Theorem proot1ex 29569

Description: The complex field has primitive N-th roots of unity for all N. (Contributed by Stefan O'Rear, 12-Sep-2015.)

Hypotheses

Ref	Expression
proot1ex.g	|- G = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
proot1ex.o	|- O = ( od ` G )

Assertion

Ref	Expression
proot1ex	|- ( N e. NN -> ( -u 1 ^c ( 2 / N ) ) e. ( `' O " { N } ) )

Proof of Theorem proot1ex

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neg1cn 10425	. . . 4 |- -u 1 e. CC
2		2rp 10996	. . . . . 6 |- 2 e. RR+
3		nnrp 11000	. . . . . 6 |- ( N e. NN -> N e. RR+ )
4		rpdivcl 11013	. . . . . 6 |- ( ( 2 e. RR+ /\ N e. RR+ ) -> ( 2 / N ) e. RR+ )
5	2, 3, 4	sylancr 663	. . . . 5 |- ( N e. NN -> ( 2 / N ) e. RR+ )
6	5	rpcnd 11029	. . . 4 |- ( N e. NN -> ( 2 / N ) e. CC )
7		cxpcl 22119	. . . 4 |- ( ( -u 1 e. CC /\ ( 2 / N ) e. CC ) -> ( -u 1 ^c ( 2 / N ) ) e. CC )
8	1, 6, 7	sylancr 663	. . 3 |- ( N e. NN -> ( -u 1 ^c ( 2 / N ) ) e. CC )
9	1	a1i 11	. . . 4 |- ( N e. NN -> -u 1 e. CC )
10		neg1ne0 10427	. . . . 5 |- -u 1 =/= 0
11	10	a1i 11	. . . 4 |- ( N e. NN -> -u 1 =/= 0 )
12	9, 11, 6	cxpne0d 22158	. . 3 |- ( N e. NN -> ( -u 1 ^c ( 2 / N ) ) =/= 0 )
13		eldifsn 4000	. . 3 |- ( ( -u 1 ^c ( 2 / N ) ) e. ( CC \ { 0 } ) <-> ( ( -u 1 ^c ( 2 / N ) ) e. CC /\ ( -u 1 ^c ( 2 / N ) ) =/= 0 ) )
14	8, 12, 13	sylanbrc 664	. 2 |- ( N e. NN -> ( -u 1 ^c ( 2 / N ) ) e. ( CC \ { 0 } ) )
15	1	a1i 11	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> -u 1 e. CC )
16	10	a1i 11	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> -u 1 =/= 0 )
17		nn0cn 10589	. . . . . . . . . 10 |- ( x e. NN0 -> x e. CC )
18		mulcl 9366	. . . . . . . . . 10 |- ( ( ( 2 / N ) e. CC /\ x e. CC ) -> ( ( 2 / N ) x. x ) e. CC )
19	6, 17, 18	syl2an 477	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( 2 / N ) x. x ) e. CC )
20	15, 16, 19	cxpefd 22157	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( -u 1 ^c ( ( 2 / N ) x. x ) ) = ( exp ` ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) ) )
21	20	eqeq1d 2451	. . . . . . 7 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( -u 1 ^c ( ( 2 / N ) x. x ) ) = 1 <-> ( exp ` ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) ) = 1 ) )
22		logcl 22020	. . . . . . . . . 10 |- ( ( -u 1 e. CC /\ -u 1 =/= 0 ) -> ( log ` -u 1 ) e. CC )
23	1, 10, 22	mp2an 672	. . . . . . . . 9 |- ( log ` -u 1 ) e. CC
24		mulcl 9366	. . . . . . . . 9 |- ( ( ( ( 2 / N ) x. x ) e. CC /\ ( log ` -u 1 ) e. CC ) -> ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) e. CC )
25	19, 23, 24	sylancl 662	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) e. CC )
26		efeq1 21985	. . . . . . . 8 |- ( ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) e. CC -> ( ( exp ` ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) ) = 1 <-> ( ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) / ( _i x. ( 2 x. pi ) ) ) e. ZZ ) )
27	25, 26	syl 16	. . . . . . 7 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( exp ` ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) ) = 1 <-> ( ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) / ( _i x. ( 2 x. pi ) ) ) e. ZZ ) )
28		2cn 10392	. . . . . . . . . . . . . 14 |- 2 e. CC
