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Theorem proot1ex 31402
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 10635 . . . 4  |-  -u 1  e.  CC
2 2rp 11226 . . . . . 6  |-  2  e.  RR+
3 nnrp 11230 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  RR+ )
4 rpdivcl 11244 . . . . . 6  |-  ( ( 2  e.  RR+  /\  N  e.  RR+ )  ->  (
2  /  N )  e.  RR+ )
52, 3, 4sylancr 661 . . . . 5  |-  ( N  e.  NN  ->  (
2  /  N )  e.  RR+ )
65rpcnd 11261 . . . 4  |-  ( N  e.  NN  ->  (
2  /  N )  e.  CC )
7 cxpcl 23223 . . . 4  |-  ( (
-u 1  e.  CC  /\  ( 2  /  N
)  e.  CC )  ->  ( -u 1  ^c  ( 2  /  N ) )  e.  CC )
81, 6, 7sylancr 661 . . 3  |-  ( N  e.  NN  ->  ( -u 1  ^c  ( 2  /  N ) )  e.  CC )
91a1i 11 . . . 4  |-  ( N  e.  NN  ->  -u 1  e.  CC )
10 neg1ne0 10637 . . . . 5  |-  -u 1  =/=  0
1110a1i 11 . . . 4  |-  ( N  e.  NN  ->  -u 1  =/=  0 )
129, 11, 6cxpne0d 23262 . . 3  |-  ( N  e.  NN  ->  ( -u 1  ^c  ( 2  /  N ) )  =/=  0 )
13 eldifsn 4141 . . 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 662 . 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 10801 . . . . . . . . . 10  |-  ( x  e.  NN0  ->  x  e.  CC )
18 mulcl 9565 . . . . . . . . . 10  |-  ( ( ( 2  /  N
)  e.  CC  /\  x  e.  CC )  ->  ( ( 2  /  N )  x.  x
)  e.  CC )
196, 17, 18syl2an 475 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  e.  CC )
2015, 16, 19cxpefd 23261 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^c 
( ( 2  /  N )  x.  x
) )  =  ( exp `  ( ( ( 2  /  N
)  x.  x )  x.  ( log `  -u 1
) ) ) )
2120eqeq1d 2456 . . . . . . 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 23122 . . . . . . . . . 10  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0
)  ->  ( log `  -u 1 )  e.  CC )
231, 10, 22mp2an 670 . . . . . . . . 9  |-  ( log `  -u 1 )  e.  CC
24 mulcl 9565 . . . . . . . . 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 660 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  e.  CC )
26 efeq1 23082 . . . . . . . 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 10602 . . . . . . . . . . . . . 14  |-  2  e.  CC
2928a1i 11 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
2  e.  CC )
30 nncn 10539 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  CC )
3130adantr 463 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  CC )
3217adantl 464 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  CC )
33 nnne0 10564 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  =/=  0 )
3433adantr 463 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  =/=  0 )
3529, 31, 32, 34div13d 10340 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  =  ( ( x  /  N )  x.  2 ) )
36 logm1 23142 . . . . . . . . . . . . 13  |-  ( log `  -u 1 )  =  ( _i  x.  pi )
3736a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( log `  -u 1
)  =  ( _i  x.  pi ) )
3835, 37oveq12d 6288 . . . . . . . . . . 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 10316 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x  /  N
)  e.  CC )
40 ax-icn 9540 . . . . . . . . . . . . . 14  |-  _i  e.  CC
41 picn 23018 . . . . . . . . . . . . . 14  |-  pi  e.  CC
4240, 41mulcli 9590 . . . . . . . . . . . . 13  |-  ( _i  x.  pi )  e.  CC
4342a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  pi )  e.  CC )
4439, 29, 43mulassd 9608 . . . . . . . . . . 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 9778 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( 2  x.  (
_i  x.  pi )
)  =  ( _i  x.  ( 2  x.  pi ) ) )
4847oveq2d 6286 . . . . . . . . . . 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 2499 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  =  ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) ) )
5049oveq1d 6285 . . . . . . . . 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 9590 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  e.  CC
5240, 51mulcli 9590 . . . . . . . . . . 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 9988 . . . . . . . . . . . 12  |-  _i  =/=  0
55 2ne0 10624 . . . . . . . . . . . . 13  |-  2  =/=  0
56 pire 23017 . . . . . . . . . . . . . 14  |-  pi  e.  RR
57 pipos 23019 . . . . . . . . . . . . . 14  |-  0  <  pi
5856, 57gt0ne0ii 10085 . . . . . . . . . . . . 13  |-  pi  =/=  0
5928, 41, 55, 58mulne0i 10188 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  =/=  0
6040, 51, 54, 59mulne0i 10188 . . . . . . . . . . 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 10322 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6350, 62eqtrd 2495 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1
) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6463eleq1d 2523 . . . . . . 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 463 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( 2  /  N
