efopnlem2 - Metamath Proof Explorer

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Theorem efopnlem2 26853

Description: Lemma for efopn 26854. (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 26760	. . . . . . . 8 ⊢ log:(ℂ ∖ {0})–1-1-onto→ran log
2		f1orn 6835	. . . . . . . . 9 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log ↔ (log Fn (ℂ ∖ {0}) ∧ Fun ^◡log))
3	2	simprbi 503	. . . . . . . 8 ⊢ (log:(ℂ ∖ {0})–1-1-onto→ran log → Fun ^◡log)
4		funcnvres 6618	. . . . . . . 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 26752	. . . . . . . . . 10 ⊢ log = ^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
7	6	cnveqi 5862	. . . . . . . . 9 ⊢ ^◡log = ^◡^◡(exp ↾ (^◡ℑ “ (-π(,]π)))
8		relres 6006	. . . . . . . . . 10 ⊢ Rel (exp ↾ (^◡ℑ “ (-π(,]π)))
9		dfrel2 6189	. . . . . . . . . 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 2788	. . . . . . . 8 ⊢ ^◡log = (exp ↾ (^◡ℑ “ (-π(,]π)))
12	11	reseq1i 5976	. . . . . . 7 ⊢ (^◡log ↾ (log “ (ℂ ∖ (-∞(,]0)))) = ((exp ↾ (^◡ℑ “ (-π(,]π))) ↾ (log “ (ℂ ∖ (-∞(,]0))))
13		imassrn 6075	. . . . . . . . 9 ⊢ (log “ (ℂ ∖ (-∞(,]0))) ⊆ ran log
14		logrn 26754	. . . . . . . . 9 ⊢ ran log = (^◡ℑ “ (-π(,]π))
15	13, 14	sseqtri 3986	. . . . . . . 8 ⊢ (log “ (ℂ ∖ (-∞(,]0))) ⊆ (^◡ℑ “ (-π(,]π))
16		resabs1 6007	. . . . . . . 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 2792	. . . . . 6 ⊢ ^◡(log ↾ (ℂ ∖ (-∞(,]0))) = (exp ↾ (log “ (ℂ ∖ (-∞(,]0))))
19	18	imaeq1i 6061	. . . . 5 ⊢ (^◡(log ↾ (ℂ ∖ (-∞(,]0))) “ (0(ball‘(abs ∘ − ))𝑅)) = ((exp ↾ (log “ (ℂ ∖ (-∞(,]0)))) “ (0(ball‘(abs ∘ − ))𝑅))
20		cnxmet 24960	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
21		0cnd 11210	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → 0 ∈ ℂ)
22		rpxr 13038	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ ℝ⁺ → 𝑅 ∈ ℝ^*)
23	22	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → 𝑅 ∈ ℝ^*)
24		blssm 24606	. . . . . . . . . . . 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 3938	. . . . . . . . . 10 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → 𝑥 ∈ ℂ)
27	26	imcld 15266	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (ℑ‘𝑥) ∈ ℝ)
28		efopnlem1 26852	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (abs‘(ℑ‘𝑥)) < π)
29		pire 26650	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
30		abslt 15386	. . . . . . . . . . . . . 14 ⊢ (((ℑ‘𝑥) ∈ ℝ ∧ π ∈ ℝ) → ((abs‘(ℑ‘𝑥)) < π ↔ (-π < (ℑ‘𝑥) ∧ (ℑ‘𝑥) < π)))
31	27, 29, 30	sylancl 598	. . . . . . . . . . . . 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 500	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → -π < (ℑ‘𝑥))
34	32	simprd 501	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (ℑ‘𝑥) < π)
35	29	renegcli 11530	. . . . . . . . . . . . 13 ⊢ -π ∈ ℝ
36	35	rexri 11278	. . . . . . . . . . . 12 ⊢ -π ∈ ℝ^*
37	29	rexri 11278	. . . . . . . . . . . 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 705	. . . . . . . . . . 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 15184	. . . . . . . . . . 11 ⊢ ℑ:ℂ⟶ℝ
42		ffn 6709	. . . . . . . . . . 11 ⊢ (ℑ:ℂ⟶ℝ → ℑ Fn ℂ)
43		elpreima 7057	. . . . . . . . . . 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 595	. . . . . . . . 9 ⊢ (((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) ∧ 𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅)) → 𝑥 ∈ (^◡ℑ “ (-π(,)π)))
46	45	ex 418	. . . . . . . 8 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (𝑥 ∈ (0(ball‘(abs ∘ − ))𝑅) → 𝑥 ∈ (^◡ℑ “ (-π(,)π))))
47	46	ssrdv 3944	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ⊆ (^◡ℑ “ (-π(,)π)))