29	28	a1i 11	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> 2 e. CC )
30		nncn 10330	. . . . . . . . . . . . . 14 |- ( N e. NN -> N e. CC )
31	30	adantr 465	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> N e. CC )
32	17	adantl 466	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> x e. CC )
33		nnne0 10354	. . . . . . . . . . . . . 14 |- ( N e. NN -> N =/= 0 )
34	33	adantr 465	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> N =/= 0 )
35	29, 31, 32, 34	div13d 10131	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( 2 / N ) x. x ) = ( ( x / N ) x. 2 ) )
36		logm1 22037	. . . . . . . . . . . . 13 |- ( log ` -u 1 ) = ( _i x. pi )
37	36	a1i 11	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( log ` -u 1 ) = ( _i x. pi ) )
38	35, 37	oveq12d 6109	. . . . . . . . . . 11 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) = ( ( ( x / N ) x. 2 ) x. ( _i x. pi ) ) )
39	32, 31, 34	divcld 10107	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( x / N ) e. CC )
40		ax-icn 9341	. . . . . . . . . . . . . 14 |- _i e. CC
41		pire 21921	. . . . . . . . . . . . . . 15 |- pi e. RR
42	41	recni 9398	. . . . . . . . . . . . . 14 |- pi e. CC
43	40, 42	mulcli 9391	. . . . . . . . . . . . 13 |- ( _i x. pi ) e. CC
44	43	a1i 11	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( _i x. pi ) e. CC )
45	39, 29, 44	mulassd 9409	. . . . . . . . . . 11 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( ( x / N ) x. 2 ) x. ( _i x. pi ) ) = ( ( x / N ) x. ( 2 x. ( _i x. pi ) ) ) )
46	40	a1i 11	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> _i e. CC )
47	42	a1i 11	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> pi e. CC )
48	29, 46, 47	mul12d 9578	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( 2 x. ( _i x. pi ) ) = ( _i x. ( 2 x. pi ) ) )
49	48	oveq2d 6107	. . . . . . . . . . 11 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( x / N ) x. ( 2 x. ( _i x. pi ) ) ) = ( ( x / N ) x. ( _i x. ( 2 x. pi ) ) ) )
50	38, 45, 49	3eqtrd 2479	. . . . . . . . . 10 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) = ( ( x / N ) x. ( _i x. ( 2 x. pi ) ) ) )
51	50	oveq1d 6106	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) / ( _i x. ( 2 x. pi ) ) ) = ( ( ( x / N ) x. ( _i x. ( 2 x. pi ) ) ) / ( _i x. ( 2 x. pi ) ) ) )
52	28, 42	mulcli 9391	. . . . . . . . . . . 12 |- ( 2 x. pi ) e. CC
53	40, 52	mulcli 9391	. . . . . . . . . . 11 |- ( _i x. ( 2 x. pi ) ) e. CC
54	53	a1i 11	. . . . . . . . . 10 |- ( ( N e. NN /\ x e. NN0 ) -> ( _i x. ( 2 x. pi ) ) e. CC )
55		ine0 9780	. . . . . . . . . . . 12 |- _i =/= 0
56		2ne0 10414	. . . . . . . . . . . . 13 |- 2 =/= 0
57		pipos 21923	. . . . . . . . . . . . . 14 |- 0 < pi
58	41, 57	gt0ne0ii 9876	. . . . . . . . . . . . 13 |- pi =/= 0
59	28, 42, 56, 58	mulne0i 9979	. . . . . . . . . . . 12 |- ( 2 x. pi ) =/= 0
60	40, 52, 55, 59	mulne0i 9979	. . . . . . . . . . 11 |- ( _i x. ( 2 x. pi ) ) =/= 0
61	60	a1i 11	. . . . . . . . . 10 |- ( ( N e. NN /\ x e. NN0 ) -> ( _i x. ( 2 x. pi ) ) =/= 0 )
62	39, 54, 61	divcan4d 10113	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( ( x / N ) x. ( _i x. ( 2 x. pi ) ) ) / ( _i x. ( 2 x. pi ) ) ) = ( x / N ) )
