efopnlem2 - Metamath Proof Explorer

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Theorem efopnlem2 25892

Description: Lemma for efopn 25893. (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 25800	. . . . . . . 8 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
2		f1orn 6763	. . . . . . . . 9 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log ↔ (log Fn (ℂ ∖ {0}) ∧ Fun ^◡log))
3	2	simprbi 497	. . . . . . . 8 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → Fun ^◡log)
4		funcnvres 6548	. . . . . . . 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 25792	. . . . . . . . . 10 ⊢ log = ^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
7	6	cnveqi 5803	. . . . . . . . 9 ⊢ ^◡log = ^◡^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
8		relres 5939	. . . . . . . . . 10 ⊢ Rel (exp ↾ (^◡ℑ “ (-π(,]π)))
9		dfrel2 6114	. . . . . . . . . 10 ⊢ (Rel (exp ↾ (^◡ℑ “ (-π(,]π))) ↔ ^◡^◡(exp ↾ (^◡ℑ “ (-π(,]π))) = (exp ↾ (^◡ℑ “ (-π(,]π))))
10	8, 9	mpbi 229	. . . . . . . . 9 ⊢ ^◡^◡(exp ↾ (^◡ℑ “ (-π(,]π))) = (exp ↾ (^◡ℑ “ (-π(,]π)))
11	7, 10	eqtri 2764	. . . . . . . 8 ⊢ ^◡log = (exp ↾ (^◡ℑ “ (-π(,]π)))
12	11	reseq1i 5906	. . . . . . 7 ⊢ (^◡log ↾ (log “ (ℂ ∖ (-∞(,]0)))) = ((exp ↾ (^◡ℑ “ (-π(,]π))) ↾ (log “ (ℂ ∖ (-∞(,]0))))
13		imassrn 5997	. . . . . . . . 9 ⊢ (log “ (ℂ ∖ (-∞(,]0))) ⊆ ran log
14		logrn 25794	. . . . . . . . 9 ⊢ ran log = (^◡ℑ “ (-π(,]π))
15	13, 14	sseqtri 3966	. . . . . . . 8 ⊢ (log “ (ℂ ∖ (-∞(,]0))) ⊆ (^◡ℑ “ (-π(,]π))
16		resabs1 5940	. . . . . . . 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 2768	. . . . . 6 ⊢ ^◡(log ↾ (ℂ ∖ (-∞(,]0))) = (exp ↾ (log “ (ℂ ∖ (-∞(,]0))))
19	18	imaeq1i 5983	. . . . 5 ⊢ (^◡(log ↾ (ℂ ∖ (-∞(,]0))) “ (0(ball‘(abs ∘ − ))𝑅)) = ((exp ↾ (log “ (ℂ ∖ (-∞(,]0)))) “ (0(ball‘(abs ∘ − ))𝑅))
20		cnxmet 24016	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
21		0cnd 11047	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → 0 ∈ ℂ)
22		rpxr 12818	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ ℝ⁺ → 𝑅 ∈ ℝ^*)
23	22	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → 𝑅 ∈ ℝ^*)
24		blssm 23651	. . . . . . . . . . . 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 3930	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → 𝑥 ∈ ℂ)
27	26	imcld 14982	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (ℑ‘𝑥) ∈ ℝ)
28		efopnlem1 25891	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (abs‘(ℑ‘𝑥)) < π)
29		pire 25695	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
30		abslt 15102	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π))
33	32	simpld 495	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → -π < (ℑ‘𝑥))
34	32	simprd 496	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (ℑ‘𝑥) < π)
35	29	renegcli 11361	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ
36	35	rexri 11112	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ^*
37	29	rexri 11112	. . . . . . . . . . . 12 ⊢ π ∈ ℝ^*
38		elioo2 13199	. . . . . . . . . . . 12 ⊢ ((-π ∈ ℝ^* ∧ π ∈ ℝ^*) → ((ℑ‘𝑥) ∈ (-π(,)π) ↔ ((ℑ‘𝑥) ∈ ℝ ∧ -π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π)))
39	36, 37, 38	mp2an 689	. . . . . . . . . . 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 14900	. . . . . . . . . . 11 ⊢ ℑ:ℂ⟶ℝ
42		ffn 6637	. . . . . . . . . . 11 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
43		elpreima 6974	. . . . . . . . . . 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 413	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅) → 𝑥 ∈ (^◡ℑ “ (-π(,)π))))
