efopnlem2 - Metamath Proof Explorer

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Theorem efopnlem2 26714

Description: Lemma for efopn 26715. (Contributed by Mario Carneiro, 2-May-2015.)

**Hypothesis**
Ref	Expression
efopn.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
efopnlem2	⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽)

Proof of Theorem efopnlem2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		logf1o 26621	. . . . . . . 8 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
2		f1orn 6859	. . . . . . . . 9 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log ↔ (log Fn (ℂ ∖ {0}) ∧ Fun ^◡log))
3	2	simprbi 496	. . . . . . . 8 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → Fun ^◡log)
4		funcnvres 6646	. . . . . . . 8 ⊢ (Fun ^◡log → ^◡(log ↾ (ℂ ∖ (-∞(,]0))) = (^◡log ↾ (log “ (ℂ ∖ (-∞(,]0)))))
5	1, 3, 4	mp2b 10	. . . . . . 7 ⊢ ^◡(log ↾ (ℂ ∖ (-∞(,]0))) = (^◡log ↾ (log “ (ℂ ∖ (-∞(,]0))))
6		df-log 26613	. . . . . . . . . 10 ⊢ log = ^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
7	6	cnveqi 5888	. . . . . . . . 9 ⊢ ^◡log = ^◡^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
8		relres 6026	. . . . . . . . . 10 ⊢ Rel (exp ↾ (^◡ℑ “ (-π(,]π)))
9		dfrel2 6211	. . . . . . . . . 10 ⊢ (Rel (exp ↾ (^◡ℑ “ (-π(,]π))) ↔ ^◡^◡(exp ↾ (^◡ℑ “ (-π(,]π))) = (exp ↾ (^◡ℑ “ (-π(,]π))))
10	8, 9	mpbi 230	. . . . . . . . 9 ⊢ ^◡^◡(exp ↾ (^◡ℑ “ (-π(,]π))) = (exp ↾ (^◡ℑ “ (-π(,]π)))
11	7, 10	eqtri 2763	. . . . . . . 8 ⊢ ^◡log = (exp ↾ (^◡ℑ “ (-π(,]π)))
12	11	reseq1i 5996	. . . . . . 7 ⊢ (^◡log ↾ (log “ (ℂ ∖ (-∞(,]0)))) = ((exp ↾ (^◡ℑ “ (-π(,]π))) ↾ (log “ (ℂ ∖ (-∞(,]0))))
13		imassrn 6091	. . . . . . . . 9 ⊢ (log “ (ℂ ∖ (-∞(,]0))) ⊆ ran log
14		logrn 26615	. . . . . . . . 9 ⊢ ran log = (^◡ℑ “ (-π(,]π))
15	13, 14	sseqtri 4032	. . . . . . . 8 ⊢ (log “ (ℂ ∖ (-∞(,]0))) ⊆ (^◡ℑ “ (-π(,]π))
16		resabs1 6027	. . . . . . . 8 ⊢ ((log “ (ℂ ∖ (-∞(,]0))) ⊆ (^◡ℑ “ (-π(,]π)) → ((exp ↾ (^◡ℑ “ (-π(,]π))) ↾ (log “ (ℂ ∖ (-∞(,]0)))) = (exp ↾ (log “ (ℂ ∖ (-∞(,]0)))))
17	15, 16	ax-mp 5	. . . . . . 7 ⊢ ((exp ↾ (^◡ℑ “ (-π(,]π))) ↾ (log “ (ℂ ∖ (-∞(,]0)))) = (exp ↾ (log “ (ℂ ∖ (-∞(,]0))))
18	5, 12, 17	3eqtri 2767	. . . . . 6 ⊢ ^◡(log ↾ (ℂ ∖ (-∞(,]0))) = (exp ↾ (log “ (ℂ ∖ (-∞(,]0))))
19	18	imaeq1i 6077	. . . . 5 ⊢ (^◡(log ↾ (ℂ ∖ (-∞(,]0))) “ (0(ball‘(abs ∘ − ))𝑅)) = ((exp ↾ (log “ (ℂ ∖ (-∞(,]0)))) “ (0(ball‘(abs ∘ − ))𝑅))
20		cnxmet 24809	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
21		0cnd 11252	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → 0 ∈ ℂ)
22		rpxr 13042	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ ℝ⁺ → 𝑅 ∈ ℝ^*)
23	22	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → 𝑅 ∈ ℝ^*)
24		blssm 24444	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑅 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))𝑅) ⊆ ℂ)
25	20, 21, 23, 24	mp3an2i 1465	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ⊆ ℂ)
26	25	sselda 3995	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → 𝑥 ∈ ℂ)
27	26	imcld 15231	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (ℑ‘𝑥) ∈ ℝ)
28		efopnlem1 26713	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (abs‘(ℑ‘𝑥)) < π)
29		pire 26515	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
30		abslt 15350	. . . . . . . . . . . . . 14 ⊢ (((ℑ‘𝑥) ∈ ℝ ∧ π ∈ ℝ) → ((abs‘(ℑ‘𝑥)) < π ↔ (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π)))
