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Theorem proot1ex 36080
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 10702 . . . 4  |-  -u 1  e.  CC
2 2rp 11297 . . . . . 6  |-  2  e.  RR+
3 nnrp 11301 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  RR+ )
4 rpdivcl 11315 . . . . . 6  |-  ( ( 2  e.  RR+  /\  N  e.  RR+ )  ->  (
2  /  N )  e.  RR+ )
52, 3, 4sylancr 674 . . . . 5  |-  ( N  e.  NN  ->  (
2  /  N )  e.  RR+ )
65rpcnd 11333 . . . 4  |-  ( N  e.  NN  ->  (
2  /  N )  e.  CC )
7 cxpcl 23631 . . . 4  |-  ( (
-u 1  e.  CC  /\  ( 2  /  N
)  e.  CC )  ->  ( -u 1  ^c  ( 2  /  N ) )  e.  CC )
81, 6, 7sylancr 674 . . 3  |-  ( N  e.  NN  ->  ( -u 1  ^c  ( 2  /  N ) )  e.  CC )
91a1i 11 . . . 4  |-  ( N  e.  NN  ->  -u 1  e.  CC )
10 neg1ne0 10704 . . . . 5  |-  -u 1  =/=  0
1110a1i 11 . . . 4  |-  ( N  e.  NN  ->  -u 1  =/=  0 )
129, 11, 6cxpne0d 23670 . . 3  |-  ( N  e.  NN  ->  ( -u 1  ^c  ( 2  /  N ) )  =/=  0 )
13 eldifsn 4066 . . 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 675 . 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 10869 . . . . . . . . . 10  |-  ( x  e.  NN0  ->  x  e.  CC )
18 mulcl 9610 . . . . . . . . . 10  |-  ( ( ( 2  /  N
)  e.  CC  /\  x  e.  CC )  ->  ( ( 2  /  N )  x.  x
)  e.  CC )
196, 17, 18syl2an 484 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  e.  CC )
2015, 16, 19cxpefd 23669 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^c 
( ( 2  /  N )  x.  x
) )  =  ( exp `  ( ( ( 2  /  N
)  x.  x )  x.  ( log `  -u 1
) ) ) )
2120eqeq1d 2454 . . . . . . 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 23530 . . . . . . . . . 10  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0
)  ->  ( log `  -u 1 )  e.  CC )
231, 10, 22mp2an 683 . . . . . . . . 9  |-  ( log `  -u 1 )  e.  CC
24 mulcl 9610 . . . . . . . . 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 673 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  e.  CC )
26 efeq1 23490 . . . . . . . 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 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 10669 . . . . . . . . . . . . . 14  |-  2  e.  CC
2928a1i 11 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
2  e.  CC )
30 nncn 10606 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  CC )
3130adantr 471 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  CC )
3217adantl 472 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  CC )
33 nnne0 10631 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  =/=  0 )
3433adantr 471 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  =/=  0 )
3529, 31, 32, 34div13d 10396 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  =  ( ( x  /  N )  x.  2 ) )
36 logm1 23550 . . . . . . . . . . . . 13  |-  ( log `  -u 1 )  =  ( _i  x.  pi )
3736a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( log `  -u 1
)  =  ( _i  x.  pi ) )
3835, 37oveq12d 6294 . . . . . . . . . . 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 10372 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x  /  N
)  e.  CC )
40 ax-icn 9585 . . . . . . . . . . . . . 14  |-  _i  e.  CC
41 picn 23426 . . . . . . . . . . . . . 14  |-  pi  e.  CC
4240, 41mulcli 9635 . . . . . . . . . . . . 13  |-  ( _i  x.  pi )  e.  CC
4342a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  pi )  e.  CC )
4439, 29, 43mulassd 9653 . . . . . . . . . . 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 9829 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( 2  x.  (
_i  x.  pi )
)  =  ( _i  x.  ( 2  x.  pi ) ) )
4847oveq2d 6292 . . . . . . . . . . 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 2490 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  =  ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) ) )
5049oveq1d 6291 . . . . . . . . 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 9635 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  e.  CC
5240, 51mulcli 9635 . . . . . . . . . . 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 10043 . . . . . . . . . . . 12  |-  _i  =/=  0
55 2ne0 10691 . . . . . . . . . . . . 13  |-  2  =/=  0
56 pire 23425 . . . . . . . . . . . . . 14  |-  pi  e.  RR
57 pipos 23427 . . . . . . . . . . . . . 14  |-  0  <  pi
5856, 57gt0ne0ii 10139 . . . . . . . . . . . . 13  |-  pi  =/=  0
5928, 41, 55, 58mulne0i 10244 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  =/=  0