47	46	ssrdv 3936	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ⊆ (^◡ℑ “ (-π(,)π)))
48		df-ima 5620	. . . . . . . 8 ⊢ (log “ (ℂ ∖ (-∞(,]0))) = ran (log ↾ (ℂ ∖ (-∞(,]0)))
49		eqid 2736	. . . . . . . . . 10 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
50	49	logf1o2 25885	. . . . . . . . 9 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–1-1-onto→(^◡ℑ “ (-π(,)π))
51		f1ofo 6760	. . . . . . . . 9 ⊢ ((log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–1-1-onto→(^◡ℑ “ (-π(,)π)) → (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–onto→(^◡ℑ “ (-π(,)π)))
52		forn 6728	. . . . . . . . 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 2764	. . . . . . 7 ⊢ (log “ (ℂ ∖ (-∞(,]0))) = (^◡ℑ “ (-π(,)π))
55	47, 54	sseqtrrdi 3981	. . . . . 6 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ⊆ (log “ (ℂ ∖ (-∞(,]0))))
56		resima2 5945	. . . . . 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 2788	. . . 4 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (^◡(log ↾ (ℂ ∖ (-∞(,]0))) “ (0(ball‘(abs ∘ − ))𝑅)) = (exp “ (0(ball‘(abs ∘ − ))𝑅)))
59	49	logcn 25882	. . . . . 6 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))) ∈ ((ℂ ∖ (-∞(,]0))–cn→ℂ)
60		difss 4076	. . . . . . 7 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ
61		ssid 3952	. . . . . . 7 ⊢ ℂ ⊆ ℂ
62		efopn.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
63		eqid 2736	. . . . . . . 8 ⊢ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) = (𝐽 ↾_t (ℂ ∖ (-∞(,]0)))
64	62	cnfldtopon 24026	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ)
65	64	toponrestid 22150	. . . . . . . 8 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
66	62, 63, 65	cncfcn 24153	. . . . . . 7 ⊢ (((ℂ ∖ (-∞(,]0)) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((ℂ ∖ (-∞(,]0))–cn→ℂ) = ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) Cn 𝐽))
67	60, 61, 66	mp2an 689	. . . . . 6 ⊢ ((ℂ ∖ (-∞(,]0))–cn→ℂ) = ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) Cn 𝐽)
68	59, 67	eleqtri 2835	. . . . 5 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))) ∈ ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) Cn 𝐽)
69	62	cnfldtopn 24025	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
70	69	blopn 23736	. . . . . 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 22496	. . . . 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 2838	. . 3 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))))
75	62	cnfldtop 24027	. . . 4 ⊢ 𝐽 ∈ Top
76	49	logdmopn 25884	. . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ∈ (TopOpen‘ℂ_fld)
77	76, 62	eleqtrri 2836	. . . 4 ⊢ (ℂ ∖ (-∞(,]0)) ∈ 𝐽
78		restopn2 22408	. . . 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 689	. . 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 217	. 2 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → ((exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽 ∧ (exp “ (0(ball‘(abs ∘ − ))𝑅)) ⊆ (ℂ ∖ (-∞(,]0))))
81	80	simpld 495	1 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∖ cdif 3893 ⊆ wss 3896 {csn 4570 class class class wbr 5086 ^◡ccnv 5606 ran crn 5608 ↾ cres 5609 “ cima 5610 ∘ ccom 5611 Rel wrel 5612 Fun wfun 6459 Fn wfn 6460 ⟶wf 6461 –onto→wfo 6463 –1-1-onto→wf1o 6464 ‘cfv 6465 (class class class)co 7316 ℂcc 10948 ℝcr 10949 0cc0 10950 -∞cmnf 11086 ℝ^*cxr 11087 < clt 11088 − cmin 11284 -cneg 11285 ℝ⁺crp 12809 (,)cioo 13158 (,]cioc 13159 ℑcim 14885 abscabs 15021 expce 15847 πcpi 15852 ↾_t crest 17205 TopOpenctopn 17206 ∞Metcxmet 20662 ballcbl 20664 ℂ_fldccnfld 20677 Topctop 22122 Cn ccn 22455 –cn→ccncf 24119 logclog 25790

