efopnlem2 - Metamath Proof Explorer

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Theorem efopnlem2 26822

Description: Lemma for efopn 26823. (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 26729	. . . . . . . 8 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
2		f1orn 6831	. . . . . . . . 9 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log ↔ (log Fn (ℂ ∖ {0}) ∧ Fun ^◡log))
3	2	simprbi 502	. . . . . . . 8 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → Fun ^◡log)
4		funcnvres 6614	. . . . . . . 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 26721	. . . . . . . . . 10 ⊢ log = ^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
7	6	cnveqi 5860	. . . . . . . . 9 ⊢ ^◡log = ^◡^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
8		relres 6004	. . . . . . . . . 10 ⊢ Rel (exp ↾ (^◡ℑ “ (-π(,]π)))
9		dfrel2 6187	. . . . . . . . . 10 ⊢ (Rel (exp ↾ (^◡ℑ “ (-π(,]π))) ↔ ^◡^◡(exp ↾ (^◡ℑ “ (-π(,]π))) = (exp ↾ (^◡ℑ “ (-π(,]π))))
10	8, 9	mpbi 233	. . . . . . . . 9 ⊢ ^◡^◡(exp ↾ (^◡ℑ “ (-π(,]π))) = (exp ↾ (^◡ℑ “ (-π(,]π)))
11	7, 10	eqtri 2786	. . . . . . . 8 ⊢ ^◡log = (exp ↾ (^◡ℑ “ (-π(,]π)))
12	11	reseq1i 5974	. . . . . . 7 ⊢ (^◡log ↾ (log “ (ℂ ∖ (-∞(,]0)))) = ((exp ↾ (^◡ℑ “ (-π(,]π))) ↾ (log “ (ℂ ∖ (-∞(,]0))))
13		imassrn 6073	. . . . . . . . 9 ⊢ (log “ (ℂ ∖ (-∞(,]0))) ⊆ ran log
14		logrn 26723	. . . . . . . . 9 ⊢ ran log = (^◡ℑ “ (-π(,]π))
15	13, 14	sseqtri 3985	. . . . . . . 8 ⊢ (log “ (ℂ ∖ (-∞(,]0))) ⊆ (^◡ℑ “ (-π(,]π))
16		resabs1 6005	. . . . . . . 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 2790	. . . . . 6 ⊢ ^◡(log ↾ (ℂ ∖ (-∞(,]0))) = (exp ↾ (log “ (ℂ ∖ (-∞(,]0))))
19	18	imaeq1i 6059	. . . . 5 ⊢ (^◡(log ↾ (ℂ ∖ (-∞(,]0))) “ (0(ball‘(abs ∘ − ))𝑅)) = ((exp ↾ (log “ (ℂ ∖ (-∞(,]0)))) “ (0(ball‘(abs ∘ − ))𝑅))
20		cnxmet 24929	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
21		0cnd 11194	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → 0 ∈ ℂ)
22		rpxr 13021	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ ℝ⁺ → 𝑅 ∈ ℝ^*)
23	22	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → 𝑅 ∈ ℝ^*)
24		blssm 24575	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑅 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))𝑅) ⊆ ℂ)
25	20, 21, 23, 24	mp3an2i 1495	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ⊆ ℂ)
26	25	sselda 3937	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → 𝑥 ∈ ℂ)
27	26	imcld 15242	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (ℑ‘𝑥) ∈ ℝ)
28		efopnlem1 26821	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (abs‘(ℑ‘𝑥)) < π)
29		pire 26619	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
30		abslt 15362	. . . . . . . . . . . . . 14 ⊢ (((ℑ‘𝑥) ∈ ℝ ∧ π ∈ ℝ) → ((abs‘(ℑ‘𝑥)) < π ↔ (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π)))
31	27, 29, 30	sylancl 597	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → ((abs‘(ℑ‘𝑥)) < π ↔ (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π)))
32	28, 31	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
33	32	simpld 499	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → -π < (ℑ‘𝑥))
34	32	simprd 500	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (ℑ‘𝑥) < π)
35	29	renegcli 11514	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ
36	35	rexri 11262	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ^*
37	29	rexri 11262	. . . . . . . . . . . 12 ⊢ π ∈ ℝ^*
38		elioo2 13408	. . . . . . . . . . . 12 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^*) → ((ℑ‘𝑥) ∈ (-π(,)π) ↔ ((ℑ‘𝑥) ∈ ℝ ∧ -π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π)))
39	36, 37, 38	mp2an 704	. . . . . . . . . . 11 ⊢ ((ℑ‘𝑥) ∈ (-π(,)π) ↔ ((ℑ‘𝑥) ∈ ℝ ∧ -π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
40	27, 33, 34, 39	syl3anbrc 1362	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (ℑ‘𝑥) ∈ (-π(,)π))
41		imf 15160	. . . . . . . . . . 11 ⊢ ℑ:ℂ⟶ℝ
42		ffn 6705	. . . . . . . . . . 11 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
43		elpreima 7053	. . . . . . . . . . 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 594	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → 𝑥 ∈ (^◡ℑ “ (-π(,)π)))
46	45	ex 417	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅) → 𝑥 ∈ (^◡ℑ “ (-π(,)π))))
47	46	ssrdv 3943	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ⊆ (^◡ℑ “ (-π(,)π)))
48		df-ima 5674	. . . . . . . 8 ⊢ (log “ (ℂ ∖ (-∞(,]0))) = ran (log ↾ (ℂ ∖ (-∞(,]0)))
49		eqid 2763	. . . . . . . . . 10 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