6040, 51, 54, 59mulne0i 10244 . . . . . . . . . . 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 10378 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6350, 62eqtrd 2486 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1
) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6463eleq1d 2514 . . . . . . 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 287 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( -u 1  ^c  ( (
2  /  N )  x.  x ) )  =  1  <->  ( x  /  N )  e.  ZZ ) )
666adantr 471 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( 2  /  N
)  e.  CC )
67 simpr 467 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  NN0 )
6815, 66, 67cxpmul2d 23666 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^c 
( ( 2  /  N )  x.  x
) )  =  ( ( -u 1  ^c  ( 2  /  N ) ) ^
x ) )
69 cnfldexp 19012 . . . . . . . . 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 478 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  (mulGrp ` fld ) ) ( -u 1  ^c  ( 2  /  N ) ) )  =  ( (
-u 1  ^c 
( 2  /  N
) ) ^ x
) )
71 cnring 19001 . . . . . . . . . 10  |-fld  e.  Ring
72 cnfldbas 18985 . . . . . . . . . . . 12  |-  CC  =  ( Base ` fld )
73 cnfld0 19003 . . . . . . . . . . . 12  |-  0  =  ( 0g ` fld )
74 cndrng 19008 . . . . . . . . . . . 12  |-fld  e.  DivRing
7572, 73, 74drngui 17992 . . . . . . . . . . 11  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
76 eqid 2452 . . . . . . . . . . 11  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
7775, 76unitsubm 17909 . . . . . . . . . 10  |-  (fld  e.  Ring  -> 
( CC  \  {
0 } )  e.  (SubMnd `  (mulGrp ` fld ) ) )
7871, 77mp1i 13 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( CC  \  {
0 } )  e.  (SubMnd `  (mulGrp ` fld ) ) )
7914adantr 471 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^c 
( 2  /  N
) )  e.  ( CC  \  { 0 } ) )
80 eqid 2452 . . . . . . . . . 10  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
81 proot1ex.g . . . . . . . . . 10  |-  G  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
82 eqid 2452 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
8380, 81, 82submmulg 16804 . . . . . . . . 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 1271 . . . . . . . 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 2493 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  G
) ( -u 1  ^c  ( 2  /  N ) ) )  =  ( -u
1  ^c  ( ( 2  /  N
)  x.  x ) ) )
8685eqeq1d 2454 . . . . . 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 10949 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  ZZ )
8887adantr 471 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  ZZ )
89 nn0z 10950 . . . . . . . 8  |-  ( x  e.  NN0  ->  x  e.  ZZ )
9089adantl 472 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  ZZ )
91 dvdsval2 14319 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0  /\  x  e.  ZZ )  ->  ( N  ||  x  <->  ( x  /  N )  e.  ZZ ) )
9288, 34, 90, 91syl3anc 1271 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( N  ||  x  <->  ( x  /  N )  e.  ZZ ) )
9365, 86, 923bitr4rd 294 . . . . 5  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( N  ||  x  <->  ( x (.g `  G ) (
-u 1  ^c 
( 2  /  N
) ) )  =  1 ) )
9493ralrimiva 2790 . . . 4  |-  ( N  e.  NN  ->  A. x  e.  NN0  ( N  ||  x 
<->  ( x (.g `  G
) ( -u 1  ^c  ( 2  /  N ) ) )  =  1 ) )
9575, 81unitgrp 17906 . . . . . 6  |-  (fld  e.  Ring  ->  G  e.  Grp )
9671, 95mp1i 13 . . . . 5  |-  ( N  e.  NN  ->  G  e.  Grp )
97 nnnn0 10866 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
9875, 81unitgrpbas 17905 . . . . . 6  |-  ( CC 
\  { 0 } )  =  ( Base `  G )
99 proot1ex.o . . . . . 6  |-  O  =  ( od `  G
)
100 cnfld1 19004 . . . . . . . 8  |-  1  =  ( 1r ` fld )
10175, 81, 100unitgrpid 17908 . . . . . . 7  |-  (fld  e.  Ring  -> 
1  =  ( 0g
`  G ) )
10271, 101ax-mp 5 . . . . . 6  |-  1  =  ( 0g `  G )
10398, 99, 82, 102odeq 17210 . . . . 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 1271 . . . 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 240 . . 3  |-  ( N  e.  NN  ->  N  =  ( O `  ( -u 1  ^c 
( 2  /  N
) ) ) )
106105eqcomd 2458 . 2  |-  ( N  e.  NN  ->  ( O `  ( -u 1  ^c  ( 2  /  N ) ) )  =  N )
10798, 99odf 17197 . . . 4  |-  O :
( CC  \  {
0 } ) --> NN0
108 ffn 5711 . . . 4  |-  ( O : ( CC  \  { 0 } ) --> NN0  ->  O  Fn  ( CC  \  { 0 } ) )
109107, 108ax-mp 5 . . 3  |-  O  Fn  ( CC  \  { 0 } )
