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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 ` fld )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.
StepHypRef 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+ )
52, 3, 4sylancr 658 . . . . 5  |-  ( N  e.  NN  ->  (
2  /  N )  e.  RR+ )
65rpcnd 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 )
81, 6, 7sylancr 658 . . 3  |-  ( N  e.  NN  ->  ( -u 1  ^c  ( 2  /  N ) )  e.  CC )
91a1i 11 . . . 4  |-  ( N  e.  NN  ->  -u 1  e.  CC )
10 neg1ne0 10423 . . . . 5  |-  -u 1  =/=  0
1110a1i 11 . . . 4  |-  ( N  e.  NN  ->  -u 1  =/=  0 )
129, 11, 6cxpne0d 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
) )
148, 12, 13sylanbrc 659 . 2  |-  ( N  e.  NN  ->  ( -u 1  ^c  ( 2  /  N ) )  e.  ( CC 
\  { 0 } ) )
151a1i 11 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  -u 1  e.  CC )
1610a1i 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 )
196, 17, 18syl2an 474 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  e.  CC )
2015, 16, 19cxpefd 22116 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^c 
( ( 2  /  N )  x.  x
) )  =  ( exp `  ( ( ( 2  /  N
)  x.  x )  x.  ( log `  -u 1
) ) ) )
2120eqeq1d 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 )
231, 10, 22mp2an 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 )
2519, 23, 24sylancl 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 ) )
2725, 26syl 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
2928a1i 11 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
2  e.  CC )
30 nncn 10326 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  CC )
3130adantr 462 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  CC )
3217adantl 463 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  CC )
33 nnne0 10350 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  =/=  0 )
3433adantr 462 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  =/=  0 )
3529, 31, 32, 34div13d 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 )
3736a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( log `  -u 1
)  =  ( _i  x.  pi ) )
3835, 37oveq12d 6108 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  =  ( ( ( x  /  N )  x.  2 )  x.  ( _i  x.  pi ) ) )
3932, 31, 34divcld 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
4241recni 9394 . . . . . . . . . . . . . 14  |-  pi  e.  CC
4340, 42mulcli 9387 . . . . . . . . . . . . 13  |-  ( _i  x.  pi )  e.  CC
4443a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  pi )  e.  CC )
4539, 29, 44mulassd 9405 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( x  /  N )  x.  2 )  x.  (
_i  x.  pi )
)  =  ( ( x  /  N )  x.  ( 2  x.  ( _i  x.  pi ) ) ) )
4640a1i 11 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  _i  e.  CC )
4742a1i 11 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  pi  e.  CC )
4829, 46, 47mul12d 9574 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( 2  x.  (
_i  x.  pi )
)  =  ( _i  x.  ( 2  x.  pi ) ) )
4948oveq2d 6106 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( x  /  N )  x.  (
2  x.  ( _i  x.  pi ) ) )  =  ( ( x  /  N )  x.  ( _i  x.  ( 2  x.  pi ) ) ) )
5038, 45, 493eqtrd 2477 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  =  ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) ) )
5150oveq1d 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 ) ) ) )
5228, 42mulcli 9387 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  e.  CC
5340, 52mulcli 9387 . . . . . . . . . . 11  |-  ( _i  x.  ( 2  x.  pi ) )  e.  CC
5453a1i 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
5841, 57gt0ne0ii 9872 . . . . . . . . . . . . 13  |-  pi  =/=  0
5928, 42, 56, 58mulne0i 9975 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  =/=  0
6040, 52, 55, 59mulne0i 9975 . . . . . . . . . . 11  |-  ( _i  x.  ( 2  x.  pi ) )  =/=  0
6160a1i 11 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  (
2  x.  pi ) )  =/=  0 )
6239, 54, 61divcan4d 10109 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6351, 62eqtrd 2473 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1
) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6463eleq1d 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 ) )
6521, 27, 643bitrd 279 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( -u 1  ^c  ( (
