Theorem proot1ex 37779

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 11124	. . . 4 |- -u 1 e. CC
2		2rp 11837	. . . . . 6 |- 2 e. RR+
3		nnrp 11842	. . . . . 6 |- ( N e. NN -> N e. RR+ )
4		rpdivcl 11856	. . . . . 6 |- ( ( 2 e. RR+ /\ N e. RR+ ) -> ( 2 / N ) e. RR+ )
5	2, 3, 4	sylancr 695	. . . . 5 |- ( N e. NN -> ( 2 / N ) e. RR+ )
6	5	rpcnd 11874	. . . 4 |- ( N e. NN -> ( 2 / N ) e. CC )
7		cxpcl 24420	. . . 4 |- ( ( -u 1 e. CC /\ ( 2 / N ) e. CC ) -> ( -u 1 ^c ( 2 / N ) ) e. CC )
8	1, 6, 7	sylancr 695	. . 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 11126	. . . . 5 |- -u 1 =/= 0
11	10	a1i 11	. . . 4 |- ( N e. NN -> -u 1 =/= 0 )
12	9, 11, 6	cxpne0d 24459	. . 3 |- ( N e. NN -> ( -u 1 ^c ( 2 / N ) ) =/= 0 )
13		eldifsn 4317	. . 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 698	. 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 11302	. . . . . . . . . 10 |- ( x e. NN0 -> x e. CC )
18		mulcl 10020	. . . . . . . . . 10 |- ( ( ( 2 / N ) e. CC /\ x e. CC ) -> ( ( 2 / N ) x. x ) e. CC )
19	6, 17, 18	syl2an 494	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( 2 / N ) x. x ) e. CC )
20	15, 16, 19	cxpefd 24458	. . . . . . . 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 2624	. . . . . . 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 24315	. . . . . . . . . 10 |- ( ( -u 1 e. CC /\ -u 1 =/= 0 ) -> ( log ` -u 1 ) e. CC )
23	1, 10, 22	mp2an 708	. . . . . . . . 9 |- ( log ` -u 1 ) e. CC
24		mulcl 10020	. . . . . . . . 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 694	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( ( 2 / N ) x. x ) x. ( log ` -u 1 ) ) e. CC )
26		efeq1 24275	. . . . . . . 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 17	. . . . . . 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 11091	. . . . . . . . . . . . . 14 |- 2 e. CC
29	28	a1i 11	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> 2 e. CC )
30		nncn 11028	. . . . . . . . . . . . . 14 |- ( N e. NN -> N e. CC )
31	30	adantr 481	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> N e. CC )
32	17	adantl 482	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> x e. CC )
33		nnne0 11053	. . . . . . . . . . . . . 14 |- ( N e. NN -> N =/= 0 )
34	33	adantr 481	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ x e. NN0 ) -> N =/= 0 )
35	29, 31, 32, 34	div13d 10825	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( 2 / N ) x. x ) = ( ( x / N ) x. 2 ) )
36		logm1 24335	. . . . . . . . . . . . 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 6668	. . . . . . . . . . 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 10801	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( x / N ) e. CC )
40		ax-icn 9995	. . . . . . . . . . . . . 14 |- _i e. CC
41		picn 24211	. . . . . . . . . . . . . 14 |- pi e. CC
42	40, 41	mulcli 10045	. . . . . . . . . . . . 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 10063	. . . . . . . . . . 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 10245	. . . . . . . . . . . 12 |- ( ( N e. NN /\ x e. NN0 ) -> ( 2 x. ( _i x. pi ) ) = ( _i x. ( 2 x. pi ) ) )
48	47	oveq2d 6666	. . . . . . . . . . 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 2660	. . . . . . . . . 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 6665	. . . . . . . . 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 10045	. . . . . . . . . . . 12 |- ( 2 x. pi ) e. CC
52	40, 51	mulcli 10045	. . . . . . . . . . 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 10465	. . . . . . . . . . . 12 |- _i =/= 0
55		2ne0 11113	. . . . . . . . . . . . 13 |- 2 =/= 0
56		pire 24210	. . . . . . . . . . . . . 14 |- pi e. RR
57		pipos 24212	. . . . . . . . . . . . . 14 |- 0 < pi
58	56, 57	gt0ne0ii 10564	. . . . . . . . . . . . 13 |- pi =/= 0
59	28, 41, 55, 58	mulne0i 10670	. . . . . . . . . . . 12 |- ( 2 x. pi ) =/= 0
60	40, 51, 54, 59	mulne0i 10670	. . . . . . . . . . 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 10807	. . . . . . . . 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 2656	. . . . . . . 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 2686	. . . . . . 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 294	. . . . . 6 |- ( ( N e. NN /\ x e. NN0 ) -> ( ( -u 1 ^c ( ( 2 / N ) x. x ) ) = 1 <-> ( x / N ) e. ZZ ) )
66	6	adantr 481	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( 2 / N ) e. CC )
