Theorem proot1ex 29494

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 10421	. . . 4 |- -u 1 e. CC
2		2rp 10992	. . . . . 6 |- 2 e. RR+
3		nnrp 10996	. . . . . 6 |- ( N e. NN -> N e. RR+ )
4		rpdivcl 11009	. . . . . 6 |- ( ( 2 e. RR+ /\ N e. RR+ ) -> ( 2 / N ) e. RR+ )
5	2, 3, 4	sylancr 658	. . . . 5 |- ( N e. NN -> ( 2 / N ) e. RR+ )
6	5	rpcnd 11025	. . . 4 |- ( N e. NN -> ( 2 / N ) e. CC )
7		cxpcl 22078	. . . 4 |- ( ( -u 1 e. CC /\ ( 2 / N ) e. CC ) -> ( -u 1 ^c ( 2 / N ) ) e. CC )
8	1, 6, 7	sylancr 658	. . 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 10423	. . . . 5 |- -u 1 =/= 0
11	10	a1i 11	. . . 4 |- ( N e. NN -> -u 1 =/= 0 )
12	9, 11, 6	cxpne0d 22117	. . 3 |- ( N e. NN -> ( -u 1 ^c ( 2 / N ) ) =/= 0 )
13		eldifsn 3997	. . 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 659	. 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 10585	. . . . . . . . . 10 |- ( x e. NN0 -> x e. CC )
18		mulcl 9362	. . . . . . . . . 10 |- ( ( ( 2 / N ) e. CC /\ x e. CC ) -> ( ( 2 / N ) x. x ) e. CC )
19	6, 17, 18	syl2an 474	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( 2 / N ) x. x ) e. CC )
20	15, 16, 19	cxpefd 22116	. . . . . . . 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 2449	. . . . . . 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 21979	. . . . . . . . . 10 |- ( ( -u 1 e. CC /\ -u 1 =/= 0 ) -> ( log ` -u 1 ) e. CC )
23	1, 10, 22	mp2an 667	. . . . . . . . 9 |- ( log ` -u 1 ) e. CC
24		mulcl 9362	. . . . . . . . 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 657	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) e. CC )
26		efeq1 21944	. . . . . . . 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 10388	. . . . . . . . . . . . . 14 |- 2 e. CC
29	28	a1i 11	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> 2 e. CC )
30		nncn 10326	. . . . . . . . . . . . . 14 |- ( N e. NN -> N e. CC )
31	30	adantr 462	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> N e. CC )
32	17	adantl 463	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> x e. CC )
33		nnne0 10350	. . . . . . . . . . . . . 14 |- ( N e. NN -> N =/= 0 )
34	33	adantr 462	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> N =/= 0 )
35	29, 31, 32, 34	div13d 10127	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( 2 / N ) x. x ) = ( ( x / N ) x. 2 ) )
36		logm1 21996	. . . . . . . . . . . . 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 6108	. . . . . . . . . . 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 10103	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( x / N ) e. CC )
40		ax-icn 9337	. . . . . . . . . . . . . 14 |- _i e. CC
41		pire 21880	. . . . . . . . . . . . . . 15 |- pi e. RR
42	41	recni 9394	. . . . . . . . . . . . . 14 |- pi e. CC
43	40, 42	mulcli 9387	. . . . . . . . . . . . 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 9405	. . . . . . . . . . 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 9574	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( 2 x. ( _i x. pi ) ) = ( _i x. ( 2 x. pi ) ) )
49	48	oveq2d 6106	. . . . . . . . . . 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 2477	. . . . . . . . . 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 6105	. . . . . . . . 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 9387	. . . . . . . . . . . 12 |- ( 2 x. pi ) e. CC
53	40, 52	mulcli 9387	. . . . . . . . . . 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 9776	. . . . . . . . . . . 12 |- _i =/= 0
56		2ne0 10410	. . . . . . . . . . . . 13 |- 2 =/= 0
57		pipos 21882	. . . . . . . . . . . . . 14 |- 0 < pi
58	41, 57	gt0ne0ii 9872	. . . . . . . . . . . . 13 |- pi =/= 0
59	28, 42, 56, 58	mulne0i 9975	. . . . . . . . . . . 12 |- ( 2 x. pi ) =/= 0
60	40, 52, 55, 59	mulne0i 9975	. . . . . . . . . . 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 10109	. . . . . . . . 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 2473	. . . . . . . 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 2507	. . . . . . 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 462	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( 2 / N ) e. CC )
67		simpr 458	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> x e. NN0 )
