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Theorem dvcnvrelem2 26228

Description: Lemma for dvcnvre 26229. (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 24969	. . . . 5 ⊢ (topGen‘ran (,)) ∈ Top
3	1, 2	eqeltri 2861	. . . 4 ⊢ 𝑇 ∈ Top
4		dvcnvre.1	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
5		f1ofo 6832	. . . . . 6 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
6		forn 6799	. . . . . 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 25103	. . . . . 6 ⊢ (𝐹 ∈ (𝑋–cn→ℝ) → 𝐹:𝑋⟶ℝ)
10		frn 6717	. . . . . 6 ⊢ (𝐹:𝑋⟶ℝ → ran 𝐹 ⊆ ℝ)
11	8, 9, 10	3syl 19	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
12	7, 11	eqsstrrd 3973	. . . 4 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
13		imassrn 6075	. . . . 5 ⊢ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ran 𝐹
14	13, 7	sseqtrid 3980	. . . 4 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ 𝑌)
15		uniretop 24970	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
16	1	unieqi 4886	. . . . . 6 ⊢ ∪ 𝑇 = ∪ (topGen‘ran (,))
17	15, 16	eqtr4i 2791	. . . . 5 ⊢ ℝ = ∪ 𝑇
18	17	ntrss 23262	. . . 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 26227	. . . 4 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
26	1	fveq2i 6888	. . . . 5 ⊢ (int‘𝑇) = (int‘(topGen‘ran (,)))
27	26	fveq1i 6886	. . . 4 ⊢ ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((int‘(topGen‘ran (,)))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
28	25, 27	eleqtrrdi 2876	. . 3 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑇)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
29	19, 28	sseldd 3939	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑇)‘𝑌))
30		f1ocnv 6837	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
31		f1of 6824	. . . . . . 7 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹:𝑌⟶𝑋)
32	4, 30, 31	3syl 19	. . . . . 6 ⊢ (𝜑 → ^◡𝐹:𝑌⟶𝑋)
33		ffun 6712	. . . . . 6 ⊢ (^◡𝐹:𝑌⟶𝑋 → Fun ^◡𝐹)
34		funcnvres 6618	. . . . . 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 26112	. . . . . . . . . . 11 ⊢ dom (ℝ D 𝐹) ⊆ ℝ
37	20, 36	eqsstrrdi 3983	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ℝ)
38		ax-resscn 11172	. . . . . . . . . 10 ⊢ ℝ ⊆ ℂ
39	37, 38	sstrdi 3950	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
40		cncfss 25109	. . . . . . . . 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 6823	. . . . . . . . . . 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 6839	. . . . . . . . . 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 25011	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)
48	1, 47	eqtri 2788	. . . . . . . . . . . . 13 ⊢ 𝑇 = (𝐽 ↾_t ℝ)
49	48	oveq1i 7429	. . . . . . . . . . . 12 ⊢ (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = ((𝐽 ↾_t ℝ) ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
50	46	cnfldtop 24991	. . . . . . . . . . . . 13 ⊢ 𝐽 ∈ Top
51	24, 37	sstrd 3948	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ ℝ)
52		reex 11206	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
53	52	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ∈ V)
54		restabs 23372	. . . . . . . . . . . . 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 2812	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
57	37, 22	sseldd 3939	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ)
58	23	rpred 13076	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℝ)
59	57, 58	resubcld 11657	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 − 𝑅) ∈ ℝ)
60	57, 58	readdcld 11253	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 + 𝑅) ∈ ℝ)
61		eqid 2765	. . . . . . . . . . . . 13 ⊢ (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝑇 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
62	1, 61	icccmp 25034	. . . . . . . . . . . 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 2866	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ Comp)
65		f1of 6824	. . . . . . . . . . . 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 3950	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐹 ⊆ ℂ)
68	13, 67	sstrid 3949	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℂ)
69		rescncf 25107	. . . . . . . . . . . . 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 25108	. . . . . . . . . . . 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 2765	. . . . . . . . . . 11 ⊢ (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) = (𝐽 ↾_t ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))
75	46, 74	cncfcnvcn 25135	. . . . . . . . . 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 3939	. . . . . . 7 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))–cn→𝑋))
79		eqid 2765	. . . . . . . . 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 25120	. . . . . . . 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 2867	. . . . . 6 ⊢ (𝜑 → ^◡(𝐹 ↾ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∈ ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) Cn 𝑀))
84	57, 23	ltsubrpd 13108	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 − 𝑅) < 𝐶)
85	59, 57, 84	ltled 11373	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 − 𝑅) ≤ 𝐶)
86	57, 23	ltaddrpd 13109	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 < (𝐶 + 𝑅))
87	57, 60, 86	ltled 11373	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ≤ (𝐶 + 𝑅))
