Theorem proot1ex 31402

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 10635	. . . 4 |- -u 1 e. CC
2		2rp 11226	. . . . . 6 |- 2 e. RR+
3		nnrp 11230	. . . . . 6 |- ( N e. NN -> N e. RR+ )
4		rpdivcl 11244	. . . . . 6 |- ( ( 2 e. RR+ /\ N e. RR+ ) -> ( 2 / N ) e. RR+ )
5	2, 3, 4	sylancr 661	. . . . 5 |- ( N e. NN -> ( 2 / N ) e. RR+ )
6	5	rpcnd 11261	. . . 4 |- ( N e. NN -> ( 2 / N ) e. CC )
7		cxpcl 23223	. . . 4 |- ( ( -u 1 e. CC /\ ( 2 / N ) e. CC ) -> ( -u 1 ^c ( 2 / N ) ) e. CC )
8	1, 6, 7	sylancr 661	. . 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 10637	. . . . 5 |- -u 1 =/= 0
11	10	a1i 11	. . . 4 |- ( N e. NN -> -u 1 =/= 0 )
12	9, 11, 6	cxpne0d 23262	. . 3 |- ( N e. NN -> ( -u 1 ^c ( 2 / N ) ) =/= 0 )
13		eldifsn 4141	. . 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 662	. 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 10801	. . . . . . . . . 10 |- ( x e. NN0 -> x e. CC )
18		mulcl 9565	. . . . . . . . . 10 |- ( ( ( 2 / N ) e. CC /\ x e. CC ) -> ( ( 2 / N ) x. x ) e. CC )
19	6, 17, 18	syl2an 475	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( 2 / N ) x. x ) e. CC )
20	15, 16, 19	cxpefd 23261	. . . . . . . 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 2456	. . . . . . 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 23122	. . . . . . . . . 10 |- ( ( -u 1 e. CC /\ -u 1 =/= 0 ) -> ( log ` -u 1 ) e. CC )
23	1, 10, 22	mp2an 670	. . . . . . . . 9 |- ( log ` -u 1 ) e. CC
24		mulcl 9565	. . . . . . . . 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 660	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) e. CC )
26		efeq1 23082	. . . . . . . 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 10602	. . . . . . . . . . . . . 14 |- 2 e. CC
29	28	a1i 11	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> 2 e. CC )
30		nncn 10539	. . . . . . . . . . . . . 14 |- ( N e. NN -> N e. CC )
31	30	adantr 463	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> N e. CC )
32	17	adantl 464	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> x e. CC )
33		nnne0 10564	. . . . . . . . . . . . . 14 |- ( N e. NN -> N =/= 0 )
34	33	adantr 463	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> N =/= 0 )
35	29, 31, 32, 34	div13d 10340	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( 2 / N ) x. x ) = ( ( x / N ) x. 2 ) )
36		logm1 23142	. . . . . . . . . . . . 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 6288	. . . . . . . . . . 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 10316	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( x / N ) e. CC )
40		ax-icn 9540	. . . . . . . . . . . . . 14 |- _i e. CC
41		picn 23018	. . . . . . . . . . . . . 14 |- pi e. CC
42	40, 41	mulcli 9590	. . . . . . . . . . . . 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 9608	. . . . . . . . . . 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 9778	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( 2 x. ( _i x. pi ) ) = ( _i x. ( 2 x. pi ) ) )
48	47	oveq2d 6286	. . . . . . . . . . 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 2499	. . . . . . . . . 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 6285	. . . . . . . . 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 9590	. . . . . . . . . . . 12 |- ( 2 x. pi ) e. CC
52	40, 51	mulcli 9590	. . . . . . . . . . 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 9988	. . . . . . . . . . . 12 |- _i =/= 0
55		2ne0 10624	. . . . . . . . . . . . 13 |- 2 =/= 0
56		pire 23017	. . . . . . . . . . . . . 14 |- pi e. RR
57		pipos 23019	. . . . . . . . . . . . . 14 |- 0 < pi
58	56, 57	gt0ne0ii 10085	. . . . . . . . . . . . 13 |- pi =/= 0
59	28, 41, 55, 58	mulne0i 10188	. . . . . . . . . . . 12 |- ( 2 x. pi ) =/= 0
60	40, 51, 54, 59	mulne0i 10188	. . . . . . . . . . 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 10322	. . . . . . . . 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 2495	. . . . . . . 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 2523	. . . . . . 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 463	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( 2 / N ) e. CC )
67		simpr 459	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> x e. NN0 )