48		df-ima 5676	. . . . . . . 8 ⊢ (log “ (ℂ ∖ (-∞(,]0))) = ran (log ↾ (ℂ ∖ (-∞(,]0)))
49		eqid 2765	. . . . . . . . . 10 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0))
50	49	logf1o2 26846	. . . . . . . . 9 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–1-1-onto→(^◡ℑ “ (-π(,)π))
51		f1ofo 6832	. . . . . . . . 9 ⊢ ((log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–1-1-onto→(^◡ℑ “ (-π(,)π)) → (log ↾ (ℂ ∖ (-∞(,]0))):(ℂ ∖ (-∞(,]0))–onto→(^◡ℑ “ (-π(,)π)))
52		forn 6799	. . . . . . . . 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 2788	. . . . . . 7 ⊢ (log “ (ℂ ∖ (-∞(,]0))) = (^◡ℑ “ (-π(,)π))
55	47, 54	sseqtrrdi 3979	. . . . . 6 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (0(ball‘(abs ∘ − ))𝑅) ⊆ (log “ (ℂ ∖ (-∞(,]0))))
56		resima2 6017	. . . . . 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 2812	. . . 4 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (^◡(log ↾ (ℂ ∖ (-∞(,]0))) “ (0(ball‘(abs ∘ − ))𝑅)) = (exp “ (0(ball‘(abs ∘ − ))𝑅)))
59	49	logcn 26843	. . . . . 6 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))) ∈ ((ℂ ∖ (-∞(,]0))–cn→ℂ)
60		difss 4090	. . . . . . 7 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ
61		ssid 3960	. . . . . . 7 ⊢ ℂ ⊆ ℂ
62		efopn.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
63		eqid 2765	. . . . . . . 8 ⊢ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))) = (𝐽 ↾_t (ℂ ∖ (-∞(,]0)))
64	62	cnfldtopon 24970	. . . . . . . . 9 ⊢ 𝐽 ∈ (TopOn‘ℂ)
65	64	toponrestid 23108	. . . . . . . 8 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
66	62, 63, 65	cncfcn 25100	. . . . . . 7 ⊢ (((ℂ ∖ (-∞(,]0)) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((ℂ ∖ (-∞(,]0))–cn→ℂ) = ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) Cn 𝐽))
67	60, 61, 66	mp2an 705	. . . . . 6 ⊢ ((ℂ ∖ (-∞(,]0))–cn→ℂ) = ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) Cn 𝐽)
68	59, 67	eleqtri 2863	. . . . 5 ⊢ (log ↾ (ℂ ∖ (-∞(,]0))) ∈ ((𝐽 ↾_t (ℂ ∖ (-∞(,]0))) Cn 𝐽)
69	62	cnfldtopn 24969	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
70	69	blopn 24688	. . . . . 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 23452	. . . . 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 599	. . . 4 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (^◡(log ↾ (ℂ ∖ (-∞(,]0))) “ (0(ball‘(abs ∘ − ))𝑅)) ∈ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))))
74	58, 73	eqeltrrd 2866	. . 3 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ (𝐽 ↾_t (ℂ ∖ (-∞(,]0))))
75	62	cnfldtop 24971	. . . 4 ⊢ 𝐽 ∈ Top
76	49	logdmopn 26845	. . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ∈ (TopOpen‘ℂ_fld)
77	76, 62	eleqtrri 2864	. . . 4 ⊢ (ℂ ∖ (-∞(,]0)) ∈ 𝐽
78		restopn2 23364	. . . 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 705	. . 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 500	1 ⊢ ((𝑅 ∈ ℝ⁺ ∧ 𝑅 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∖ cdif 3903 ⊆ wss 3906 {csn 4591 class class class wbr 5111 ^◡ccnv 5662 ran crn 5664 ↾ cres 5665 “ cima 5666 ∘ ccom 5667 Rel wrel 5668 Fun wfun 6534 Fn wfn 6535 ⟶wf 6536 –onto→wfo 6538 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7416 ℂcc 11109 ℝcr 11110 0cc0 11111 -∞cmnf 11252 ℝ^*cxr 11253 < clt 11254 − cmin 11452 -cneg 11453 ℝ⁺crp 13028 (,)cioo 13384 (,]cioc 13385 ℑcim 15169 abscabs 15305 expce 16133 πcpi 16138 ↾_t crest 17491 TopOpenctopn 17492 ∞Metcxmet 21537 ballcbl 21539 ℂ_fldccnfld 21552 Topctop 23080 Cn ccn 23411 –cn→ccncf 25066 logclog 26750