88		elicc2 13454	. . . . . . . . . 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 6712	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶ℝ → Fun 𝐹)
92	8, 9, 91	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐹)
93		fdm 6719	. . . . . . . . . . 11 ⊢ (𝐹:𝑋⟶ℝ → dom 𝐹 = 𝑋)
94	8, 9, 93	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝑋)
95	24, 94	sseqtrrd 3975	. . . . . . . . 9 ⊢ (𝜑 → ((𝐶 − 𝑅)[,](𝐶 + 𝑅)) ⊆ dom 𝐹)
96		funfvima2 7236	. . . . . . . . 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 24990	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ)
100		resttopon 23368	. . . . . . . . 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 23121	. . . . . . . 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 2867	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
105		eqid 2765	. . . . . . 7 ⊢ ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ∪ (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
106	105	cncnpi 23485	. . . . . 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 2866	. . . 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 7429	. . . . . . 7 ⊢ (𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((𝐽 ↾_t 𝑌) ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))))
111		ssexg 5292	. . . . . . . . 9 ⊢ ((𝑌 ⊆ ℝ ∧ ℝ ∈ V) → 𝑌 ∈ V)
112	12, 52, 111	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ V)
113		restabs 23372	. . . . . . . 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 2812	. . . . . 6 ⊢ (𝜑 → (𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = (𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
116	115	oveq1d 7434	. . . . 5 ⊢ (𝜑 → ((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀) = ((𝐽 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀))
117	116	fveq1d 6887	. . . 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 2868	. . 3 ⊢ (𝜑 → (^◡𝐹 ↾ (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) ∈ (((𝑁 ↾_t (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) CnP 𝑀)‘(𝐹‘𝐶)))
119	12, 38	sstrdi 3950	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ ℂ)
120		resttopon 23368	. . . . . . 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 2869	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘𝑌))
123		topontop 23120	. . . . 5 ⊢ (𝑁 ∈ (TopOn‘𝑌) → 𝑁 ∈ Top)
124	122, 123	syl 18	. . . 4 ⊢ (𝜑 → 𝑁 ∈ Top)
125		toponuni 23121	. . . . . 6 ⊢ (𝑁 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑁)
126	122, 125	syl 18	. . . . 5 ⊢ (𝜑 → 𝑌 = ∪ 𝑁)
127	14, 126	sseqtrd 3974	. . . 4 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ∪ 𝑁)
128	14, 12	sstrd 3948	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ⊆ ℝ)
129		difssd 4091	. . . . . . . . 9 ⊢ (𝜑 → (ℝ ∖ 𝑌) ⊆ ℝ)
130	128, 129	unssd 4145	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌)) ⊆ ℝ)
131		ssun1 4131	. . . . . . . . 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 23262	. . . . . . . 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 3939	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))))
136		f1of 6824	. . . . . . . 8 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋⟶𝑌)
137	4, 136	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
138	137, 22	ffvelcdmd 7084	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝑌)
139	135, 138	elind 4153	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) ∩ 𝑌))
140		eqid 2765	. . . . . . . 8 ⊢ (𝑇 ↾_t 𝑌) = (𝑇 ↾_t 𝑌)
141	17, 140	restntr 23389	. . . . . . 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 23372	. . . . . . . . . 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 7429	. . . . . . . . 9 ⊢ (𝑇 ↾_t 𝑌) = ((𝐽 ↾_t ℝ) ↾_t 𝑌)
146	144, 145, 109	3eqtr4g 2825	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ↾_t 𝑌) = 𝑁)
147	146	fveq2d 6889	. . . . . . 7 ⊢ (𝜑 → (int‘(𝑇 ↾_t 𝑌)) = (int‘𝑁))
148	147	fveq1d 6887	. . . . . 6 ⊢ (𝜑 → ((int‘(𝑇 ↾_t 𝑌))‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))) = ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
149	142, 148	eqtr3d 2802	. . . . 5 ⊢ (𝜑 → (((int‘𝑇)‘((𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅))) ∪ (ℝ ∖ 𝑌))) ∩ 𝑌) = ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
150	139, 149	eleqtrd 2867	. . . 4 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((int‘𝑁)‘(𝐹 “ ((𝐶 − 𝑅)[,](𝐶 + 𝑅)))))
151	126	feq2d 6693	. . . . . 6 ⊢ (𝜑 → (^◡𝐹:𝑌⟶𝑋 ↔ ^◡𝐹:∪ 𝑁⟶𝑋))
152	32, 151	mpbid 235	. . . . 5 ⊢ (𝜑 → ^◡𝐹:∪ 𝑁⟶𝑋)
153		resttopon 23368	. . . . . . . 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 2869	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑋))
156		toponuni 23121	. . . . . 6 ⊢ (𝑀 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑀)
157		feq3 6689	. . . . . 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 2765	. . . . 5 ⊢ ∪ 𝑁 = ∪ 𝑁
161		eqid 2765	. . . . 5 ⊢ ∪ 𝑀 = ∪ 𝑀
162	160, 161	cnprest 23496	. . . 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 2146 Vcvv 3457 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∪ cuni 4874 class class class wbr 5111 ^◡ccnv 5662 dom cdm 5663 ran crn 5664 ↾ cres 5665 “ cima 5666 Fun wfun 6534 ⟶wf 6536 –1-1→wf1 6537 –onto→wfo 6538 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7419 ℂcc 11113 ℝcr 11114 0cc0 11115 + caddc 11118 ≤ cle 11259 − cmin 11456 ℝ⁺crp 13032 (,)cioo 13388 [,]cicc 13391 ↾_t crest 17495 TopOpenctopn 17496 topGenctg 17512 ℂ_fldccnfld 21572 Topctop 23100 TopOnctopon 23117 intcnt 23224 Cn ccn 23431 CnP ccnp 23432 Compccmp 23593 –cn→ccncf 25086 D cdv 26073

