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Theorem dvcnvrelem2 26245

Description: Lemma for dvcnvre 26246. (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 24987	. . . . 5 ⊢ (topGen‘ran (,)) ∈ Top
3	1, 2	eqeltri 2856	. . . 4 ⊢ 𝑇 ∈ Top
4		dvcnvre.1	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
5		f1ofo 6825	. . . . . 6 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
6		forn 6792	. . . . . 6 ⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌)
7	4, 5, 6	3syl 19	. . . . 5 ⊢ (𝜑 → ran 𝐹 = 𝑌)
8		dvcnvre.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋–cn→ℝ))
9		cncff 25121	. . . . . 6 ⊢ (𝐹 ∈ (𝑋–cn→ℝ) → 𝐹:𝑋⟶ℝ)
10		frn 6710	. . . . . 6 ⊢ (𝐹:𝑋⟶ℝ → ran 𝐹 ⊆ ℝ)
11	8, 9, 10	3syl 19	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
12	7, 11	eqsstrrd 3966	. . . 4 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
13		imassrn 6067	. . . . 5 ⊢ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ran 𝐹
14	13, 7	sseqtrid 3973	. . . 4 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ 𝑌)
15		uniretop 24988	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
16	1	unieqi 4879	. . . . . 6 ⊢ ∪ 𝑇 = ∪ (topGen‘ran (,))
17	15, 16	eqtr4i 2786	. . . . 5 ⊢ ℝ = ∪ 𝑇
18	17	ntrss 23280	. . . 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 1495	. . 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 26244	. . . 4 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
26	1	fveq2i 6881	. . . . 5 ⊢ (int‘𝑇) = (int‘(topGen‘ran (,)))
27	26	fveq1i 6879	. . . 4 ⊢ ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
28	25, 27	eleqtrrdi 2871	. . 3 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
29	19, 28	sseldd 3932	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑇)‘𝑌))
30		f1ocnv 6830	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
31		f1of 6817	. . . . . . 7 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹:𝑌⟶𝑋)
32	4, 30, 31	3syl 19	. . . . . 6 ⊢ (𝜑 → ^◡𝐹:𝑌⟶𝑋)
33		ffun 6705	. . . . . 6 ⊢ (^◡𝐹:𝑌⟶𝑋 → Fun ^◡𝐹)
34		funcnvres 6611	. . . . . 6 ⊢ (Fun ^◡𝐹 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
35	32, 33, 34	3syl 19	. . . . 5 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
36		dvbsss 26129	. . . . . . . . . . 11 ⊢ dom (ℝ D 𝐹) ⊆ ℝ
37	20, 36	eqsstrrdi 3976	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ℝ)
38		ax-resscn 11181	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
39	37, 38	sstrdi 3943	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
40		cncfss 25127	. . . . . . . . 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 596	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→𝑋))
42		f1of1 6816	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–1-1→𝑌)
43	4, 42	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑋–1-1→𝑌)
44		f1ores 6832	. . . . . . . . . 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 596	. . . . . . . . 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 25029	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)
48	1, 47	eqtri 2783	. . . . . . . . . . . . 13 ⊢ 𝑇 = (𝐽 ↾_t ℝ)
49	48	oveq1i 7423	. . . . . . . . . . . 12 ⊢ (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ((𝐽 ↾_t ℝ) ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
50	46	cnfldtop 25009	. . . . . . . . . . . . 13 ⊢ 𝐽 ∈ Top
51	24, 37	sstrd 3941	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ ℝ)
52		reex 11215	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
53	52	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ∈ V)
54		restabs 23390	. . . . . . . . . . . . 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 1495	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐽 ↾_t ℝ) ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
56	49, 55	eqtrid 2807	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
57	37, 22	sseldd 3932	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ)
58	23	rpred 13086	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℝ)
59	57, 58	resubcld 11666	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ)
60	57, 58	readdcld 11262	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ)
61		eqid 2760	. . . . . . . . . . . . 13 ⊢ (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
62	1, 61	icccmp 25052	. . . . . . . . . . . 12 ⊢ (((𝐶 − 𝑅) ∈ ℝ ∧ (𝐶 + 𝑅) ∈ ℝ) → (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ Comp)
63	59, 60, 62	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ Comp)
64	56, 63	eqeltrrd 2861	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ Comp)
65		f1of 6817	. . . . . . . . . . . 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 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))):((𝐶 − 𝑅)[,](𝐶 + 𝑅))⟶(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
