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Theorem dvcnvrelem2 25899

Description: Lemma for dvcnvre 25900. (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 24625	. . . . 5 ⊢ (topGen‘ran (,)) ∈ Top
3	1, 2	eqeltri 2824	. . . 4 ⊢ 𝑇 ∈ Top
4		dvcnvre.1	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
5		f1ofo 6789	. . . . . 6 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
6		forn 6757	. . . . . 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 24762	. . . . . 6 ⊢ (𝐹 ∈ (𝑋–cn→ℝ) → 𝐹:𝑋⟶ℝ)
10		frn 6677	. . . . . 6 ⊢ (𝐹:𝑋⟶ℝ → ran 𝐹 ⊆ ℝ)
11	8, 9, 10	3syl 18	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
12	7, 11	eqsstrrd 3979	. . . 4 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
13		imassrn 6031	. . . . 5 ⊢ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ran 𝐹
14	13, 7	sseqtrid 3986	. . . 4 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ 𝑌)
15		uniretop 24626	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
16	1	unieqi 4879	. . . . . 6 ⊢ ∪ 𝑇 = ∪ (topGen‘ran (,))
17	15, 16	eqtr4i 2755	. . . . 5 ⊢ ℝ = ∪ 𝑇
18	17	ntrss 22918	. . . 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 25898	. . . 4 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
26	1	fveq2i 6843	. . . . 5 ⊢ (int‘𝑇) = (int‘(topGen‘ran (,)))
27	26	fveq1i 6841	. . . 4 ⊢ ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
28	25, 27	eleqtrrdi 2839	. . 3 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
29	19, 28	sseldd 3944	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑇)‘𝑌))
30		f1ocnv 6794	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
31		f1of 6782	. . . . . . 7 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹:𝑌⟶𝑋)
32	4, 30, 31	3syl 18	. . . . . 6 ⊢ (𝜑 → ^◡𝐹:𝑌⟶𝑋)
33		ffun 6673	. . . . . 6 ⊢ (^◡𝐹:𝑌⟶𝑋 → Fun ^◡𝐹)
34		funcnvres 6578	. . . . . 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 25779	. . . . . . . . . . 11 ⊢ dom (ℝ D 𝐹) ⊆ ℝ
37	20, 36	eqsstrrdi 3989	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ℝ)
38		ax-resscn 11101	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
39	37, 38	sstrdi 3956	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
40		cncfss 24768	. . . . . . . . 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 6781	. . . . . . . . . . 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 6796	. . . . . . . . . 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 24667	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)
48	1, 47	eqtri 2752	. . . . . . . . . . . . 13 ⊢ 𝑇 = (𝐽 ↾_t ℝ)
49	48	oveq1i 7379	. . . . . . . . . . . 12 ⊢ (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ((𝐽 ↾_t ℝ) ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
50	46	cnfldtop 24647	. . . . . . . . . . . . 13 ⊢ 𝐽 ∈ Top
51	24, 37	sstrd 3954	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ ℝ)
52		reex 11135	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
53	52	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ∈ V)
54		restabs 23028	. . . . . . . . . . . . 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 2776	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
57	37, 22	sseldd 3944	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ)
58	23	rpred 12971	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℝ)
59	57, 58	resubcld 11582	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ)
60	57, 58	readdcld 11179	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ)
61		eqid 2729	. . . . . . . . . . . . 13 ⊢ (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
62	1, 61	icccmp 24690	. . . . . . . . . . . 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 2829	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ Comp)
65		f1of 6782	. . . . . . . . . . . 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 3956	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
68	13, 67	sstrid 3955	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℂ)
69		rescncf 24766	. . . . . . . . . . . . 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 24767	. . . . . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
75	46, 74	cncfcnvcn 24795	. . . . . . . . . 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 3944	. . . . . . 7 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→𝑋))
79		eqid 2729	. . . . . . . . 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 24779	. . . . . . . 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 2830	. . . . . 6 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) Cn 𝑀))
84	57, 23	ltsubrpd 13003	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 − 𝑅) < 𝐶)
85	59, 57, 84	ltled 11298	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 − 𝑅) ≤ 𝐶)
86	57, 23	ltaddrpd 13004	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 < (𝐶 + 𝑅))
87	57, 60, 86	ltled 11298	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ≤ (𝐶 + 𝑅))
88		elicc2 13348	. . . . . . . . . 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 6673	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶ℝ → Fun 𝐹)
92	8, 9, 91	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐹)
93		fdm 6679	. . . . . . . . . . 11 ⊢ (𝐹:𝑋⟶ℝ → dom 𝐹 = 𝑋)
