≤ (𝐶 + 𝑅))))
89	59, 60, 88	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ↔ (𝐶 ∈ ℝ ∧ (𝐶 − 𝑅) ≤ 𝐶 ∧ 𝐶 ≤ (𝐶 + 𝑅))))
90	57, 85, 87, 89	mpbir3and 1343	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
91		ffun 6665	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶ℝ → Fun 𝐹)
92	8, 9, 91	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐹)
93		fdm 6671	. . . . . . . . . . 11 ⊢ (𝐹:𝑋⟶ℝ → dom 𝐹 = 𝑋)
94	8, 9, 93	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝑋)
95	24, 94	sseqtrrd 3971	. . . . . . . . 9 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹)
96		funfvima2 7177	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹) → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐹‘𝐶) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
97	92, 95, 96	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐹‘𝐶) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
98	90, 97	mpd 15	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
99	46	cnfldtopon 24726	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ)
100		resttopon 23105	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℂ) → (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (TopOn‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
101	99, 68, 100	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (TopOn‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
102		toponuni 22858	. . . . . . . 8 ⊢ ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (TopOn‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
103	101, 102	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
104	98, 103	eleqtrd 2838	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
105		eqid 2736	. . . . . . 7 ⊢ ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
106	105	cncnpi 23222	. . . . . 6 ⊢ ((^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) Cn 𝑀) ∧ (𝐹‘𝐶) ∈ ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))) → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)))
107	83, 104, 106	syl2anc 584	. . . . 5 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)))
108	35, 107	eqeltrrd 2837	. . . 4 ⊢ (𝜑 → (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)))
109		dvcnvre.n	. . . . . . . 8 ⊢ 𝑁 = (𝐽 ↾_t 𝑌)
110	109	oveq1i 7368	. . . . . . 7 ⊢ (𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((𝐽 ↾_t 𝑌) ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
111		ssexg 5268	. . . . . . . . 9 ⊢ ((𝑌 ⊆ ℝ ∧ ℝ ∈ V) → 𝑌 ∈ V)
112	12, 52, 111	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ V)
113		restabs 23109	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ 𝑌 ∧ 𝑌 ∈ V) → ((𝐽 ↾_t 𝑌) ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
114	50, 14, 112, 113	mp3an2i 1468	. . . . . . 7 ⊢ (𝜑 → ((𝐽 ↾_t 𝑌) ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
115	110, 114	eqtrid 2783	. . . . . 6 ⊢ (𝜑 → (𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
116	115	oveq1d 7373	. . . . 5 ⊢ (𝜑 → ((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀) = ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀))
117	116	fveq1d 6836	. . . 4 ⊢ (𝜑 → (((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)) = (((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)))
118	108, 117	eleqtrrd 2839	. . 3 ⊢ (𝜑 → (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)))
119	12, 38	sstrdi 3946	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
120		resttopon 23105	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑌 ⊆ ℂ) → (𝐽 ↾_t 𝑌) ∈ (TopOn‘𝑌))
121	99, 119, 120	sylancr 587	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾_t 𝑌) ∈ (TopOn‘𝑌))
122	109, 121	eqeltrid 2840	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘𝑌))
123		topontop 22857	. . . . 5 ⊢ (𝑁 ∈ (TopOn‘𝑌) → 𝑁 ∈ Top)
124	122, 123	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ Top)
125		toponuni 22858	. . . . . 6 ⊢ (𝑁 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑁)
126	122, 125	syl 17	. . . . 5 ⊢ (𝜑 → 𝑌 = ∪ 𝑁)
127	14, 126	sseqtrd 3970	. . . 4 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ∪ 𝑁)
128	14, 12	sstrd 3944	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℝ)
129		difssd 4089	. . . . . . . . 9 ⊢ (𝜑 → (ℝ ∖ 𝑌) ⊆ ℝ)
130	128, 129	unssd 4144	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌)) ⊆ ℝ)