67	11, 38	sstrdi 3943	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
68	13, 67	sstrid 3942	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℂ)
69		rescncf 25125	. . . . . . . . . . . . 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 66	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→ℝ))
71		cncfcdm 25126	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 260	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐶 − 𝑅)[,](𝐶 + 𝑅))–cn→(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
74		eqid 2760	. . . . . . . . . . 11 ⊢ (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
75	46, 74	cncfcnvcn 25153	. . . . . . . . . 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 596	. . . . . . . . 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 235	. . . . . . . 8 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
78	41, 77	sseldd 3932	. . . . . . 7 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→𝑋))
79		eqid 2760	. . . . . . . . 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 25138	. . . . . . . 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 596	. . . . . . 7 ⊢ (𝜑 → ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→𝑋) = ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) Cn 𝑀))
83	78, 82	eleqtrd 2862	. . . . . 6 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) Cn 𝑀))
84	57, 23	ltsubrpd 13118	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 − 𝑅) < 𝐶)
85	59, 57, 84	ltled 11382	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 − 𝑅) ≤ 𝐶)
86	57, 23	ltaddrpd 13119	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 < (𝐶 + 𝑅))
87	57, 60, 86	ltled 11382	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ≤ (𝐶 + 𝑅))
88		elicc2 13464	. . . . . . . . . 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 596	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ↔ (𝐶 ∈ ℝ ∧ (𝐶 − 𝑅) ≤ 𝐶 ∧ 𝐶 ≤ (𝐶 + 𝑅))))
90	57, 85, 87, 89	mpbir3and 1361	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
91		ffun 6705	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶ℝ → Fun 𝐹)
92	8, 9, 91	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐹)
93		fdm 6712	. . . . . . . . . . 11 ⊢ (𝐹:𝑋⟶ℝ → dom 𝐹 = 𝑋)
94	8, 9, 93	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝑋)
95	24, 94	sseqtrrd 3968	. . . . . . . . 9 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹)
96		funfvima2 7230	. . . . . . . . 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 596	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∈ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) → (𝐹‘𝐶) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
98	90, 97	mpd 16	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
99	46	cnfldtopon 25008	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ)
100		resttopon 23386	. . . . . . . . 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 599	. . . . . . . 8 ⊢ (𝜑 → (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (TopOn‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
102		toponuni 23139	. . . . . . . 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 18	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
104	98, 103	eleqtrd 2862	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
105		eqid 2760	. . . . . . 7 ⊢ ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
106	105	cncnpi 23503	. . . . . 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 596	. . . . 5 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ (((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)))
108	35, 107	eqeltrrd 2861	. . . 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 7423	. . . . . . 7 ⊢ (𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((𝐽 ↾_t 𝑌) ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
111		ssexg 5284	. . . . . . . . 9 ⊢ ((𝑌 ⊆ ℝ ∧ ℝ ∈ V) → 𝑌 ∈ V)
112	12, 52, 111	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ V)
113		restabs 23390	. . . . . . . 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 1495	. . . . . . 7 ⊢ (𝜑 → ((𝐽 ↾_t 𝑌) ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
115	110, 114	eqtrid 2807	. . . . . 6 ⊢ (𝜑 → (𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
116	115	oveq1d 7428	. . . . 5 ⊢ (𝜑 → ((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀) = ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀))
117	116	fveq1d 6880	. . . 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 2863	. . 3 ⊢ (𝜑 → (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)))
119	12, 38	sstrdi 3943	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
120		resttopon 23386	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑌 ⊆ ℂ) → (𝐽 ↾_t 𝑌) ∈ (TopOn‘𝑌))
121	99, 119, 120	sylancr 599	. . . . . 6 ⊢ (𝜑 → (𝐽 ↾_t 𝑌) ∈ (TopOn‘𝑌))
