Theorem proot1ex 31137

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 10645	. . . 4 |- -u 1 e. CC
2		2rp 11234	. . . . . 6 |- 2 e. RR+
3		nnrp 11238	. . . . . 6 |- ( N e. NN -> N e. RR+ )
4		rpdivcl 11251	. . . . . 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 11267	. . . 4 |- ( N e. NN -> ( 2 / N ) e. CC )
7		cxpcl 22927	. . . 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 10647	. . . . 5 |- -u 1 =/= 0
11	10	a1i 11	. . . 4 |- ( N e. NN -> -u 1 =/= 0 )
12	9, 11, 6	cxpne0d 22966	. . 3 |- ( N e. NN -> ( -u 1 ^c ( 2 / N ) ) =/= 0 )
13		eldifsn 4140	. . 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 10811	. . . . . . . . . 10 |- ( x e. NN0 -> x e. CC )
18		mulcl 9579	. . . . . . . . . 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 22965	. . . . . . . 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 2445	. . . . . . 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 22828	. . . . . . . . . 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 9579	. . . . . . . . 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 22788	. . . . . . . 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 10612	. . . . . . . . . . . . . 14 |- 2 e. CC
29	28	a1i 11	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> 2 e. CC )
30		nncn 10550	. . . . . . . . . . . . . 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 10574	. . . . . . . . . . . . . 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 10350	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( 2 / N ) x. x ) = ( ( x / N ) x. 2 ) )
36		logm1 22845	. . . . . . . . . . . . 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 6299	. . . . . . . . . . 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 10326	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( x / N ) e. CC )
40		ax-icn 9554	. . . . . . . . . . . . . 14 |- _i e. CC
41		picn 22724	. . . . . . . . . . . . . 14 |- pi e. CC
42	40, 41	mulcli 9604	. . . . . . . . . . . . 13 |- ( _i x. pi ) e. CC
43	42	a1i 11	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( _i x. pi ) e. CC )
44	39, 29, 43	mulassd 9622	. . . . . . . . . . 11 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( ( x / N ) x. 2 ) x. ( _i x. pi ) ) = ( ( x / N ) x. ( 2 x. ( _i x. pi ) ) ) )
45	40	a1i 11	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> _i e. CC )
46	41	a1i 11	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> pi e. CC )
47	29, 45, 46	mul12d 9792	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( 2 x. ( _i x. pi ) ) = ( _i x. ( 2 x. pi ) ) )
48	47	oveq2d 6297	. . . . . . . . . . 11 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( x / N ) x. ( 2 x. ( _i x. pi ) ) ) = ( ( x / N ) x. ( _i x. ( 2 x. pi ) ) ) )
49	38, 44, 48	3eqtrd 2488	. . . . . . . . . 10 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) = ( ( x / N ) x. ( _i x. ( 2 x. pi ) ) ) )
50	49	oveq1d 6296	. . . . . . . . 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 ) ) ) )
51	28, 41	mulcli 9604	. . . . . . . . . . . 12 |- ( 2 x. pi ) e. CC
52	40, 51	mulcli 9604	. . . . . . . . . . 11 |- ( _i x. ( 2 x. pi ) ) e. CC
53	52	a1i 11	. . . . . . . . . 10 |- ( ( N e. NN /\ x e. NN0 ) -> ( _i x. ( 2 x. pi ) ) e. CC )
54		ine0 9998	. . . . . . . . . . . 12 |- _i =/= 0
55		2ne0 10634	. . . . . . . . . . . . 13 |- 2 =/= 0
56		pire 22723	. . . . . . . . . . . . . 14 |- pi e. RR
57		pipos 22725	. . . . . . . . . . . . . 14 |- 0 < pi
58	56, 57	gt0ne0ii 10095	. . . . . . . . . . . . 13 |- pi =/= 0
59	28, 41, 55, 58	mulne0i 10198	. . . . . . . . . . . 12 |- ( 2 x. pi ) =/= 0
60	40, 51, 54, 59	mulne0i 10198	. . . . . . . . . . 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, 53, 61	divcan4d 10332	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( ( x / N ) x. ( _i x. ( 2 x. pi ) ) ) / ( _i x. ( 2 x. pi ) ) ) = ( x / N ) )
63	50, 62	eqtrd 2484	. . . . . . . 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 2512	. . . . . . 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 22962	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( -u 1 ^c ( ( 2 / N ) x. x ) ) = ( ( -u 1 ^c ( 2 / N ) ) ^ x ) )
69		cnfldexp 18325	. . . . . . . . 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		cnring 18314	. . . . . . . . . 10 |- ℂfld e. Ring
72		cnfldbas 18298	. . . . . . . . . . . 12 |- CC = ( Base ` ℂfld )