31	27, 29, 30	sylancl 586	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → ((abs‘(ℑ‘𝑥)) < π ↔ (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π)))
32	28, 31	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
33	32	simpld 494	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → -π < (ℑ‘𝑥))
34	32	simprd 495	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (ℑ‘𝑥) < π)
35	29	renegcli 11568	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ
36	35	rexri 11317	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ^*
37	29	rexri 11317	. . . . . . . . . . . 12 ⊢ π ∈ ℝ^*
38		elioo2 13425	. . . . . . . . . . . 12 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^*) → ((ℑ‘𝑥) ∈ (-π(,)π) ↔ ((ℑ‘𝑥) ∈ ℝ ∧ -π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π)))
39	36, 37, 38	mp2an 692	. . . . . . . . . . 11 ⊢ ((ℑ‘𝑥) ∈ (-π(,)π) ↔ ((ℑ‘𝑥) ∈ ℝ ∧ -π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
40	27, 33, 34, 39	syl3anbrc 1342	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (ℑ‘𝑥) ∈ (-π(,)π))
41		imf 15149	. . . . . . . . . . 11 ⊢ ℑ:ℂ⟶ℝ
42		ffn 6737	. . . . . . . . . . 11 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
43		elpreima 7078	. . . . . . . . . . 11 ⊢ (ℑ Fn ℂ → (𝑥 ∈ (^◡ℑ “ (-π(,)π)) ↔ (𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π))))
44	41, 42, 43	mp2b 10	. . . . . . . . . 10 ⊢ (𝑥 ∈ (^◡ℑ “ (-π(,)π)) ↔ (𝑥 ∈ ℂ ∧ (ℑ‘𝑥) ∈ (-π(,)π)))
45	26, 40, 44	sylanbrc 583	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → 𝑥 ∈ (^◡ℑ “ (-π(,)π)))
46	45	ex 412	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅) → 𝑥 ∈ (^◡ℑ “ (-π(,)π))))
47	46	ssrdv 4001	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ⊆ (^◡ℑ “ (-π(,)π)))
48		df-ima 5702	. . . . . . . 8 ⊢ (log “ (ℂ ∖ (-∞(,]0))) = ran (log ↾ (ℂ ∖ (-∞(,]0)))
49		eqid 2735	. . . . . . . . . 10 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
50	49	logf1o2 26707	. . . . . . . . 9 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–1-1-onto→(^◡ℑ “ (-π(,)π))
51		f1ofo 6856	. . . . . . . . 9 ⊢ ((log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–1-1-onto→(^◡ℑ “ (-π(,)π)) → (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–onto→(^◡ℑ “ (-π(,)π)))
52		forn 6824	. . . . . . . . 9 ⊢ ((log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–onto→(^◡ℑ “ (-π(,)π)) → ran (log ↾ (ℂ ∖ (-∞(,]0))) = (^◡ℑ “ (-π(,)π)))
53	50, 51, 52	mp2b 10	. . . . . . . 8 ⊢ ran (log ↾ (ℂ ∖ (-∞(,]0))) = (^◡ℑ “ (-π(,)π))
54	48, 53	eqtri 2763	. . . . . . 7 ⊢ (log “ (ℂ ∖ (-∞(,]0))) = (^◡ℑ “ (-π(,)π))
55	47, 54	sseqtrrdi 4047	. . . . . 6 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ⊆ (log “ (ℂ ∖ (-∞(,]0))))
56		resima2 6036	. . . . . 6 ⊢ ((0(ball‘(abs ∘ − ))𝑅) ⊆ (log “ (ℂ ∖ (-∞(,]0))) → ((exp ↾ (log “ (ℂ ∖ (-∞(,]0)))) “ (0(ball‘(abs ∘ − ))𝑅)) = (exp “ (0(ball‘(abs ∘ − ))𝑅)))
57	55, 56	syl 17	. . . . 5 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → ((exp ↾ (log “ (ℂ ∖ (-∞(,]0)))) “ (0(ball‘(abs ∘ − ))𝑅)) = (exp “ (0(ball‘(abs ∘ − ))𝑅)))
58	19, 57	eqtrid 2787	. . . 4 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (^◡(log ↾ (ℂ ∖ (-∞(,]0))) “ (0(ball‘(abs ∘ − ))𝑅)) = (exp “ (0(ball‘(abs ∘ − ))𝑅)))
59	49	logcn 26704	. . . . . 6 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))) ∈ ((ℂ ∖ (-∞(,]0))–cn→ℂ)
60		difss 4146	. . . . . . 7 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ
61		ssid 4018	. . . . . . 7 ⊢ ℂ ⊆ ℂ