122	109, 121	eqeltrid 2864	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘𝑌))
123		topontop 23138	. . . . 5 ⊢ (𝑁 ∈ (TopOn‘𝑌) → 𝑁 ∈ Top)
124	122, 123	syl 18	. . . 4 ⊢ (𝜑 → 𝑁 ∈ Top)
125		toponuni 23139	. . . . . 6 ⊢ (𝑁 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑁)
126	122, 125	syl 18	. . . . 5 ⊢ (𝜑 → 𝑌 = ∪ 𝑁)
127	14, 126	sseqtrd 3967	. . . 4 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ∪ 𝑁)
128	14, 12	sstrd 3941	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℝ)
129		difssd 4084	. . . . . . . . 9 ⊢ (𝜑 → (ℝ ∖ 𝑌) ⊆ ℝ)
130	128, 129	unssd 4138	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌)) ⊆ ℝ)
131		ssun1 4124	. . . . . . . . 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 23280	. . . . . . . 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 1495	. . . . . . 7 ⊢ (𝜑 → ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ⊆ ((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))))
135	134, 28	sseldd 3932	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))))
136		f1of 6817	. . . . . . . 8 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋⟶𝑌)
137	4, 136	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
138	137, 22	ffvelcdmd 7078	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝑌)
139	135, 138	elind 4146	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) ∩ 𝑌))
140		eqid 2760	. . . . . . . 8 ⊢ (𝑇 ↾_t 𝑌) = (𝑇 ↾_t 𝑌)
141	17, 140	restntr 23407	. . . . . . 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 1495	. . . . . 6 ⊢ (𝜑 → ((int‘(𝑇 ↾_t 𝑌))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) ∩ 𝑌))
143		restabs 23390	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ ℝ ∧ ℝ ∈ V) → ((𝐽 ↾_t ℝ) ↾_t 𝑌) = (𝐽 ↾_t 𝑌))
144	50, 12, 53, 143	mp3an2i 1495	. . . . . . . . 9 ⊢ (𝜑 → ((𝐽 ↾_t ℝ) ↾_t 𝑌) = (𝐽 ↾_t 𝑌))
145	48	oveq1i 7423	. . . . . . . . 9 ⊢ (𝑇 ↾_t 𝑌) = ((𝐽 ↾_t ℝ) ↾_t 𝑌)
146	144, 145, 109	3eqtr4g 2820	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ↾_t 𝑌) = 𝑁)
147	146	fveq2d 6882	. . . . . . 7 ⊢ (𝜑 → (int‘(𝑇 ↾_t 𝑌)) = (int‘𝑁))
148	147	fveq1d 6880	. . . . . 6 ⊢ (𝜑 → ((int‘(𝑇 ↾_t 𝑌))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
149	142, 148	eqtr3d 2797	. . . . 5 ⊢ (𝜑 → (((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) ∩ 𝑌) = ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
150	139, 149	eleqtrd 2862	. . . 4 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
151	126	feq2d 6686	. . . . . 6 ⊢ (𝜑 → (^◡𝐹:𝑌⟶𝑋 ↔ ^◡𝐹:∪ 𝑁⟶𝑋))
152	32, 151	mpbid 235	. . . . 5 ⊢ (𝜑 → ^◡𝐹:∪ 𝑁⟶𝑋)
153		resttopon 23386	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑋 ⊆ ℂ) → (𝐽 ↾_t 𝑋) ∈ (TopOn‘𝑋))
154	99, 39, 153	sylancr 599	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾_t 𝑋) ∈ (TopOn‘𝑋))
155	80, 154	eqeltrid 2864	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑋))
156		toponuni 23139	. . . . . 6 ⊢ (𝑀 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑀)
157		feq3 6682	. . . . . 6 ⊢ (𝑋 = ∪ 𝑀 → (^◡𝐹:∪ 𝑁⟶𝑋 ↔ ^◡𝐹:∪ 𝑁⟶∪ 𝑀))
158	155, 156, 157	3syl 19	. . . . 5 ⊢ (𝜑 → (^◡𝐹:∪ 𝑁⟶𝑋 ↔ ^◡𝐹:∪ 𝑁⟶∪ 𝑀))
159	152, 158	mpbid 235	. . . 4 ⊢ (𝜑 → ^◡𝐹:∪ 𝑁⟶∪ 𝑀)
160		eqid 2760	. . . . 5 ⊢ ∪ 𝑁 = ∪ 𝑁
161		eqid 2760	. . . . 5 ⊢ ∪ 𝑀 = ∪ 𝑀
162	160, 161	cnprest 23514	. . . 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 852	. . 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 260	. 2 ⊢ (𝜑 → ^◡𝐹 ∈ ((𝑁 CnP 𝑀)‘(𝐹‘𝐶)))
165	29, 164	jca 521	1 ⊢ (𝜑 → ((𝐹‘𝐶) ∈ ((int‘𝑇)‘𝑌) ∧ ^◡𝐹 ∈ ((𝑁 CnP 𝑀)‘(𝐹‘𝐶))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 Vcvv 3450 ∖ cdif 3896 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 ∪ cuni 4867 class class class wbr 5103 ^◡ccnv 5654 dom cdm 5655 ran crn 5656 ↾ cres 5657 “ cima 5658 Fun wfun 6527 ⟶wf 6529 –1-1→wf1 6530 –onto→wfo 6531 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7413 ℂcc 11122 ℝcr 11123 0cc0 11124 + caddc 11127 ≤ cle 11268 − cmin 11465 ℝ⁺crp 13042 (,)cioo 13398 [,]cicc 13401 ↾_t crest 17505 TopOpenctopn 17506 topGenctg 17522 ℂ_fldccnfld 21585 Topctop 23118 TopOnctopon 23135 intcnt 23242 Cn ccn 23449 CnP ccnp 23450 Compccmp 23611 –cn→ccncf 25104 D cdv 26090

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 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 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-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-fbas 21582 df-fg 21583 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-ntr 23245 df-cls 23246 df-nei 23323 df-lp 23361 df-perf 23362 df-cn 23452 df-cnp 23453 df-haus 23540 df-cmp 23612 df-tx 23788 df-hmeo 23981 df-fil 24072 df-fm 24164 df-flim 24165 df-flf 24166 df-xms 24546 df-ms 24547 df-tms 24548 df-cncf 25106 df-limc 26093 df-dv 26094

This theorem is used by: dvcnvre 26246

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