73		cnfld0 18316	. . . . . . . . . . . 12 |- 0 = ( 0g ` ℂfld )
74		cndrng 18321	. . . . . . . . . . . 12 |- ℂfld e. DivRing
75	72, 73, 74	drngui 17276	. . . . . . . . . . 11 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
76		eqid 2443	. . . . . . . . . . 11 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
77	75, 76	unitsubm 17193	. . . . . . . . . 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 16051	. . . . . . . . 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 1229	. . . . . . . 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 2491	. . . . . . 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 2445	. . . . . 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 10892	. . . . . . . 8 |- ( N e. NN -> N e. ZZ )
88	87	adantr 465	. . . . . . 7 |- ( ( N e. NN /\ x e. NN0 ) -> N e. ZZ )
89		nn0z 10893	. . . . . . . 8 |- ( x e. NN0 -> x e. ZZ )
90	89	adantl 466	. . . . . . 7 |- ( ( N e. NN /\ x e. NN0 ) -> x e. ZZ )
91		dvdsval2 13866	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 /\ x e. ZZ ) -> ( N || x <-> ( x / N ) e. ZZ ) )
92	88, 34, 90, 91	syl3anc 1229	. . . . . 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 2857	. . . 4 |- ( N e. NN -> A. x e. NN0 ( N || x <-> ( x (.g ` G ) ( -u 1 ^c ( 2 / N ) ) ) = 1 ) )
95	75, 81	unitgrp 17190	. . . . . 6 |- (ℂfld e. Ring -> G e. Grp )
96	71, 95	mp1i 12	. . . . 5 |- ( N e. NN -> G e. Grp )
97		nnnn0 10808	. . . . 5 |- ( N e. NN -> N e. NN0 )
98	75, 81	unitgrpbas 17189	. . . . . 6 |- ( CC \ { 0 } ) = ( Base ` G )
99		proot1ex.o	. . . . . 6 |- O = ( od ` G )
100		cnfld1 18317	. . . . . . . 8 |- 1 = ( 1r ` ℂfld )
101	75, 81, 100	unitgrpid 17192	. . . . . . 7 |- (ℂfld e. Ring -> 1 = ( 0g ` G ) )
102	71, 101	ax-mp 5	. . . . . 6 |- 1 = ( 0g ` G )
103	98, 99, 82, 102	odeq 16448	. . . . 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 1229	. . . 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 2451	. 2 |- ( N e. NN -> ( O ` ( -u 1 ^c ( 2 / N ) ) ) = N )
107	98, 99	odf 16435	. . . 4 |- O : ( CC \ { 0 } ) --> NN0
108		ffn 5721	. . . 4 |- ( O : ( CC \ { 0 } ) --> NN0 -> O Fn ( CC \ { 0 } ) )
109	107, 108	ax-mp 5	. . 3 |- O Fn ( CC \ { 0 } )
110		fniniseg 5993	. . 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 922	1 |- ( N e. NN -> ( -u 1 ^c ( 2 / N ) ) e. ( `' O " { N } ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1383   e. wcel 1804   =/= wne 2638   A.wral 2793   \ cdif 3458   {csn 4014   class class class wbr 4437   `'ccnv 4988   "cima 4992   Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496   _ici 9497   x. cmul 9500   -ucneg 9811   / cdiv 10212   NNcn 10542   2c2 10591   NN0cn0 10801   ZZcz 10870   RR+crp 11229   ^cexp 12145   expce 13675   picpi 13680   || cdvds 13863   ↾_s cress 14510   0gc0g 14714  SubMndcsubmnd 15839   Grpcgrp 15927  ._gcmg 15930   odcod 16423  mulGrpcmgp 17015   Ringcrg 17072  ℂ_fldccnfld 18294   logclog 22814   ^c ccxp 22815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-ioc 11543  df-ico 11544  df-icc 11545  df-fz 11682  df-fzo 11804  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-fac 12333  df-bc 12360  df-hash 12385  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-sum 13488  df-ef 13681  df-sin 13683  df-cos 13684  df-pi 13686  df-dvds 13864  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-hom 14598  df-cco 14599  df-rest 14697  df-topn 14698  df-0g 14716  df-gsum 14717  df-topgen 14718  df-pt 14719  df-prds 14722  df-xrs 14776  df-qtop 14781  df-imas 14782  df-xps 14784  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-grp 15931  df-minusg 15932  df-sbg 15933  df-mulg 15934  df-cntz 16229  df-od 16427  df-cmn 16674  df-mgp 17016  df-ur 17028  df-ring 17074  df-cring 17075  df-oppr 17146  df-dvdsr 17164  df-unit 17165  df-invr 17195  df-dvr 17206  df-drng 17272  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-fbas 18290  df-fg 18291  df-cnfld 18295  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cld 19393  df-ntr 19394  df-cls 19395  df-nei 19472  df-lp 19510  df-perf 19511  df-cn 19601  df-cnp 19602  df-haus 19689  df-tx 19936  df-hmeo 20129  df-fil 20220  df-fm 20312  df-flim 20313  df-flf 20314  df-xms 20696  df-ms 20697  df-tms 20698  df-cncf 21255  df-limc 22143  df-dv 22144  df-log 22816  df-cxp 22817

This theorem is referenced by: (None)