63	51, 62	eqtrd 2475	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) / ( _i x. ( 2 x. pi ) ) ) = ( x / N ) )
64	63	eleq1d 2509	. . . . . . 7 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) / ( _i x. ( 2 x. pi ) ) ) e. ZZ <-> ( x / N ) e. ZZ ) )
65	21, 27, 64	3bitrd 279	. . . . . 6 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( -u 1 ^c ( ( 2 / N ) x. x ) ) = 1 <-> ( x / N ) e. ZZ ) )
66	6	adantr 465	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( 2 / N ) e. CC )
67		simpr 461	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> x e. NN0 )
68	15, 66, 67	cxpmul2d 22154	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( -u 1 ^c ( ( 2 / N ) x. x ) ) = ( ( -u 1 ^c ( 2 / N ) ) ^ x ) )
69		cnfldexp 17849	. . . . . . . . 9 |- ( ( ( -u 1 ^c ( 2 / N ) ) e. CC /\ x e. NN0 ) -> ( x (.g ` (mulGrp ` ℂfld ) ) ( -u 1 ^c ( 2 / N ) ) ) = ( ( -u 1 ^c ( 2 / N ) ) ^ x ) )
70	8, 69	sylan 471	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( x (.g ` (mulGrp ` ℂfld ) ) ( -u 1 ^c ( 2 / N ) ) ) = ( ( -u 1 ^c ( 2 / N ) ) ^ x ) )
71		cnrng 17838	. . . . . . . . . 10 |- ℂfld e. Ring
72		cnfldbas 17822	. . . . . . . . . . . 12 |- CC = ( Base ` ℂfld )
73		cnfld0 17840	. . . . . . . . . . . 12 |- 0 = ( 0g ` ℂfld )
74		cndrng 17845	. . . . . . . . . . . 12 |- ℂfld e. DivRing
75	72, 73, 74	drngui 16838	. . . . . . . . . . 11 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
76		eqid 2443	. . . . . . . . . . 11 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
77	75, 76	unitsubm 16762	. . . . . . . . . 10 |- (ℂfld e. Ring -> ( CC \ { 0 } ) e. (SubMnd ` (mulGrp ` ℂfld ) ) )
78	71, 77	mp1i 12	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( CC \ { 0 } ) e. (SubMnd ` (mulGrp ` ℂfld ) ) )
79	14	adantr 465	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( -u 1 ^c ( 2 / N ) ) e. ( CC \ { 0 } ) )
80		eqid 2443	. . . . . . . . . 10 |- (.g ` (mulGrp ` ℂfld ) ) = (.g ` (mulGrp ` ℂfld ) )
81		proot1ex.g	. . . . . . . . . 10 |- G = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
82		eqid 2443	. . . . . . . . . 10 |- (.g ` G ) = (.g ` G )
83	80, 81, 82	submmulg 15662	. . . . . . . . 9 |- ( ( ( CC \ { 0 } ) e. (SubMnd ` (mulGrp ` ℂfld ) ) /\ x e. NN0 /\ ( -u 1 ^c ( 2 / N ) ) e. ( CC \ { 0 } ) ) -> ( x (.g ` (mulGrp ` ℂfld ) ) ( -u 1 ^c ( 2 / N ) ) ) = ( x (.g ` G ) ( -u 1 ^c ( 2 / N ) ) ) )
84	78, 67, 79, 83	syl3anc 1218	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( x (.g ` (mulGrp ` ℂfld ) ) ( -u 1 ^c ( 2 / N ) ) ) = ( x (.g ` G ) ( -u 1 ^c ( 2 / N ) ) ) )
85	68, 70, 84	3eqtr2rd 2482	. . . . . . 7 |- ( ( N e. NN /\ x e. NN0 ) -> ( x (.g ` G ) ( -u 1 ^c ( 2 / N ) ) ) = ( -u 1 ^c ( ( 2 / N ) x. x ) ) )
86	85	eqeq1d 2451	. . . . . 6 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( x (.g ` G ) ( -u 1 ^c ( 2 / N ) ) ) = 1 <-> ( -u 1 ^c ( ( 2 / N ) x. x ) ) = 1 ) )
87		nnz 10668	. . . . . . . 8 |- ( N e. NN -> N e. ZZ )
88	87	adantr 465	. . . . . . 7 |- ( ( N e. NN /\ x e. NN0 ) -> N e. ZZ )
89		nn0z 10669	. . . . . . . 8 |- ( x e. NN0 -> x e. ZZ )
90	89	adantl 466	. . . . . . 7 |- ( ( N e. NN /\ x e. NN0 ) -> x e. ZZ )
91		dvdsval2 13538	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 /\ x e. ZZ ) -> ( N || x <-> ( x / N ) e. ZZ ) )