2  /  N )  x.  x ) )  =  1  <->  ( x  /  N )  e.  ZZ ) )
666adantr 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 )
6815, 66, 67cxpmul2d 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 ` fld ) ) ( -u
1  ^c  ( 2  /  N ) ) )  =  ( ( -u 1  ^c  ( 2  /  N ) ) ^
x ) )
708, 69sylan 468 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  (mulGrp ` fld ) ) ( -u 1  ^c  ( 2  /  N ) ) )  =  ( (
-u 1  ^c 
( 2  /  N
) ) ^ x
) )
71 cnrng 17797 . . . . . . . . . 10  |-fld  e.  Ring
72 cnfldbas 17781 . . . . . . . . . . . 12  |-  CC  =  ( Base ` fld )
73 cnfld0 17799 . . . . . . . . . . . 12  |-  0  =  ( 0g ` fld )
74 cndrng 17804 . . . . . . . . . . . 12  |-fld  e.  DivRing
7572, 73, 74drngui 16818 . . . . . . . . . . 11  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
76 eqid 2441 . . . . . . . . . . 11  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
7775, 76unitsubm 16752 . . . . . . . . . 10  |-  (fld  e.  Ring  -> 
( CC  \  {
0 } )  e.  (SubMnd `  (mulGrp ` fld ) ) )
7871, 77mp1i 12 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( CC  \  {
0 } )  e.  (SubMnd `  (mulGrp ` fld ) ) )
7914adantr 462 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^c 
( 2  /  N
) )  e.  ( CC  \  { 0 } ) )
80 eqid 2441 . . . . . . . . . 10  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
81 proot1ex.g . . . . . . . . . 10  |-  G  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
82 eqid 2441 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
8380, 81, 82submmulg 15655 . . . . . . . . 9  |-  ( ( ( CC  \  {
0 } )  e.  (SubMnd `  (mulGrp ` fld ) )  /\  x  e.  NN0  /\  ( -u
1  ^c  ( 2  /  N ) )  e.  ( CC 
\  { 0 } ) )  ->  (
x (.g `  (mulGrp ` fld ) ) ( -u
1  ^c  ( 2  /  N ) ) )  =  ( x (.g `  G ) (
-u 1  ^c 
( 2  /  N
) ) ) )
8478, 67, 79, 83syl3anc 1213 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  (mulGrp ` fld ) ) ( -u 1  ^c  ( 2  /  N ) ) )  =  ( x (.g `  G ) (
-u 1  ^c 
( 2  /  N
) ) ) )
8568, 70, 843eqtr2rd 2480 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  G
) ( -u 1  ^c  ( 2  /  N ) ) )  =  ( -u
1  ^c  ( ( 2  /  N
)  x.  x ) ) )
8685eqeq1d 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 )
8887adantr 462 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  ZZ )
89 nn0z 10665 . . . . . . . 8  |-  ( x  e.  NN0  ->  x  e.  ZZ )
9089adantl 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 ) )
9288, 34, 90, 91syl3anc 1213 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( N  ||  x  <->  ( x  /  N )  e.  ZZ ) )
9365, 86, 923bitr4rd 286 . . . . 5  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( N  ||  x  <->  ( x (.g `  G ) (
-u 1  ^c 
( 2  /  N
) ) )  =  1 ) )
9493ralrimiva 2797 . . . 4  |-  ( N  e.  NN  ->  A. x  e.  NN0  ( N  ||  x 
<->  ( x (.g `  G
) ( -u 1  ^c  ( 2  /  N ) ) )  =  1 ) )
9575, 81unitgrp 16749 . . . . . 6  |-  (fld  e.  Ring  ->  G  e.  Grp )
9671, 95mp1i 12 . . . . 5  |-  ( N  e.  NN  ->  G  e.  Grp )
97 nnnn0 10582 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
9875, 81unitgrpbas 16748 . . . . . 6  |-  ( CC 
\  { 0 } )  =  ( Base `  G )
99 proot1ex.o . . . . . 6  |-  O  =  ( od `  G
)
100 cnfld1 17800 . . . . . . . 8  |-  1  =  ( 1r ` fld )
10175, 81, 100unitgrpid 16751 . . . . . . 7  |-  (fld  e.  Ring  -> 
1  =  ( 0g
`  G ) )
10271, 101ax-mp 5 . . . . . 6  |-  1  =  ( 0g `  G )
10398, 99, 82, 102odeq 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 ) ) )
10496, 14, 97, 103syl3anc 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 ) ) )
10594, 104mpbird 232 . . 3  |-  ( N  e.  NN  ->  N  =  ( O `  ( -u 1  ^c 
( 2  /  N
) ) ) )
106105eqcomd 2446 . 2  |-  ( N  e.  NN  ->  ( O `  ( -u 1  ^c  ( 2  /  N ) ) )  =  N )
10798, 99odf 16033 . . . 4  |-  O :
( CC  \  {
0 } ) --> NN0
108 ffn 5556 . . . 4  |-  ( O : ( CC  \  { 0 } ) --> NN0  ->  O  Fn  ( CC  \  { 0 } ) )
109107, 108ax-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 ) ) )
111109, 110mp1i 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 ) ) )
11214, 106, 111mpbir2and 908 1  |-  ( N  e.  NN  ->  ( -u 1  ^c  ( 2  /  N ) )  e.  ( `' O " { N } ) )
Colors of variables: wff setvar class
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  ℂfldccnfld 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)
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