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-addf 11194

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-z 12607 df-dec 12728 df-uz 12879 df-q 12989 df-rp 13033 df-xneg 13153 df-xadd 13154 df-xmul 13155 df-ioo 13392 df-ico 13394 df-icc 13395 df-fz 13552 df-fzo 13700 df-seq 14056 df-exp 14116 df-hash 14385 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-starv 17347 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-hom 17356 df-cco 17357 df-rest 17497 df-topn 17498 df-0g 17516 df-gsum 17517 df-topgen 17518 df-pt 17519 df-prds 17522 df-xrs 17578 df-qtop 17583 df-imas 17584 df-xps 17586 df-mre 17660 df-mrc 17661 df-acs 17663 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-submnd 18879 df-mulg 19178 df-cntz 19431 df-cmn 19896 df-psmet 21564 df-xmet 21565 df-met 21566 df-bl 21567 df-mopn 21568 df-fbas 21569 df-fg 21570 df-cnfld 21573 df-top 23101 df-topon 23118 df-topsp 23140 df-bases 23153 df-cld 23226 df-ntr 23227 df-cls 23228 df-nei 23305 df-lp 23343 df-perf 23344 df-cn 23434 df-cnp 23435 df-haus 23522 df-cmp 23594 df-tx 23770 df-hmeo 23963 df-fil 24054 df-fm 24146 df-flim 24147 df-flf 24148 df-xms 24528 df-ms 24529 df-tms 24530 df-cncf 25088 df-limc 26076 df-dv 26077

This theorem is used by: dvcnvre 26229

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