92	88, 34, 90, 91	syl3anc 1218	. . . . . 6 |- ( ( N e. NN /\ x e. NN0 ) -> ( N || x <-> ( x / N ) e. ZZ ) )
93	65, 86, 92	3bitr4rd 286	. . . . 5 |- ( ( N e. NN /\ x e. NN0 ) -> ( N || x <-> ( x (.g ` G ) ( -u 1 ^c ( 2 / N ) ) ) = 1 ) )
94	93	ralrimiva 2799	. . . 4 |- ( N e. NN -> A. x e. NN0 ( N || x <-> ( x (.g ` G ) ( -u 1 ^c ( 2 / N ) ) ) = 1 ) )
95	75, 81	unitgrp 16759	. . . . . 6 |- (ℂfld e. Ring -> G e. Grp )
96	71, 95	mp1i 12	. . . . 5 |- ( N e. NN -> G e. Grp )
97		nnnn0 10586	. . . . 5 |- ( N e. NN -> N e. NN0 )
98	75, 81	unitgrpbas 16758	. . . . . 6 |- ( CC \ { 0 } ) = ( Base ` G )
99		proot1ex.o	. . . . . 6 |- O = ( od ` G )
100		cnfld1 17841	. . . . . . . 8 |- 1 = ( 1r ` ℂfld )
101	75, 81, 100	unitgrpid 16761	. . . . . . 7 |- (ℂfld e. Ring -> 1 = ( 0g ` G ) )
102	71, 101	ax-mp 5	. . . . . 6 |- 1 = ( 0g ` G )
103	98, 99, 82, 102	odeq 16053	. . . . 5 |- ( ( G e. Grp /\ ( -u 1 ^c ( 2 / N ) ) e. ( CC \ { 0 } ) /\ N e. NN0 ) -> ( N = ( O ` ( -u 1 ^c ( 2 / N ) ) ) <-> A. x e. NN0 ( N || x <-> ( x (.g ` G ) ( -u 1 ^c ( 2 / N ) ) ) = 1 ) ) )
104	96, 14, 97, 103	syl3anc 1218	. . . 4 |- ( N e. NN -> ( N = ( O ` ( -u 1 ^c ( 2 / N ) ) ) <-> A. x e. NN0 ( N || x <-> ( x (.g ` G ) ( -u 1 ^c ( 2 / N ) ) ) = 1 ) ) )
105	94, 104	mpbird 232	. . 3 |- ( N e. NN -> N = ( O ` ( -u 1 ^c ( 2 / N ) ) ) )
106	105	eqcomd 2448	. 2 |- ( N e. NN -> ( O ` ( -u 1 ^c ( 2 / N ) ) ) = N )
107	98, 99	odf 16040	. . . 4 |- O : ( CC \ { 0 } ) --> NN0
108		ffn 5559	. . . 4 |- ( O : ( CC \ { 0 } ) --> NN0 -> O Fn ( CC \ { 0 } ) )
109	107, 108	ax-mp 5	. . 3 |- O Fn ( CC \ { 0 } )
110		fniniseg 5824	. . 3 |- ( O Fn ( CC \ { 0 } ) -> ( ( -u 1 ^c ( 2 / N ) ) e. ( `' O " { N } ) <-> ( ( -u 1 ^c ( 2 / N ) ) e. ( CC \ { 0 } ) /\ ( O ` ( -u 1 ^c ( 2 / N ) ) ) = N ) ) )
111	109, 110	mp1i 12	. 2 |- ( N e. NN -> ( ( -u 1 ^c ( 2 / N ) ) e. ( `' O " { N } ) <-> ( ( -u 1 ^c ( 2 / N ) ) e. ( CC \ { 0 } ) /\ ( O ` ( -u 1 ^c ( 2 / N ) ) ) = N ) ) )
112	14, 106, 111	mpbir2and 913	1 |- ( N e. NN -> ( -u 1 ^c ( 2 / N ) ) e. ( `' O " { N } ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2606   A.wral 2715   \ cdif 3325   {csn 3877   class class class wbr 4292   `'ccnv 4839   "cima 4843   Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091   CCcc 9280   0cc0 9282   1c1 9283   _ici 9284   x. cmul 9287   -ucneg 9596   / cdiv 9993   NNcn 10322   2c2 10371   NN0cn0 10579   ZZcz 10646   RR+crp 10991   ^cexp 11865   expce 13347   picpi 13352   || cdivides 13535   ↾_s cress 14175   0gc0g 14378   Grpcgrp 15410  ._gcmg 15414  SubMndcsubmnd 15463   odcod 16028  mulGrpcmgp 16591   Ringcrg 16645  ℂ_fldccnfld 17818   logclog 22006   ^c ccxp 22007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-tpos 6745  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-pi 13358  df-dvds 13536  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-mulg 15548  df-cntz 15835  df-od 16032  df-cmn 16279  df-mgp 16592  df-ur 16604  df-rng 16647  df-cring 16648  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-dvr 16775  df-drng 16834  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342  df-log 22008  df-cxp 22009

This theorem is referenced by: (None)