67		simpr 477	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> x e. NN0 )
68	15, 66, 67	cxpmul2d 24455	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( -u 1 ^c ( ( 2 / N ) x. x ) ) = ( ( -u 1 ^c ( 2 / N ) ) ^ x ) )
69		cnfldexp 19779	. . . . . . . . 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 488	. . . . . . . 8 |- ( ( N e. NN /\ x e. NN0 ) -> ( x (.g ` (mulGrp ` ℂfld ) ) ( -u 1 ^c ( 2 / N ) ) ) = ( ( -u 1 ^c ( 2 / N ) ) ^ x ) )
71		cnring 19768	. . . . . . . . . 10 |- ℂfld e. Ring
72		cnfldbas 19750	. . . . . . . . . . . 12 |- CC = ( Base ` ℂfld )
73		cnfld0 19770	. . . . . . . . . . . 12 |- 0 = ( 0g ` ℂfld )
74		cndrng 19775	. . . . . . . . . . . 12 |- ℂfld e. DivRing
75	72, 73, 74	drngui 18753	. . . . . . . . . . 11 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
76		eqid 2622	. . . . . . . . . . 11 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
77	75, 76	unitsubm 18670	. . . . . . . . . 10 |- (ℂfld e. Ring -> ( CC \ { 0 } ) e. (SubMnd ` (mulGrp ` ℂfld ) ) )
78	71, 77	mp1i 13	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( CC \ { 0 } ) e. (SubMnd ` (mulGrp ` ℂfld ) ) )
79	14	adantr 481	. . . . . . . . 9 |- ( ( N e. NN /\ x e. NN0 ) -> ( -u 1 ^c ( 2 / N ) ) e. ( CC \ { 0 } ) )
80		eqid 2622	. . . . . . . . . 10 |- (.g ` (mulGrp ` ℂfld ) ) = (.g ` (mulGrp ` ℂfld ) )
81		proot1ex.g	. . . . . . . . . 10 |- G = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
82		eqid 2622	. . . . . . . . . 10 |- (.g ` G ) = (.g ` G )
83	80, 81, 82	submmulg 17586	. . . . . . . . 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 1326	. . . . . . . 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 2663	. . . . . . 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 2624	. . . . . 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 11399	. . . . . . . 8 |- ( N e. NN -> N e. ZZ )
88	87	adantr 481	. . . . . . 7 |- ( ( N e. NN /\ x e. NN0 ) -> N e. ZZ )
89		nn0z 11400	. . . . . . . 8 |- ( x e. NN0 -> x e. ZZ )
90	89	adantl 482	. . . . . . 7 |- ( ( N e. NN /\ x e. NN0 ) -> x e. ZZ )
91		dvdsval2 14986	. . . . . . 7 |- ( ( N e. ZZ /\ N =/= 0 /\ x e. ZZ ) -> ( N || x <-> ( x / N ) e. ZZ ) )
92	88, 34, 90, 91	syl3anc 1326	. . . . . 6 |- ( ( N e. NN /\ x e. NN0 ) -> ( N || x <-> ( x / N ) e. ZZ ) )
93	65, 86, 92	3bitr4rd 301	. . . . 5 |- ( ( N e. NN /\ x e. NN0 ) -> ( N || x <-> ( x (.g ` G ) ( -u 1 ^c ( 2 / N ) ) ) = 1 ) )
94	93	ralrimiva 2966	. . . 4 |- ( N e. NN -> A. x e. NN0 ( N || x <-> ( x (.g ` G ) ( -u 1 ^c ( 2 / N ) ) ) = 1 ) )
95	75, 81	unitgrp 18667	. . . . . 6 |- (ℂfld e. Ring -> G e. Grp )
96	71, 95	mp1i 13	. . . . 5 |- ( N e. NN -> G e. Grp )
97		nnnn0 11299	. . . . 5 |- ( N e. NN -> N e. NN0 )
98	75, 81	unitgrpbas 18666	. . . . . 6 |- ( CC \ { 0 } ) = ( Base ` G )
99		proot1ex.o	. . . . . 6 |- O = ( od ` G )
100		cnfld1 19771	. . . . . . . 8 |- 1 = ( 1r ` ℂfld )
101	75, 81, 100	unitgrpid 18669	. . . . . . 7 |- (ℂfld e. Ring -> 1 = ( 0g ` G ) )
102	71, 101	ax-mp 5	. . . . . 6 |- 1 = ( 0g ` G )
103	98, 99, 82, 102	odeq 17969	. . . . 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 1326	. . . 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 247	. . 3 |- ( N e. NN -> N = ( O ` ( -u 1 ^c ( 2 / N ) ) ) )
106	105	eqcomd 2628	. 2 |- ( N e. NN -> ( O ` ( -u 1 ^c ( 2 / N ) ) ) = N )
107	98, 99	odf 17956	. . . 4 |- O : ( CC \ { 0 } ) --> NN0
108		ffn 6045	. . . 4 |- ( O : ( CC \ { 0 } ) --> NN0 -> O Fn ( CC \ { 0 } ) )
109	107, 108	ax-mp 5	. . 3 |- O Fn ( CC \ { 0 } )
110		fniniseg 6338	. . 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 13	. 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 957	1 |- ( N e. NN -> ( -u 1 ^c ( 2 / N ) ) e. ( `' O " { N } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   {csn 4177   class class class wbr 4653   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   _ici 9938   x. cmul 9941   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   RR+crp 11832   ^cexp 12860   expce 14792   picpi 14797   || cdvds 14983   ↾_s cress 15858   0gc0g 16100  SubMndcsubmnd 17334   Grpcgrp 17422  ._gcmg 17540   odcod 17944  mulGrpcmgp 18489   Ringcrg 18547  ℂ_fldccnfld 19746   logclog 24301   ^c ccxp 24302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-cntz 17750  df-od 17948  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304

This theorem is referenced by: (None)