94	8, 9, 93	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝑋)
95	24, 94	sseqtrrd 3981	. . . . . . . . 9 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹)
96		funfvima2 7187	. . . . . . . . 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 24646	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ)
100		resttopon 23024	. . . . . . . . 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 22777	. . . . . . . 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 2830	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
105		eqid 2729	. . . . . . 7 ⊢ ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
106	105	cncnpi 23141	. . . . . 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 2829	. . . 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 7379	. . . . . . 7 ⊢ (𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((𝐽 ↾_t 𝑌) ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
111		ssexg 5273	. . . . . . . . 9 ⊢ ((𝑌 ⊆ ℝ ∧ ℝ ∈ V) → 𝑌 ∈ V)
112	12, 52, 111	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ V)
113		restabs 23028	. . . . . . . 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 2776	. . . . . 6 ⊢ (𝜑 → (𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
116	115	oveq1d 7384	. . . . 5 ⊢ (𝜑 → ((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀) = ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀))
117	116	fveq1d 6842	. . . 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 2831	. . 3 ⊢ (𝜑 → (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)))
119	12, 38	sstrdi 3956	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
120		resttopon 23024	. . . . . . 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 2832	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘𝑌))
123		topontop 22776	. . . . 5 ⊢ (𝑁 ∈ (TopOn‘𝑌) → 𝑁 ∈ Top)
124	122, 123	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ∈ Top)
125		toponuni 22777	. . . . . 6 ⊢ (𝑁 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑁)
126	122, 125	syl 17	. . . . 5 ⊢ (𝜑 → 𝑌 = ∪ 𝑁)
127	14, 126	sseqtrd 3980	. . . 4 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ∪ 𝑁)
128	14, 12	sstrd 3954	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℝ)
129		difssd 4096	. . . . . . . . 9 ⊢ (𝜑 → (ℝ ∖ 𝑌) ⊆ ℝ)
130	128, 129	unssd 4151	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌)) ⊆ ℝ)
131		ssun1 4137	. . . . . . . . 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 22918	. . . . . . . 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 3944	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))))
136		f1of 6782	. . . . . . . 8 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋⟶𝑌)
137	4, 136	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
138	137, 22	ffvelcdmd 7039	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝑌)
139	135, 138	elind 4159	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) ∩ 𝑌))
140		eqid 2729	. . . . . . . 8 ⊢ (𝑇 ↾_t 𝑌) = (𝑇 ↾_t 𝑌)
141	17, 140	restntr 23045	. . . . . . 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 23028	. . . . . . . . . 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 7379	. . . . . . . . 9 ⊢ (𝑇 ↾_t 𝑌) = ((𝐽 ↾_t ℝ) ↾_t 𝑌)
146	144, 145, 109	3eqtr4g 2789	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ↾_t 𝑌) = 𝑁)
147	146	fveq2d 6844	. . . . . . 7 ⊢ (𝜑 → (int‘(𝑇 ↾_t 𝑌)) = (int‘𝑁))
148	147	fveq1d 6842	. . . . . 6 ⊢ (𝜑 → ((int‘(𝑇 ↾_t 𝑌))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
149	142, 148	eqtr3d 2766	. . . . 5 ⊢ (𝜑 → (((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) ∩ 𝑌) = ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
150	139, 149	eleqtrd 2830	. . . 4 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
151	126	feq2d 6654	. . . . . 6 ⊢ (𝜑 → (^◡𝐹:𝑌⟶𝑋 ↔ ^◡𝐹:∪ 𝑁⟶𝑋))
152	32, 151	mpbid 232	. . . . 5 ⊢ (𝜑 → ^◡𝐹:∪ 𝑁⟶𝑋)
153		resttopon 23024	. . . . . . . 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 2832	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑋))
156		toponuni 22777	. . . . . 6 ⊢ (𝑀 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑀)
157		feq3 6650	. . . . . 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 2729	. . . . 5 ⊢ ∪ 𝑁 = ∪ 𝑁
161		eqid 2729	. . . . 5 ⊢ ∪ 𝑀 = ∪ 𝑀
162	160, 161	cnprest 23152	. . . 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 1540 ∈ wcel 2109 Vcvv 3444 ∖ cdif 3908 ∪ cun 3909 ∩ cin 3910 ⊆ wss 3911 ∪ cuni 4867 class class class wbr 5102 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 ↾ cres 5633 “ cima 5634 Fun wfun 6493 ⟶wf 6495 –1-1→wf1 6496 –onto→wfo 6497 –1-1-onto→wf1o 6498 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 + caddc 11047 ≤ cle 11185 − cmin 11381 ℝ⁺crp 12927 (,)cioo 13282 [,]cicc 13285 ↾_t crest 17359 TopOpenctopn 17360 topGenctg 17376 ℂ_fldccnfld 21240 Topctop 22756 TopOnctopon 22773 intcnt 22880 Cn ccn 23087 CnP ccnp 23088 Compccmp 23249 –cn→ccncf 24745 D cdv 25740

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-cmp 23250 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-limc 25743 df-dv 25744

This theorem is referenced by: dvcnvre 25900

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