68	15, 66, 67	cxpmul2d 22113	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( -u 1 ^c ( ( 2 / N ) x. x ) ) = ( ( -u 1 ^c ( 2 / N ) ) ^ x ) )
69		cnfldexp 17808	. . . . . . . . 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 468	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( x (.g ` (mulGrp ` ℂfld ) ) ( -u 1 ^c ( 2 / N ) ) ) = ( ( -u 1 ^c ( 2 / N ) ) ^ x ) )
71		cnrng 17797	. . . . . . . . . 10 |- ℂfld e. Ring
72		cnfldbas 17781	. . . . . . . . . . . 12 |- CC = ( Base ` ℂfld )
73		cnfld0 17799	. . . . . . . . . . . 12 |- 0 = ( 0g ` ℂfld )
74		cndrng 17804	. . . . . . . . . . . 12 |- ℂfld e. DivRing
75	72, 73, 74	drngui 16818	. . . . . . . . . . 11 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
76		eqid 2441	. . . . . . . . . . 11 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
77	75, 76	unitsubm 16752	. . . . . . . . . 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 462	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( -u 1 ^c ( 2 / N ) ) e. ( CC \ { 0 } ) )
80		eqid 2441	. . . . . . . . . 10 |- (.g ` (mulGrp ` ℂfld ) ) = (.g ` (mulGrp ` ℂfld ) )
81		proot1ex.g	. . . . . . . . . 10 |- G = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
82		eqid 2441	. . . . . . . . . 10 |- (.g ` G ) = (.g ` G )
83	80, 81, 82	submmulg 15655	. . . . . . . . 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 1213	. . . . . . . 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 2480	. . . . . . 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 2449	. . . . . 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 10664	. . . . . . . 8 |- ( N e. NN -> N e. ZZ )
88	87	adantr 462	. . . . . . 7 |- ( ( N e. NN /\ x e. NN0 ) -> N e. ZZ )
89		nn0z 10665	. . . . . . . 8 |- ( x e. NN0 -> x e. ZZ )
90	89	adantl 463	. . . . . . 7 |- ( ( N e. NN /\ x e. NN0 ) -> x e. ZZ )
91		dvdsval2 13534	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 /\ x e. ZZ ) -> ( N || x <-> ( x / N ) e. ZZ ) )
92	88, 34, 90, 91	syl3anc 1213	. . . . . 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 2797	. . . 4 |- ( N e. NN -> A. x e. NN0 ( N || x <-> ( x (.g ` G ) ( -u 1 ^c ( 2 / N ) ) ) = 1 ) )
95	75, 81	unitgrp 16749	. . . . . 6 |- (ℂfld e. Ring -> G e. Grp )
96	71, 95	mp1i 12	. . . . 5 |- ( N e. NN -> G e. Grp )
97		nnnn0 10582	. . . . 5 |- ( N e. NN -> N e. NN0 )
98	75, 81	unitgrpbas 16748	. . . . . 6 |- ( CC \ { 0 } ) = ( Base ` G )
99		proot1ex.o	. . . . . 6 |- O = ( od ` G )
100		cnfld1 17800	. . . . . . . 8 |- 1 = ( 1r ` ℂfld )
101	75, 81, 100	unitgrpid 16751	. . . . . . 7 |- (ℂfld e. Ring -> 1 = ( 0g ` G ) )
102	71, 101	ax-mp 5	. . . . . 6 |- 1 = ( 0g ` G )
103	98, 99, 82, 102	odeq 16046	. . . . 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 1213	. . . 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 2446	. 2 |- ( N e. NN -> ( O ` ( -u 1 ^c ( 2 / N ) ) ) = N )
107	98, 99	odf 16033	. . . 4 |- O : ( CC \ { 0 } ) --> NN0
108		ffn 5556	. . . 4 |- ( O : ( CC \ { 0 } ) --> NN0 -> O Fn ( CC \ { 0 } ) )
109	107, 108	ax-mp 5	. . 3 |- O Fn ( CC \ { 0 } )
110		fniniseg 5821	. . 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 908	1 |- ( N e. NN -> ( -u 1 ^c ( 2 / N ) ) e. ( `' O " { N } ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1364   e. wcel 1761   =/= wne 2604   A.wral 2713   \ cdif 3322   {csn 3874   class class class wbr 4289   `'ccnv 4835   "cima 4839   Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279   _ici 9280   x. cmul 9283   -ucneg 9592   / cdiv 9989   NNcn 10318   2c2 10367   NN0cn0 10575   ZZcz 10642   RR+crp 10987   ^cexp 11861   expce 13343   picpi 13348   || cdivides 13531   ↾_s cress 14171   0gc0g 14374   Grpcgrp 15406  ._gcmg 15410  SubMndcsubmnd 15459   odcod 16021  mulGrpcmgp 16581   Ringcrg 16635  ℂ_fldccnfld 17777   logclog 21965   ^c ccxp 21966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-dvds 13532  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-cntz 15828  df-od 16025  df-cmn 16272  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-drng 16814  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967  df-cxp 21968

This theorem is referenced by: (None)