68	15, 66, 67	cxpmul2d 23258	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( -u 1 ^c ( ( 2 / N ) x. x ) ) = ( ( -u 1 ^c ( 2 / N ) ) ^ x ) )
69		cnfldexp 18646	. . . . . . . . 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 469	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( x (.g ` (mulGrp ` ℂfld ) ) ( -u 1 ^c ( 2 / N ) ) ) = ( ( -u 1 ^c ( 2 / N ) ) ^ x ) )
71		cnring 18635	. . . . . . . . . 10 |- ℂfld e. Ring
72		cnfldbas 18619	. . . . . . . . . . . 12 |- CC = ( Base ` ℂfld )
73		cnfld0 18637	. . . . . . . . . . . 12 |- 0 = ( 0g ` ℂfld )
74		cndrng 18642	. . . . . . . . . . . 12 |- ℂfld e. DivRing
75	72, 73, 74	drngui 17597	. . . . . . . . . . 11 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
76		eqid 2454	. . . . . . . . . . 11 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
77	75, 76	unitsubm 17514	. . . . . . . . . 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 463	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( -u 1 ^c ( 2 / N ) ) e. ( CC \ { 0 } ) )
80		eqid 2454	. . . . . . . . . 10 |- (.g ` (mulGrp ` ℂfld ) ) = (.g ` (mulGrp ` ℂfld ) )
81		proot1ex.g	. . . . . . . . . 10 |- G = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
82		eqid 2454	. . . . . . . . . 10 |- (.g ` G ) = (.g ` G )
83	80, 81, 82	submmulg 16376	. . . . . . . . 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 1226	. . . . . . . 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 2502	. . . . . . 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 2456	. . . . . 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 10882	. . . . . . . 8 |- ( N e. NN -> N e. ZZ )
88	87	adantr 463	. . . . . . 7 |- ( ( N e. NN /\ x e. NN0 ) -> N e. ZZ )
89		nn0z 10883	. . . . . . . 8 |- ( x e. NN0 -> x e. ZZ )
90	89	adantl 464	. . . . . . 7 |- ( ( N e. NN /\ x e. NN0 ) -> x e. ZZ )
91		dvdsval2 14073	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 /\ x e. ZZ ) -> ( N || x <-> ( x / N ) e. ZZ ) )
92	88, 34, 90, 91	syl3anc 1226	. . . . . 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 2868	. . . 4 |- ( N e. NN -> A. x e. NN0 ( N || x <-> ( x (.g ` G ) ( -u 1 ^c ( 2 / N ) ) ) = 1 ) )
95	75, 81	unitgrp 17511	. . . . . 6 |- (ℂfld e. Ring -> G e. Grp )
96	71, 95	mp1i 12	. . . . 5 |- ( N e. NN -> G e. Grp )
97		nnnn0 10798	. . . . 5 |- ( N e. NN -> N e. NN0 )
98	75, 81	unitgrpbas 17510	. . . . . 6 |- ( CC \ { 0 } ) = ( Base ` G )
99		proot1ex.o	. . . . . 6 |- O = ( od ` G )
100		cnfld1 18638	. . . . . . . 8 |- 1 = ( 1r ` ℂfld )
101	75, 81, 100	unitgrpid 17513	. . . . . . 7 |- (ℂfld e. Ring -> 1 = ( 0g ` G ) )
102	71, 101	ax-mp 5	. . . . . 6 |- 1 = ( 0g ` G )
103	98, 99, 82, 102	odeq 16773	. . . . 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 1226	. . . 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 2462	. 2 |- ( N e. NN -> ( O ` ( -u 1 ^c ( 2 / N ) ) ) = N )
107	98, 99	odf 16760	. . . 4 |- O : ( CC \ { 0 } ) --> NN0
108		ffn 5713	. . . 4 |- ( O : ( CC \ { 0 } ) --> NN0 -> O Fn ( CC \ { 0 } ) )
109	107, 108	ax-mp 5	. . 3 |- O Fn ( CC \ { 0 } )
110		fniniseg 5984	. . 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 920	1 |- ( N e. NN -> ( -u 1 ^c ( 2 / N ) ) e. ( `' O " { N } ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   \ cdif 3458   {csn 4016   class class class wbr 4439   `'ccnv 4987   "cima 4991   Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482   _ici 9483   x. cmul 9486   -ucneg 9797   / cdiv 10202   NNcn 10531   2c2 10581   NN0cn0 10791   ZZcz 10860   RR+crp 11221   ^cexp 12148   expce 13879   picpi 13884   || cdvds 14070   ↾_s cress 14717   0gc0g 14929  SubMndcsubmnd 16164   Grpcgrp 16252  ._gcmg 16255   odcod 16748  mulGrpcmgp 17336   Ringcrg 17393  ℂ_fldccnfld 18615   logclog 23108   ^c ccxp 23109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12982  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-limsup 13376  df-clim 13393  df-rlim 13394  df-sum 13591  df-ef 13885  df-sin 13887  df-cos 13888  df-pi 13890  df-dvds 14071  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-mulg 16259  df-cntz 16554  df-od 16752  df-cmn 16999  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-dvr 17527  df-drng 17593  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-limc 22436  df-dv 22437  df-log 23110  df-cxp 23111

This theorem is referenced by: (None)