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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 ` 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 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+ )
52, 3, 4sylancr 663 . . . . 5  |-  ( N  e.  NN  ->  (
2  /  N )  e.  RR+ )
65rpcnd 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 )
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 10647 . . . . 5  |-  -u 1  =/=  0
1110a1i 11 . . . 4  |-  ( N  e.  NN  ->  -u 1  =/=  0 )
129, 11, 6cxpne0d 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
) )
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 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 )
196, 17, 18syl2an 477 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  e.  CC )
2015, 16, 19cxpefd 22965 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^c 
( ( 2  /  N )  x.  x
) )  =  ( exp `  ( ( ( 2  /  N
)  x.  x )  x.  ( log `  -u 1
) ) ) )
2120eqeq1d 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 )
231, 10, 22mp2an 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 )
2519, 23, 24sylancl 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 ) )
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 10612 . . . . . . . . . . . . . 14  |-  2  e.  CC
2928a1i 11 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
2  e.  CC )
30 nncn 10550 . . . . . . . . . . . . . 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 10574 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  =/=  0 )
3433adantr 465 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  =/=  0 )
3529, 31, 32, 34div13d 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 )
3736a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( log `  -u 1
)  =  ( _i  x.  pi ) )
3835, 37oveq12d 6299 . . . . . . . . . . 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 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
4240, 41mulcli 9604 . . . . . . . . . . . . 13  |-  ( _i  x.  pi )  e.  CC
4342a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  pi )  e.  CC )
4439, 29, 43mulassd 9622 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( x  /  N )  x.  2 )  x.  (
_i  x.  pi )
)  =  ( ( x  /  N )  x.  ( 2  x.  ( _i  x.  pi ) ) ) )
4540a1i 11 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  _i  e.  CC )
4641a1i 11 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  pi  e.  CC )
4729, 45, 46mul12d 9792 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( 2  x.  (
_i  x.  pi )
)  =  ( _i  x.  ( 2  x.  pi ) ) )
4847oveq2d 6297 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( x  /  N )  x.  (
2  x.  ( _i  x.  pi ) ) )  =  ( ( x  /  N )  x.  ( _i  x.  ( 2  x.  pi ) ) ) )
4938, 44, 483eqtrd 2488 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  =  ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) ) )
5049oveq1d 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 ) ) ) )
5128, 41mulcli 9604 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  e.  CC
5240, 51mulcli 9604 . . . . . . . . . . 11  |-  ( _i  x.  ( 2  x.  pi ) )  e.  CC
5352a1i 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
5856, 57gt0ne0ii 10095 . . . . . . . . . . . . 13  |-  pi  =/=  0
5928, 41, 55, 58mulne0i 10198 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  =/=  0
6040, 51, 54, 59mulne0i 10198 . . . . . . . . . . 11  |-  ( _i  x.  ( 2  x.  pi ) )  =/=  0
6160a1i 11 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  (
2  x.  pi ) )  =/=  0 )
6239, 53, 61divcan4d 10332 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6350, 62eqtrd 2484 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1
) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6463eleq1d 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 ) )
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 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 ` 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 cnring 18314 . . . . . . . . . 10  |-fld  e.  Ring
72 cnfldbas 18298 . . . . . . . . . . . 12  |-  CC  =  ( Base ` fld )
73 cnfld0 18316 . . . . . . . . . . . 12  |-  0  =  ( 0g ` fld )
74 cndrng 18321 . . . . . . . . . . . 12  |-fld  e.  DivRing
7572, 73, 74drngui 17276 . . . . . . . . . . 11  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
76 eqid 2443 . . . . . . . . . . 11  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
7775, 76unitsubm 17193 . . . . . . . . . 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 16051 . . . . . . . . 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 1229 . . . . . . . 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 2491 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  G
) ( -u 1  ^c  ( 2  /  N ) ) )  =  ( -u
1  ^c  ( ( 2  /  N
)  x.  x ) ) )
8685eqeq1d 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 )
8887adantr 465 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  ZZ )
89 nn0z 10893 . . . . . . . 8  |-  ( x  e.  NN0  ->  x  e.  ZZ )
9089adantl 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 ) )
9288, 34, 90, 91syl3anc 1229 . . . . . 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 2857 . . . 4  |-  ( N  e.  NN  ->  A. x  e.  NN0  ( N  ||  x 
<->  ( x (.g `  G
) ( -u 1  ^c  ( 2  /  N ) ) )  =  1 ) )
9575, 81unitgrp 17190 . . . . . 6  |-  (fld  e.  Ring  ->  G  e.  Grp )
9671, 95mp1i 12 . . . . 5  |-  ( N  e.  NN  ->  G  e.  Grp )
97 nnnn0 10808 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
9875, 81unitgrpbas 17189 . . . . . 6  |-  ( CC 
\  { 0 } )  =  ( Base `  G )
99 proot1ex.o . . . . . 6  |-  O  =  ( od `  G
)
100 cnfld1 18317 . . . . . . . 8  |-  1  =  ( 1r ` fld )
10175, 81, 100unitgrpid 17192 . . . . . . 7  |-  (fld  e.  Ring  -> 
1  =  ( 0g
`  G ) )
10271, 101ax-mp 5 . . . . . 6  |-  1  =  ( 0g `  G )
10398, 99, 82, 102odeq 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 ) ) )
10496, 14, 97, 103syl3anc 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 ) ) )
10594, 104mpbird 232 . . 3  |-  ( N  e.  NN  ->  N  =  ( O `  ( -u 1  ^c 
( 2  /  N
) ) ) )
106105eqcomd 2451 . 2  |-  ( N  e.  NN  ->  ( O `  ( -u 1  ^c  ( 2  /  N ) ) )  =  N )
10798, 99odf 16435 . . . 4  |-  O :
( CC  \  {
0 } ) --> NN0
108 ffn 5721 . . . 4  |-  ( O : ( CC  \  { 0 } ) --> NN0  ->  O  Fn  ( CC  \  { 0 } ) )
109107, 108ax-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 ) ) )
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 922 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 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  ℂfldccnfld 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)
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