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 7738 ax-inf2 9613 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189 ax-addf 11190

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 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9325 df-fi 9374 df-sup 9405 df-inf 9406 df-oi 9475 df-card 9937 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-z 12603 df-dec 12724 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13149 df-xadd 13150 df-xmul 13151 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13548 df-fzo 13696 df-fl 13839 df-mod 13917 df-seq 14052 df-exp 14112 df-fac 14324 df-bc 14353 df-hash 14381 df-shft 15124 df-cj 15170 df-re 15171 df-im 15172 df-sqrt 15306 df-abs 15307 df-limsup 15542 df-clim 15559 df-rlim 15560 df-sum 15758 df-ef 16139 df-sin 16141 df-cos 16142 df-tan 16143 df-pi 16144 df-struct 17225 df-sets 17242 df-slot 17260 df-ndx 17272 df-base 17288 df-ress 17309 df-plusg 17341 df-mulr 17342 df-starv 17343 df-sca 17344 df-vsca 17345 df-ip 17346 df-tset 17347 df-ple 17348 df-ds 17350 df-unif 17351 df-hom 17352 df-cco 17353 df-rest 17493 df-topn 17494 df-0g 17512 df-gsum 17513 df-topgen 17514 df-pt 17515 df-prds 17518 df-xrs 17574 df-qtop 17579 df-imas 17580 df-xps 17582 df-mre 17656 df-mrc 17657 df-acs 17659 df-mgm 18716 df-sgrp 18799 df-mnd 18815 df-submnd 18866 df-mulg 19158 df-cntz 19411 df-cmn 19876 df-psmet 21544 df-xmet 21545 df-met 21546 df-bl 21547 df-mopn 21548 df-fbas 21549 df-fg 21550 df-cnfld 21553 df-top 23081 df-topon 23098 df-topsp 23120 df-bases 23133 df-cld 23206 df-ntr 23207 df-cls 23208 df-nei 23285 df-lp 23323 df-perf 23324 df-cn 23414 df-cnp 23415 df-haus 23502 df-cmp 23574 df-tx 23750 df-hmeo 23943 df-fil 24034 df-fm 24126 df-flim 24127 df-flf 24128 df-xms 24508 df-ms 24509 df-tms 24510 df-cncf 25068 df-limc 26056 df-dv 26057 df-log 26752

This theorem is used by: efopn 26854

W3C validator