62		efopn.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
63		eqid 2735	. . . . . . . 8 ⊢ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) = (𝐽 ↾_t (ℂ ∖ (-∞(,]0)))
64	62	cnfldtopon 24819	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ)
65	64	toponrestid 22943	. . . . . . . 8 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
66	62, 63, 65	cncfcn 24950	. . . . . . 7 ⊢ (((ℂ ∖ (-∞(,]0)) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((ℂ ∖ (-∞(,]0))–cn→ℂ) = ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) Cn 𝐽))
67	60, 61, 66	mp2an 692	. . . . . 6 ⊢ ((ℂ ∖ (-∞(,]0))–cn→ℂ) = ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) Cn 𝐽)
68	59, 67	eleqtri 2837	. . . . 5 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))) ∈ ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) Cn 𝐽)
69	62	cnfldtopn 24818	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
70	69	blopn 24529	. . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑅 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))𝑅) ∈ 𝐽)
71	20, 21, 23, 70	mp3an2i 1465	. . . . 5 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ∈ 𝐽)
72		cnima 23289	. . . . 5 ⊢ (((log ↾ (ℂ ∖ (-∞(,]0))) ∈ ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) Cn 𝐽) ∧ (0(ball‘(abs ∘ − ))𝑅) ∈ 𝐽) → (^◡(log ↾ (ℂ ∖ (-∞(,]0))) “ (0(ball‘(abs ∘ − ))𝑅)) ∈ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))))
73	68, 71, 72	sylancr 587	. . . 4 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (^◡(log ↾ (ℂ ∖ (-∞(,]0))) “ (0(ball‘(abs ∘ − ))𝑅)) ∈ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))))
74	58, 73	eqeltrrd 2840	. . 3 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))))
75	62	cnfldtop 24820	. . . 4 ⊢ 𝐽 ∈ Top
76	49	logdmopn 26706	. . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ∈ (TopOpen‘ℂ_fld)
77	76, 62	eleqtrri 2838	. . . 4 ⊢ (ℂ ∖ (-∞(,]0)) ∈ 𝐽
78		restopn2 23201	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (ℂ ∖ (-∞(,]0)) ∈ 𝐽) → ((exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ↔ ((exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽 ∧ (exp “ (0(ball‘(abs ∘ − ))𝑅)) ⊆ (ℂ ∖ (-∞(,]0)))))
79	75, 77, 78	mp2an 692	. . 3 ⊢ ((exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) ↔ ((exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽 ∧ (exp “ (0(ball‘(abs ∘ − ))𝑅)) ⊆ (ℂ ∖ (-∞(,]0))))
80	74, 79	sylib 218	. 2 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → ((exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽 ∧ (exp “ (0(ball‘(abs ∘ − ))𝑅)) ⊆ (ℂ ∖ (-∞(,]0))))
81	80	simpld 494	1 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 ⊆ wss 3963 {csn 4631 class class class wbr 5148 ^◡ccnv 5688 ran crn 5690 ↾ cres 5691 “ cima 5692 ∘ ccom 5693 Rel wrel 5694 Fun wfun 6557 Fn wfn 6558 ⟶wf 6559 –onto→wfo 6561 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 -∞cmnf 11291 ℝ^*cxr 11292 < clt 11293 − cmin 11490 -cneg 11491 ℝ⁺crp 13032 (,)cioo 13384 (,]cioc 13385 ℑcim 15134 abscabs 15270 expce 16094 πcpi 16099 ↾_t crest 17467 TopOpenctopn 17468 ∞Metcxmet 21367 ballcbl 21369 ℂ_fldccnfld 21382 Topctop 22915 Cn ccn 23248 –cn→ccncf 24916 logclog 26611

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 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 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-tan 16104 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613

This theorem is referenced by: efopn 26715

W3C validator