efopnlem2 - Metamath Proof Explorer

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Theorem efopnlem2 25545

Description: Lemma for efopn 25546. (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 25453	. . . . . . . 8 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
2		f1orn 6671	. . . . . . . . 9 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log ↔ (log Fn (ℂ ∖ {0}) ∧ Fun ^◡log))
3	2	simprbi 500	. . . . . . . 8 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → Fun ^◡log)
4		funcnvres 6458	. . . . . . . 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 25445	. . . . . . . . . 10 ⊢ log = ^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
7	6	cnveqi 5743	. . . . . . . . 9 ⊢ ^◡log = ^◡^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
8		relres 5880	. . . . . . . . . 10 ⊢ Rel (exp ↾ (^◡ℑ “ (-π(,]π)))
9		dfrel2 6052	. . . . . . . . . 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 2765	. . . . . . . 8 ⊢ ^◡log = (exp ↾ (^◡ℑ “ (-π(,]π)))
12	11	reseq1i 5847	. . . . . . 7 ⊢ (^◡log ↾ (log “ (ℂ ∖ (-∞(,]0)))) = ((exp ↾ (^◡ℑ “ (-π(,]π))) ↾ (log “ (ℂ ∖ (-∞(,]0))))
13		imassrn 5940	. . . . . . . . 9 ⊢ (log “ (ℂ ∖ (-∞(,]0))) ⊆ ran log
14		logrn 25447	. . . . . . . . 9 ⊢ ran log = (^◡ℑ “ (-π(,]π))
15	13, 14	sseqtri 3937	. . . . . . . 8 ⊢ (log “ (ℂ ∖ (-∞(,]0))) ⊆ (^◡ℑ “ (-π(,]π))
16		resabs1 5881	. . . . . . . 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 2769	. . . . . 6 ⊢ ^◡(log ↾ (ℂ ∖ (-∞(,]0))) = (exp ↾ (log “ (ℂ ∖ (-∞(,]0))))
19	18	imaeq1i 5926	. . . . 5 ⊢ (^◡(log ↾ (ℂ ∖ (-∞(,]0))) “ (0(ball‘(abs ∘ − ))𝑅)) = ((exp ↾ (log “ (ℂ ∖ (-∞(,]0)))) “ (0(ball‘(abs ∘ − ))𝑅))
20		cnxmet 23670	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
21		0cnd 10826	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → 0 ∈ ℂ)
22		rpxr 12595	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ ℝ⁺ → 𝑅 ∈ ℝ^*)
23	22	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → 𝑅 ∈ ℝ^*)
24		blssm 23316	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑅 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))𝑅) ⊆ ℂ)
25	20, 21, 23, 24	mp3an2i 1468	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ⊆ ℂ)
26	25	sselda 3901	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → 𝑥 ∈ ℂ)
27	26	imcld 14758	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (ℑ‘𝑥) ∈ ℝ)
28		efopnlem1 25544	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (abs‘(ℑ‘𝑥)) < π)
29		pire 25348	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
30		abslt 14878	. . . . . . . . . . . . . 14 ⊢ (((ℑ‘𝑥) ∈ ℝ ∧ π ∈ ℝ) → ((abs‘(ℑ‘𝑥)) < π ↔ (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π)))
31	27, 29, 30	sylancl 589	. . . . . . . . . . . . 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 498	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → -π < (ℑ‘𝑥))
34	32	simprd 499	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (ℑ‘𝑥) < π)
35	29	renegcli 11139	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ
36	35	rexri 10891	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ^*
37	29	rexri 10891	. . . . . . . . . . . 12 ⊢ π ∈ ℝ^*
38		elioo2 12976	. . . . . . . . . . . 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 1345	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (ℑ‘𝑥) ∈ (-π(,)π))
41		imf 14676	. . . . . . . . . . 11 ⊢ ℑ:ℂ⟶ℝ
42		ffn 6545	. . . . . . . . . . 11 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
43		elpreima 6878	. . . . . . . . . . 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 586	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → 𝑥 ∈ (^◡ℑ “ (-π(,)π)))
46	45	ex 416	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅) → 𝑥 ∈ (^◡ℑ “ (-π(,)π))))
47	46	ssrdv 3907	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ⊆ (^◡ℑ “ (-π(,)π)))
48		df-ima 5564	. . . . . . . 8 ⊢ (log “ (ℂ ∖ (-∞(,]0))) = ran (log ↾ (ℂ ∖ (-∞(,]0)))
49		eqid 2737	. . . . . . . . . 10 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