131		ssun1 4130	. . . . . . . . 9 ⊢ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))
132	131	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌)))
133	17	ntrss 22999	. . . . . . . 8 ⊢ ((𝑇 ∈ Top ∧ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌)) ⊆ ℝ ∧ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) → ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ ((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))))
134	3, 130, 132, 133	mp3an2i 1468	. . . . . . 7 ⊢ (𝜑 → ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ ((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))))
135	134, 28	sseldd 3934	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))))
136		f1of 6774	. . . . . . . 8 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋⟶𝑌)
137	4, 136	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
138	137, 22	ffvelcdmd 7030	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝑌)
139	135, 138	elind 4152	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) ∩ 𝑌))
140		eqid 2736	. . . . . . . 8 ⊢ (𝑇 ↾_t 𝑌) = (𝑇 ↾_t 𝑌)
141	17, 140	restntr 23126	. . . . . . 7 ⊢ ((𝑇 ∈ Top ∧ 𝑌 ⊆ ℝ ∧ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ 𝑌) → ((int‘(𝑇 ↾_t 𝑌))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) ∩ 𝑌))
142	3, 12, 14, 141	mp3an2i 1468	. . . . . 6 ⊢ (𝜑 → ((int‘(𝑇 ↾_t 𝑌))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) ∩ 𝑌))
143		restabs 23109	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ ℝ ∧ ℝ ∈ V) → ((𝐽 ↾_t ℝ) ↾_t 𝑌) = (𝐽 ↾_t 𝑌))
144	50, 12, 53, 143	mp3an2i 1468	. . . . . . . . 9 ⊢ (𝜑 → ((𝐽 ↾_t ℝ) ↾_t 𝑌) = (𝐽 ↾_t 𝑌))
145	48	oveq1i 7368	. . . . . . . . 9 ⊢ (𝑇 ↾_t 𝑌) = ((𝐽 ↾_t ℝ) ↾_t 𝑌)
146	144, 145, 109	3eqtr4g 2796	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ↾_t 𝑌) = 𝑁)
147	146	fveq2d 6838	. . . . . . 7 ⊢ (𝜑 → (int‘(𝑇 ↾_t 𝑌)) = (int‘𝑁))
148	147	fveq1d 6836	. . . . . 6 ⊢ (𝜑 → ((int‘(𝑇 ↾_t 𝑌))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
149	142, 148	eqtr3d 2773	. . . . 5 ⊢ (𝜑 → (((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) ∩ 𝑌) = ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
150	139, 149	eleqtrd 2838	. . . 4 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
151	126	feq2d 6646	. . . . . 6 ⊢ (𝜑 → (^◡𝐹:𝑌⟶𝑋 ↔ ^◡𝐹:∪ 𝑁⟶𝑋))
152	32, 151	mpbid 232	. . . . 5 ⊢ (𝜑 → ^◡𝐹:∪ 𝑁⟶𝑋)
153		resttopon 23105	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑋 ⊆ ℂ) → (𝐽 ↾_t 𝑋) ∈ (TopOn‘𝑋))
154	99, 39, 153	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾_t 𝑋) ∈ (TopOn‘𝑋))
155	80, 154	eqeltrid 2840	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑋))
156		toponuni 22858	. . . . . 6 ⊢ (𝑀 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑀)
157		feq3 6642	. . . . . 6 ⊢ (𝑋 = ∪ 𝑀 → (^◡𝐹:∪ 𝑁⟶𝑋 ↔ ^◡𝐹:∪ 𝑁⟶∪ 𝑀))
158	155, 156, 157	3syl 18	. . . . 5 ⊢ (𝜑 → (^◡𝐹:∪ 𝑁⟶𝑋 ↔ ^◡𝐹:∪ 𝑁⟶∪ 𝑀))
159	152, 158	mpbid 232	. . . 4 ⊢ (𝜑 → ^◡𝐹:∪ 𝑁⟶∪ 𝑀)
160		eqid 2736	. . . . 5 ⊢ ∪ 𝑁 = ∪ 𝑁
161		eqid 2736	. . . . 5 ⊢ ∪ 𝑀 = ∪ 𝑀
162	160, 161	cnprest 23233	. . . 4 ⊢ (((𝑁 ∈ Top ∧ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ∪ 𝑁) ∧ ((𝐹‘𝐶) ∈ ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∧ ^◡𝐹:∪ 𝑁⟶∪ 𝑀)) → (^◡𝐹 ∈ ((𝑁 CnP 𝑀)‘(𝐹‘𝐶)) ↔ (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶))))
163	124, 127, 150, 159, 162	syl22anc 838	. . 3 ⊢ (𝜑 → (^◡𝐹 ∈ ((𝑁 CnP 𝑀)‘(𝐹‘𝐶)) ↔ (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶))))
164	118, 163	mpbird 257	. 2 ⊢ (𝜑 → ^◡𝐹 ∈ ((𝑁 CnP 𝑀)‘(𝐹‘𝐶)))
165	29, 164	jca 511	1 ⊢ (𝜑 → ((𝐹‘𝐶) ∈ ((int‘𝑇)‘𝑌) ∧ ^◡𝐹 ∈ ((𝑁 CnP 𝑀)‘(𝐹‘𝐶))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 Vcvv 3440 ∖ cdif 3898 ∪ cun 3899 ∩ cin 3900 ⊆ wss 3901 ∪ cuni 4863 class class class wbr 5098 ^◡ccnv 5623 dom cdm 5624 ran crn 5625 ↾ cres 5626 “ cima 5627 Fun wfun 6486 ⟶wf 6488 –1-1→wf1 6489 –onto→wfo 6490 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 ℝcr 11025 0cc0 11026 + caddc 11029 ≤ cle 11167 − cmin 11364 ℝ⁺crp 12905 (,)cioo 13261 [,]cicc 13264 ↾_t crest 17340 TopOpenctopn 17341 topGenctg 17357 ℂ_fldccnfld 21309 Topctop 22837 TopOnctopon 22854 intcnt 22961 Cn ccn 23168 CnP ccnp 23169 Compccmp 23330 –cn→ccncf 24825 D cdv 25820

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824

This theorem is referenced by: dvcnvre 25980

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