)  e.  CC )
67 simpr 459 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  NN0 )
6815, 66, 67cxpmul2d 23258 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^c 
( ( 2  /  N )  x.  x
) )  =  ( ( -u 1  ^c  ( 2  /  N ) ) ^
x ) )
69 cnfldexp 18646 . . . . . . . . 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 469 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  (mulGrp ` fld ) ) ( -u 1  ^c  ( 2  /  N ) ) )  =  ( (
-u 1  ^c 
( 2  /  N
) ) ^ x
) )
71 cnring 18635 . . . . . . . . . 10  |-fld  e.  Ring
72 cnfldbas 18619 . . . . . . . . . . . 12  |-  CC  =  ( Base ` fld )
73 cnfld0 18637 . . . . . . . . . . . 12  |-  0  =  ( 0g ` fld )
74 cndrng 18642 . . . . . . . . . . . 12  |-fld  e.  DivRing
7572, 73, 74drngui 17597 . . . . . . . . . . 11  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
76 eqid 2454 . . . . . . . . . . 11  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
7775, 76unitsubm 17514 . . . . . . . . . 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 463 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^c 
( 2  /  N
) )  e.  ( CC  \  { 0 } ) )
80 eqid 2454 . . . . . . . . . 10  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
81 proot1ex.g . . . . . . . . . 10  |-  G  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
82 eqid 2454 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
8380, 81, 82submmulg 16376 . . . . . . . . 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 1226 . . . . . . . 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 2502 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  G
) ( -u 1  ^c  ( 2  /  N ) ) )  =  ( -u
1  ^c  ( ( 2  /  N
)  x.  x ) ) )
8685eqeq1d 2456 . . . . . 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 10882 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  ZZ )
8887adantr 463 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  ZZ )
89 nn0z 10883 . . . . . . . 8  |-  ( x  e.  NN0  ->  x  e.  ZZ )
9089adantl 464 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  ZZ )
91 dvdsval2 14073 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0  /\  x  e.  ZZ )  ->  ( N  ||  x  <->  ( x  /  N )  e.  ZZ ) )
9288, 34, 90, 91syl3anc 1226 . . . . . 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 2868 . . . 4  |-  ( N  e.  NN  ->  A. x  e.  NN0  ( N  ||  x 
<->  ( x (.g `  G
) ( -u 1  ^c  ( 2  /  N ) ) )  =  1 ) )
9575, 81unitgrp 17511 . . . . . 6  |-  (fld  e.  Ring  ->  G  e.  Grp )
9671, 95mp1i 12 . . . . 5  |-  ( N  e.  NN  ->  G  e.  Grp )
97 nnnn0 10798 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
9875, 81unitgrpbas 17510 . . . . . 6  |-  ( CC 
\  { 0 } )  =  ( Base `  G )
99 proot1ex.o . . . . . 6  |-  O  =  ( od `  G
)
100 cnfld1 18638 . . . . . . . 8  |-  1  =  ( 1r ` fld )
10175, 81, 100unitgrpid 17513 . . . . . . 7  |-  (fld  e.  Ring  -> 
1  =  ( 0g
`  G ) )
10271, 101ax-mp 5 . . . . . 6  |-  1  =  ( 0g `  G )
10398, 99, 82, 102odeq 16773 . . . . 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 1226 . . . 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 2462 . 2  |-  ( N  e.  NN  ->  ( O `  ( -u 1  ^c  ( 2  /  N ) ) )  =  N )
10798, 99odf 16760 . . . 4  |-  O :
( CC  \  {
0 } ) --> NN0
108 ffn 5713 . . . 4  |-  ( O : ( CC  \  { 0 } ) --> NN0  ->  O  Fn  ( CC  \  { 0 } ) )
109107, 108ax-mp 5 . . 3  |-  O  Fn  ( CC  \  { 0 } )
110 fniniseg 5984 . . 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 920 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804    \ cdif 3458   {csn 4016   class class class wbr 4439   `'ccnv 4987   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482   _ici 9483    x. cmul 9486   -ucneg 9797    / cdiv 10202   NNcn 10531   2c2 10581   NN0cn0 10791   ZZcz 10860   RR+crp 11221   ^cexp 12148   expce 13879   picpi 13884    || cdvds 14070   ↾s cress 14717   0gc0g 14929  SubMndcsubmnd 16164   Grpcgrp 16252  .gcmg 16255   odcod 16748  mulGrpcmgp 17336   Ringcrg 17393  ℂfldccnfld 18615   logclog 23108    ^c ccxp 23109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12982  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-limsup 13376  df-clim 13393  df-rlim 13394  df-sum 13591  df-ef 13885  df-sin 13887  df-cos 13888  df-pi 13890  df-dvds 14071  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-mulg 16259  df-cntz 16554  df-od 16752  df-cmn 16999  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-dvr 17527  df-drng 17593  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-limc 22436  df-dv 22437  df-log 23110  df-cxp 23111
This theorem is referenced by: (None)
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