50	49	logf1o2 25538	. . . . . . . . 9 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–1-1-onto→(^◡ℑ “ (-π(,)π))
51		f1ofo 6668	. . . . . . . . 9 ⊢ ((log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–1-1-onto→(^◡ℑ “ (-π(,)π)) → (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–onto→(^◡ℑ “ (-π(,)π)))
52		forn 6636	. . . . . . . . 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 2765	. . . . . . 7 ⊢ (log “ (ℂ ∖ (-∞(,]0))) = (^◡ℑ “ (-π(,)π))
55	47, 54	sseqtrrdi 3952	. . . . . 6 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ⊆ (log “ (ℂ ∖ (-∞(,]0))))
56		resima2 5886	. . . . . 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	syl5eq 2790	. . . 4 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (^◡(log ↾ (ℂ ∖ (-∞(,]0))) “ (0(ball‘(abs ∘ − ))𝑅)) = (exp “ (0(ball‘(abs ∘ − ))𝑅)))
59	49	logcn 25535	. . . . . 6 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))) ∈ ((ℂ ∖ (-∞(,]0))–cn→ℂ)
60		difss 4046	. . . . . . 7 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ
61		ssid 3923	. . . . . . 7 ⊢ ℂ ⊆ ℂ
62		efopn.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
63		eqid 2737	. . . . . . . 8 ⊢ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) = (𝐽 ↾_t (ℂ ∖ (-∞(,]0)))
64	62	cnfldtopon 23680	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ)
65	64	toponrestid 21818	. . . . . . . 8 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
66	62, 63, 65	cncfcn 23807	. . . . . . 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 2836	. . . . 5 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))) ∈ ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) Cn 𝐽)
69	62	cnfldtopn 23679	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
70	69	blopn 23398	. . . . . 6 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑅 ∈ ℝ^*) → (0(ball‘(abs ∘ − ))𝑅) ∈ 𝐽)
71	20, 21, 23, 70	mp3an2i 1468	. . . . 5 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ∈ 𝐽)
72		cnima 22162	. . . . 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 590	. . . 4 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (^◡(log ↾ (ℂ ∖ (-∞(,]0))) “ (0(ball‘(abs ∘ − ))𝑅)) ∈ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))))
74	58, 73	eqeltrrd 2839	. . 3 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))))
75	62	cnfldtop 23681	. . . 4 ⊢ 𝐽 ∈ Top
76	49	logdmopn 25537	. . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ∈ (TopOpen‘ℂ_fld)
77	76, 62	eleqtrri 2837	. . . 4 ⊢ (ℂ ∖ (-∞(,]0)) ∈ 𝐽
78		restopn2 22074	. . . 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 221	. 2 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → ((exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽 ∧ (exp “ (0(ball‘(abs ∘ − ))𝑅)) ⊆ (ℂ ∖ (-∞(,]0))))
81	80	simpld 498	1 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∖ cdif 3863 ⊆ wss 3866 {csn 4541 class class class wbr 5053 ^◡ccnv 5550 ran crn 5552 ↾ cres 5553 “ cima 5554 ∘ ccom 5555 Rel wrel 5556 Fun wfun 6374 Fn wfn 6375 ⟶wf 6376 –onto→wfo 6378 –1-1-onto→wf1o 6379 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℝcr 10728 0cc0 10729 -∞cmnf 10865 ℝ^*cxr 10866 < clt 10867 − cmin 11062 -cneg 11063 ℝ⁺crp 12586 (,)cioo 12935 (,]cioc 12936 ℑcim 14661 abscabs 14797 expce 15623 πcpi 15628 ↾_t crest 16925 TopOpenctopn 16926 ∞Metcxmet 20348 ballcbl 20350 ℂ_fldccnfld 20363 Topctop 21790 Cn ccn 22121 –cn→ccncf 23773 logclog 25443

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-seq 13575 df-exp 13636 df-fac 13840 df-bc 13869 df-hash 13897 df-shft 14630 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-limsup 15032 df-clim 15049 df-rlim 15050 df-sum 15250 df-ef 15629 df-sin 15631 df-cos 15632 df-tan 15633 df-pi 15634 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-lp 22033 df-perf 22034 df-cn 22124 df-cnp 22125 df-haus 22212 df-cmp 22284 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-xms 23218 df-ms 23219 df-tms 23220 df-cncf 23775 df-limc 24763 df-dv 24764 df-log 25445

This theorem is referenced by: efopn 25546

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