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5223 ax-sep 5237 ax-nul 5244 ax-pow 5302 ax-pr 5366 ax-un 7629 ax-inf2 9476 ax-cnex 11006 ax-resscn 11007 ax-1cn 11008 ax-icn 11009 ax-addcl 11010 ax-addrcl 11011 ax-mulcl 11012 ax-mulrcl 11013 ax-mulcom 11014 ax-addass 11015 ax-mulass 11016 ax-distr 11017 ax-i2m1 11018 ax-1ne0 11019 ax-1rid 11020 ax-rnegex 11021 ax-rrecex 11022 ax-cnre 11023 ax-pre-lttri 11024 ax-pre-lttrn 11025 ax-pre-ltadd 11026 ax-pre-mulgt0 11027 ax-pre-sup 11028 ax-addf 11029 ax-mulf 11030

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3442 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4471 df-pw 4546 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4850 df-int 4892 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5170 df-tr 5204 df-id 5506 df-eprel 5512 df-po 5520 df-so 5521 df-fr 5562 df-se 5563 df-we 5564 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-pred 6224 df-ord 6291 df-on 6292 df-lim 6293 df-suc 6294 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-f1 6470 df-fo 6471 df-f1o 6472 df-fv 6473 df-isom 6474 df-riota 7273 df-ov 7319 df-oprab 7320 df-mpo 7321 df-of 7574 df-om 7759 df-1st 7877 df-2nd 7878 df-supp 8026 df-frecs 8145 df-wrecs 8176 df-recs 8250 df-rdg 8289 df-1o 8345 df-2o 8346 df-er 8547 df-map 8666 df-pm 8667 df-ixp 8735 df-en 8783 df-dom 8784 df-sdom 8785 df-fin 8786 df-fsupp 9205 df-fi 9246 df-sup 9277 df-inf 9278 df-oi 9345 df-card 9774 df-pnf 11090 df-mnf 11091 df-xr 11092 df-ltxr 11093 df-le 11094 df-sub 11286 df-neg 11287 df-div 11712 df-nn 12053 df-2 12115 df-3 12116 df-4 12117 df-5 12118 df-6 12119 df-7 12120 df-8 12121 df-9 12122 df-n0 12313 df-z 12399 df-dec 12517 df-uz 12662 df-q 12768 df-rp 12810 df-xneg 12927 df-xadd 12928 df-xmul 12929 df-ioo 13162 df-ioc 13163 df-ico 13164 df-icc 13165 df-fz 13319 df-fzo 13462 df-fl 13591 df-mod 13669 df-seq 13801 df-exp 13862 df-fac 14067 df-bc 14096 df-hash 14124 df-shft 14854 df-cj 14886 df-re 14887 df-im 14888 df-sqrt 15022 df-abs 15023 df-limsup 15256 df-clim 15273 df-rlim 15274 df-sum 15474 df-ef 15853 df-sin 15855 df-cos 15856 df-tan 15857 df-pi 15858 df-struct 16922 df-sets 16939 df-slot 16957 df-ndx 16969 df-base 16987 df-ress 17016 df-plusg 17049 df-mulr 17050 df-starv 17051 df-sca 17052 df-vsca 17053 df-ip 17054 df-tset 17055 df-ple 17056 df-ds 17058 df-unif 17059 df-hom 17060 df-cco 17061 df-rest 17207 df-topn 17208 df-0g 17226 df-gsum 17227 df-topgen 17228 df-pt 17229 df-prds 17232 df-xrs 17287 df-qtop 17292 df-imas 17293 df-xps 17295 df-mre 17369 df-mrc 17370 df-acs 17372 df-mgm 18400 df-sgrp 18449 df-mnd 18460 df-submnd 18505 df-mulg 18774 df-cntz 18996 df-cmn 19460 df-psmet 20669 df-xmet 20670 df-met 20671 df-bl 20672 df-mopn 20673 df-fbas 20674 df-fg 20675 df-cnfld 20678 df-top 22123 df-topon 22140 df-topsp 22162 df-bases 22176 df-cld 22250 df-ntr 22251 df-cls 22252 df-nei 22329 df-lp 22367 df-perf 22368 df-cn 22458 df-cnp 22459 df-haus 22546 df-cmp 22618 df-tx 22793 df-hmeo 22986 df-fil 23077 df-fm 23169 df-flim 23170 df-flf 23171 df-xms 23553 df-ms 23554 df-tms 23555 df-cncf 24121 df-limc 25110 df-dv 25111 df-log 25792

This theorem is referenced by: efopn 25893

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