50	49	logf1o2 26815	. . . . . . . . 9 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–1-1-onto→(^◡ℑ “ (-π(,)π))
51		f1ofo 6828	. . . . . . . . 9 ⊢ ((log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–1-1-onto→(^◡ℑ “ (-π(,)π)) → (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–onto→(^◡ℑ “ (-π(,)π)))
52		forn 6795	. . . . . . . . 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 2786	. . . . . . 7 ⊢ (log “ (ℂ ∖ (-∞(,]0))) = (^◡ℑ “ (-π(,)π))
55	47, 54	sseqtrrdi 3978	. . . . . 6 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ⊆ (log “ (ℂ ∖ (-∞(,]0))))
56		resima2 6015	. . . . . 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 18	. . . . 5 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → ((exp ↾ (log “ (ℂ ∖ (-∞(,]0)))) “ (0(ball‘(abs ∘ − ))𝑅)) = (exp “ (0(ball‘(abs ∘ − ))𝑅)))
58	19, 57	eqtrid 2810	. . . 4 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (^◡(log ↾ (ℂ ∖ (-∞(,]0))) “ (0(ball‘(abs ∘ − ))𝑅)) = (exp “ (0(ball‘(abs ∘ − ))𝑅)))
59	49	logcn 26812	. . . . . 6 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))) ∈ ((ℂ ∖ (-∞(,]0))–cn→ℂ)
60		difss 4090	. . . . . . 7 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ
61		ssid 3959	. . . . . . 7 ⊢ ℂ ⊆ ℂ
62		efopn.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
63		eqid 2763	. . . . . . . 8 ⊢ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) = (𝐽 ↾_t (ℂ ∖ (-∞(,]0)))
64	62	cnfldtopon 24939	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ)
65	64	toponrestid 23078	. . . . . . . 8 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
66	62, 63, 65	cncfcn 25069	. . . . . . 7 ⊢ (((ℂ ∖ (-∞(,]0)) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((ℂ ∖ (-∞(,]0))–cn→ℂ) = ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) Cn 𝐽))
67	60, 61, 66	mp2an 704	. . . . . 6 ⊢ ((ℂ ∖ (-∞(,]0))–cn→ℂ) = ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) Cn 𝐽)
68	59, 67	eleqtri 2861	. . . . 5 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))) ∈ ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) Cn 𝐽)
69	62	cnfldtopn 24938	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
70	69	blopn 24657	. . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑅 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))𝑅) ∈ 𝐽)
71	20, 21, 23, 70	mp3an2i 1495	. . . . 5 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ∈ 𝐽)
72		cnima 23422	. . . . 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 598	. . . 4 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (^◡(log ↾ (ℂ ∖ (-∞(,]0))) “ (0(ball‘(abs ∘ − ))𝑅)) ∈ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))))
74	58, 73	eqeltrrd 2864	. . 3 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))))
75	62	cnfldtop 24940	. . . 4 ⊢ 𝐽 ∈ Top
76	49	logdmopn 26814	. . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ∈ (TopOpen‘ℂ_fld)
77	76, 62	eleqtrri 2862	. . . 4 ⊢ (ℂ ∖ (-∞(,]0)) ∈ 𝐽
78		restopn2 23334	. . . 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 704	. . 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 221	. 2 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → ((exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽 ∧ (exp “ (0(ball‘(abs ∘ − ))𝑅)) ⊆ (ℂ ∖ (-∞(,]0))))
81	80	simpld 499	1 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ∖ cdif 3902 ⊆ wss 3905 {csn 4589 class class class wbr 5109 ^◡ccnv 5660 ran crn 5662 ↾ cres 5663 “ cima 5664 ∘ ccom 5665 Rel wrel 5666 Fun wfun 6530 Fn wfn 6531 ⟶wf 6532 –onto→wfo 6534 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 0cc0 11095 -∞cmnf 11236 ℝ^*cxr 11237 < clt 11238 − cmin 11436 -cneg 11437 ℝ⁺crp 13011 (,)cioo 13367 (,]cioc 13368 ℑcim 15145 abscabs 15281 expce 16110 πcpi 16115 ↾_t crest 17468 TopOpenctopn 17469 ∞Metcxmet 21507 ballcbl 21509 ℂ_fldccnfld 21522 Topctop 23050 Cn ccn 23381 –cn→ccncf 25035 logclog 26719

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-fac 14306 df-bc 14335 df-hash 14363 df-shft 15100 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-limsup 15518 df-clim 15535 df-rlim 15536 df-sum 15734 df-ef 16116 df-sin 16118 df-cos 16119 df-tan 16120 df-pi 16121 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-fbas 21519 df-fg 21520 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-ntr 23177 df-cls 23178 df-nei 23255 df-lp 23293 df-perf 23294 df-cn 23384 df-cnp 23385 df-haus 23472 df-cmp 23544 df-tx 23719 df-hmeo 23912 df-fil 24003 df-fm 24095 df-flim 24096 df-flf 24097 df-xms 24477 df-ms 24478 df-tms 24479 df-cncf 25037 df-limc 26025 df-dv 26026 df-log 26721

This theorem is referenced by: efopn 26823

W3C validator