110 fniniseg 5987 . . 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 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 ) ) )
11214, 106, 111mpbir2and 933 1  |-  ( N  e.  NN  ->  ( -u 1  ^c  ( 2  /  N ) )  e.  ( `' O " { N } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1448    e. wcel 1891    =/= wne 2622   A.wral 2737    \ cdif 3369   {csn 3936   class class class wbr 4374   `'ccnv 4811   "cima 4815    Fn wfn 5556   -->wf 5557   ` cfv 5561  (class class class)co 6276   CCcc 9524   0cc0 9526   1c1 9527   _ici 9528    x. cmul 9531   -ucneg 9848    / cdiv 10258   NNcn 10598   2c2 10648   NN0cn0 10859   ZZcz 10927   RR+crp 11292   ^cexp 12266   expce 14125   picpi 14130    || cdvds 14316   ↾s cress 15133   0gc0g 15349  SubMndcsubmnd 16592   Grpcgrp 16680  .gcmg 16683   odcod 17176  mulGrpcmgp 17734   Ringcrg 17791  ℂfldccnfld 18981   logclog 23516    ^c ccxp 23517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-rep 4487  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571  ax-inf2 8133  ax-cnex 9582  ax-resscn 9583  ax-1cn 9584  ax-icn 9585  ax-addcl 9586  ax-addrcl 9587  ax-mulcl 9588  ax-mulrcl 9589  ax-mulcom 9590  ax-addass 9591  ax-mulass 9592  ax-distr 9593  ax-i2m1 9594  ax-1ne0 9595  ax-1rid 9596  ax-rnegex 9597  ax-rrecex 9598  ax-cnre 9599  ax-pre-lttri 9600  ax-pre-lttrn 9601  ax-pre-ltadd 9602  ax-pre-mulgt0 9603  ax-pre-sup 9604  ax-addf 9605  ax-mulf 9606
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1451  df-fal 1454  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4169  df-int 4205  df-iun 4250  df-iin 4251  df-br 4375  df-opab 4434  df-mpt 4435  df-tr 4470  df-eprel 4723  df-id 4727  df-po 4733  df-so 4734  df-fr 4771  df-se 4772  df-we 4773  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-pred 5359  df-ord 5405  df-on 5406  df-lim 5407  df-suc 5408  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-isom 5570  df-riota 6238  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-of 6519  df-om 6681  df-1st 6781  df-2nd 6782  df-supp 6903  df-tpos 6960  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-2o 7170  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-ixp 7510  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-fsupp 7871  df-fi 7912  df-sup 7943  df-inf 7944  df-oi 8012  df-card 8360  df-cda 8585  df-pnf 9664  df-mnf 9665  df-xr 9666  df-ltxr 9667  df-le 9668  df-sub 9849  df-neg 9850  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10860  df-z 10928  df-dec 11042  df-uz 11150  df-q 11255  df-rp 11293  df-xneg 11399  df-xadd 11400  df-xmul 11401  df-ioo 11629  df-ioc 11630  df-ico 11631  df-icc 11632  df-fz 11776  df-fzo 11909  df-fl 12022  df-mod 12091  df-seq 12208  df-exp 12267  df-fac 12454  df-bc 12482  df-hash 12510  df-shft 13141  df-cj 13173  df-re 13174  df-im 13175  df-sqrt 13309  df-abs 13310  df-limsup 13537  df-clim 13563  df-rlim 13564  df-sum 13764  df-ef 14132  df-sin 14134  df-cos 14135  df-pi 14137  df-dvds 14317  df-struct 15134  df-ndx 15135  df-slot 15136  df-base 15137  df-sets 15138  df-ress 15139  df-plusg 15214  df-mulr 15215  df-starv 15216  df-sca 15217  df-vsca 15218  df-ip 15219  df-tset 15220  df-ple 15221  df-ds 15223  df-unif 15224  df-hom 15225  df-cco 15226  df-rest 15332  df-topn 15333  df-0g 15351  df-gsum 15352  df-topgen 15353  df-pt 15354  df-prds 15357  df-xrs 15411  df-qtop 15417  df-imas 15418  df-xps 15421  df-mre 15503  df-mrc 15504  df-acs 15506  df-mgm 16499  df-sgrp 16538  df-mnd 16548  df-submnd 16594  df-grp 16684  df-minusg 16685  df-sbg 16686  df-mulg 16687  df-cntz 16982  df-od 17183  df-cmn 17443  df-mgp 17735  df-ur 17747  df-ring 17793  df-cring 17794  df-oppr 17862  df-dvdsr 17880  df-unit 17881  df-invr 17911  df-dvr 17922  df-drng 17988  df-psmet 18973  df-xmet 18974  df-met 18975  df-bl 18976  df-mopn 18977  df-fbas 18978  df-fg 18979  df-cnfld 18982  df-top 19932  df-bases 19933  df-topon 19934  df-topsp 19935  df-cld 20045  df-ntr 20046  df-cls 20047  df-nei 20125  df-lp 20163  df-perf 20164  df-cn 20254  df-cnp 20255  df-haus 20342  df-tx 20588  df-hmeo 20781  df-fil 20872  df-fm 20964  df-flim 20965  df-flf 20966  df-xms 21346  df-ms 21347  df-tms 21348  df-cncf 21921  df-limc 22833  df-dv 22834  df-log 23518  df-cxp 23519
This theorem is referenced by: (None)
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