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Theorem proot1ex 29569
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 10425 . . . 4  |-  -u 1  e.  CC
2 2rp 10996 . . . . . 6  |-  2  e.  RR+
3 nnrp 11000 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  RR+ )
4 rpdivcl 11013 . . . . . 6  |-  ( ( 2  e.  RR+  /\  N  e.  RR+ )  ->  (
2  /  N )  e.  RR+ )
52, 3, 4sylancr 663 . . . . 5  |-  ( N  e.  NN  ->  (
2  /  N )  e.  RR+ )
65rpcnd 11029 . . . 4  |-  ( N  e.  NN  ->  (
2  /  N )  e.  CC )
7 cxpcl 22119 . . . 4  |-  ( (
-u 1  e.  CC  /\  ( 2  /  N
)  e.  CC )  ->  ( -u 1  ^c  ( 2  /  N ) )  e.  CC )
81, 6, 7sylancr 663 . . 3  |-  ( N  e.  NN  ->  ( -u 1  ^c  ( 2  /  N ) )  e.  CC )
91a1i 11 . . . 4  |-  ( N  e.  NN  ->  -u 1  e.  CC )
10 neg1ne0 10427 . . . . 5  |-  -u 1  =/=  0
1110a1i 11 . . . 4  |-  ( N  e.  NN  ->  -u 1  =/=  0 )
129, 11, 6cxpne0d 22158 . . 3  |-  ( N  e.  NN  ->  ( -u 1  ^c  ( 2  /  N ) )  =/=  0 )
13 eldifsn 4000 . . 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 664 . 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 10589 . . . . . . . . . 10  |-  ( x  e.  NN0  ->  x  e.  CC )
18 mulcl 9366 . . . . . . . . . 10  |-  ( ( ( 2  /  N
)  e.  CC  /\  x  e.  CC )  ->  ( ( 2  /  N )  x.  x
)  e.  CC )
196, 17, 18syl2an 477 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  e.  CC )
2015, 16, 19cxpefd 22157 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^c 
( ( 2  /  N )  x.  x
) )  =  ( exp `  ( ( ( 2  /  N
)  x.  x )  x.  ( log `  -u 1
) ) ) )
2120eqeq1d 2451 . . . . . . 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 22020 . . . . . . . . . 10  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0
)  ->  ( log `  -u 1 )  e.  CC )
231, 10, 22mp2an 672 . . . . . . . . 9  |-  ( log `  -u 1 )  e.  CC
24 mulcl 9366 . . . . . . . . 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 662 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  e.  CC )
26 efeq1 21985 . . . . . . . 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 10392 . . . . . . . . . . . . . 14  |-  2  e.  CC
2928a1i 11 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
2  e.  CC )
30 nncn 10330 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  CC )
3130adantr 465 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  CC )
3217adantl 466 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  CC )
33 nnne0 10354 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  =/=  0 )
3433adantr 465 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  =/=  0 )
3529, 31, 32, 34div13d 10131 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  =  ( ( x  /  N )  x.  2 ) )
36 logm1 22037 . . . . . . . . . . . . 13  |-  ( log `  -u 1 )  =  ( _i  x.  pi )
3736a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( log `  -u 1
)  =  ( _i  x.  pi ) )
3835, 37oveq12d 6109 . . . . . . . . . . 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 10107 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x  /  N
)  e.  CC )
40 ax-icn 9341 . . . . . . . . . . . . . 14  |-  _i  e.  CC
41 pire 21921 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
4241recni 9398 . . . . . . . . . . . . . 14  |-  pi  e.  CC
4340, 42mulcli 9391 . . . . . . . . . . . . 13  |-  ( _i  x.  pi )  e.  CC
4443a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  pi )  e.  CC )
4539, 29, 44mulassd 9409 . . . . . . . . . . 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 9578 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( 2  x.  (
_i  x.  pi )
)  =  ( _i  x.  ( 2  x.  pi ) ) )
4948oveq2d 6107 . . . . . . . . . . 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 2479 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  =  ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) ) )
5150oveq1d 6106 . . . . . . . . 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 9391 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  e.  CC
5340, 52mulcli 9391 . . . . . . . . . . 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 9780 . . . . . . . . . . . 12  |-  _i  =/=  0
56 2ne0 10414 . . . . . . . . . . . . 13  |-  2  =/=  0
57 pipos 21923 . . . . . . . . . . . . . 14  |-  0  <  pi
5841, 57gt0ne0ii 9876 . . . . . . . . . . . . 13  |-  pi  =/=  0
5928, 42, 56, 58mulne0i 9979 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  =/=  0
6040, 52, 55, 59mulne0i 9979 . . . . . . . . . . 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 10113 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6351, 62eqtrd 2475 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1
) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6463eleq1d 2509 . . . . . . 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 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 )
6815, 66, 67cxpmul2d 22154 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^c 
( ( 2  /  N )  x.  x
) )  =  ( ( -u 1  ^c  ( 2  /  N ) ) ^
x ) )
69 cnfldexp 17849 . . . . . . . . 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 471 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  (mulGrp ` fld ) ) ( -u 1  ^c  ( 2  /  N ) ) )  =  ( (
-u 1  ^c 
( 2  /  N
) ) ^ x
) )
71 cnrng 17838 . . . . . . . . . 10  |-fld  e.  Ring
72 cnfldbas 17822 . . . . . . . . . . . 12  |-  CC  =  ( Base ` fld )
73 cnfld0 17840 . . . . . . . . . . . 12  |-  0  =  ( 0g ` fld )
74 cndrng 17845 . . . . . . . . . . . 12  |-fld  e.  DivRing
7572, 73, 74drngui 16838 . . . . . . . . . . 11  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
76 eqid 2443 . . . . . . . . . . 11  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
7775, 76unitsubm 16762 . . . . . . . . . 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 465 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^c 
( 2  /  N
) )  e.  ( CC  \  { 0 } ) )
80 eqid 2443 . . . . . . . . . 10  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
81 proot1ex.g . . . . . . . . . 10  |-  G  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
82 eqid 2443 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
8380, 81, 82submmulg 15662 . . . . . . . . 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 1218 . . . . . . . 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 2482 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  G
) ( -u 1  ^c  ( 2  /  N ) ) )  =  ( -u
1  ^c  ( ( 2  /  N
)  x.  x ) ) )
8685eqeq1d 2451 . . . . . 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 10668 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  ZZ )
8887adantr 465 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  ZZ )
89 nn0z 10669 . . . . . . . 8  |-  ( x  e.  NN0  ->  x  e.  ZZ )
9089adantl 466 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  ZZ )
91 dvdsval2 13538 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0  /\  x  e.  ZZ )  ->  ( N  ||  x  <->  ( x  /  N )  e.  ZZ ) )
9288, 34, 90, 91syl3anc 1218 . . . . . 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 2799 . . . 4  |-  ( N  e.  NN  ->  A. x  e.  NN0  ( N  ||  x 
<->  ( x (.g `  G
) ( -u 1  ^c  ( 2  /  N ) ) )  =  1 ) )
9575, 81unitgrp 16759 . . . . . 6  |-  (fld  e.  Ring  ->  G  e.  Grp )
9671, 95mp1i 12 . . . . 5  |-  ( N  e.  NN  ->  G  e.  Grp )
97 nnnn0 10586 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
9875, 81unitgrpbas 16758 . . . . . 6  |-  ( CC 
\  { 0 } )  =  ( Base `  G )
99 proot1ex.o . . . . . 6  |-  O  =  ( od `  G
)
100 cnfld1 17841 . . . . . . . 8  |-  1  =  ( 1r ` fld )
10175, 81, 100unitgrpid 16761 . . . . . . 7  |-  (fld  e.  Ring  -> 
1  =  ( 0g
`  G ) )
10271, 101ax-mp 5 . . . . . 6  |-  1  =  ( 0g `  G )
10398, 99, 82, 102odeq 16053 . . . . 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 1218 . . . 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 2448 . 2  |-  ( N  e.  NN  ->  ( O `  ( -u 1  ^c  ( 2  /  N ) ) )  =  N )
10798, 99odf 16040 . . . 4  |-  O :
( CC  \  {
0 } ) --> NN0
108 ffn 5559 . . . 4  |-  ( O : ( CC  \  { 0 } ) --> NN0  ->  O  Fn  ( CC  \  { 0 } ) )
109107, 108ax-mp 5 . . 3  |-  O  Fn  ( CC  \  { 0 } )
110 fniniseg 5824 . . 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 913 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 1369    e. wcel 1756    =/= wne 2606   A.wral 2715    \ cdif 3325   {csn 3877   class class class wbr 4292   `'ccnv 4839   "cima 4843    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091   CCcc 9280   0cc0 9282   1c1 9283   _ici 9284    x. cmul 9287   -ucneg 9596    / cdiv 9993   NNcn 10322   2c2 10371   NN0cn0 10579   ZZcz 10646   RR+crp 10991   ^cexp 11865   expce 13347   picpi 13352    || cdivides 13535   ↾s cress 14175   0gc0g 14378   Grpcgrp 15410  .gcmg 15414  SubMndcsubmnd 15463   odcod 16028  mulGrpcmgp 16591   Ringcrg 16645  ℂfldccnfld 17818   logclog 22006    ^c ccxp 22007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-tpos 6745  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-pi 13358  df-dvds 13536  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-mulg 15548  df-cntz 15835  df-od 16032  df-cmn 16279  df-mgp 16592  df-ur 16604  df-rng 16647  df-cring 16648  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-dvr 16775  df-drng 16834  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342  df-log 22008  df-cxp 22009
This theorem is referenced by: (None)
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