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Theorem proot1ex 27182
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 9992 . . . 4  |-  -u 1  e.  CC
2 2rp 10542 . . . . . 6  |-  2  e.  RR+
3 nnrp 10546 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  RR+ )
4 rpdivcl 10559 . . . . . 6  |-  ( ( 2  e.  RR+  /\  N  e.  RR+ )  ->  (
2  /  N )  e.  RR+ )
52, 3, 4sylancr 645 . . . . 5  |-  ( N  e.  NN  ->  (
2  /  N )  e.  RR+ )
65rpcnd 10575 . . . 4  |-  ( N  e.  NN  ->  (
2  /  N )  e.  CC )
7 cxpcl 20425 . . . 4  |-  ( (
-u 1  e.  CC  /\  ( 2  /  N
)  e.  CC )  ->  ( -u 1  ^ c  ( 2  /  N ) )  e.  CC )
81, 6, 7sylancr 645 . . 3  |-  ( N  e.  NN  ->  ( -u 1  ^ c  ( 2  /  N ) )  e.  CC )
91a1i 11 . . . 4  |-  ( N  e.  NN  ->  -u 1  e.  CC )
10 ax-1cn 8974 . . . . . 6  |-  1  e.  CC
11 ax-1ne0 8985 . . . . . 6  |-  1  =/=  0
1210, 11negne0i 9300 . . . . 5  |-  -u 1  =/=  0
1312a1i 11 . . . 4  |-  ( N  e.  NN  ->  -u 1  =/=  0 )
149, 13, 6cxpne0d 20464 . . 3  |-  ( N  e.  NN  ->  ( -u 1  ^ c  ( 2  /  N ) )  =/=  0 )
15 eldifsn 3863 . . 3  |-  ( (
-u 1  ^ c 
( 2  /  N
) )  e.  ( CC  \  { 0 } )  <->  ( ( -u 1  ^ c  ( 2  /  N ) )  e.  CC  /\  ( -u 1  ^ c 
( 2  /  N
) )  =/=  0
) )
168, 14, 15sylanbrc 646 . 2  |-  ( N  e.  NN  ->  ( -u 1  ^ c  ( 2  /  N ) )  e.  ( CC 
\  { 0 } ) )
171a1i 11 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  -u 1  e.  CC )
1812a1i 11 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  -u 1  =/=  0 )
19 nn0cn 10156 . . . . . . . . . 10  |-  ( x  e.  NN0  ->  x  e.  CC )
20 mulcl 9000 . . . . . . . . . 10  |-  ( ( ( 2  /  N
)  e.  CC  /\  x  e.  CC )  ->  ( ( 2  /  N )  x.  x
)  e.  CC )
216, 19, 20syl2an 464 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  e.  CC )
2217, 18, 21cxpefd 20463 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^ c 
( ( 2  /  N )  x.  x
) )  =  ( exp `  ( ( ( 2  /  N
)  x.  x )  x.  ( log `  -u 1
) ) ) )
2322eqeq1d 2388 . . . . . . 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 ) )
24 logcl 20326 . . . . . . . . . 10  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0
)  ->  ( log `  -u 1 )  e.  CC )
251, 12, 24mp2an 654 . . . . . . . . 9  |-  ( log `  -u 1 )  e.  CC
26 mulcl 9000 . . . . . . . . 9  |-  ( ( ( ( 2  /  N )  x.  x
)  e.  CC  /\  ( log `  -u 1
)  e.  CC )  ->  ( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1
) )  e.  CC )
2721, 25, 26sylancl 644 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  e.  CC )
28 efeq1 20291 . . . . . . . 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 ) )
2927, 28syl 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 ) )
30 2cn 9995 . . . . . . . . . . . . . 14  |-  2  e.  CC
3130a1i 11 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
2  e.  CC )
32 nncn 9933 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  CC )
3332adantr 452 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  CC )
3419adantl 453 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  CC )
35 nnne0 9957 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  =/=  0 )
3635adantr 452 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  =/=  0 )
3731, 33, 34, 36div13d 9739 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  =  ( ( x  /  N )  x.  2 ) )
38 logm1 20343 . . . . . . . . . . . . 13  |-  ( log `  -u 1 )  =  ( _i  x.  pi )
3938a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( log `  -u 1
)  =  ( _i  x.  pi ) )
4037, 39oveq12d 6031 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  =  ( ( ( x  /  N )  x.  2 )  x.  ( _i  x.  pi ) ) )
4134, 33, 36divcld 9715 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x  /  N
)  e.  CC )
42 ax-icn 8975 . . . . . . . . . . . . . 14  |-  _i  e.  CC
43 pire 20232 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
4443recni 9028 . . . . . . . . . . . . . 14  |-  pi  e.  CC
4542, 44mulcli 9021 . . . . . . . . . . . . 13  |-  ( _i  x.  pi )  e.  CC
4645a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  pi )  e.  CC )
4741, 31, 46mulassd 9037 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( x  /  N )  x.  2 )  x.  (
_i  x.  pi )
)  =  ( ( x  /  N )  x.  ( 2  x.  ( _i  x.  pi ) ) ) )
4842a1i 11 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  _i  e.  CC )
4944a1i 11 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  pi  e.  CC )
5031, 48, 49mul12d 9200 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( 2  x.  (
_i  x.  pi )
)  =  ( _i  x.  ( 2  x.  pi ) ) )
5150oveq2d 6029 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( x  /  N )  x.  (
2  x.  ( _i  x.  pi ) ) )  =  ( ( x  /  N )  x.  ( _i  x.  ( 2  x.  pi ) ) ) )
5240, 47, 513eqtrd 2416 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  =  ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) ) )
5352oveq1d 6028 . . . . . . . . 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 ) ) ) )
5430, 44mulcli 9021 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  e.  CC
5542, 54mulcli 9021 . . . . . . . . . . 11  |-  ( _i  x.  ( 2  x.  pi ) )  e.  CC
5655a1i 11 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  (
2  x.  pi ) )  e.  CC )
57 ine0 9394 . . . . . . . . . . . 12  |-  _i  =/=  0
58 2ne0 10008 . . . . . . . . . . . . 13  |-  2  =/=  0
59 pipos 20233 . . . . . . . . . . . . . 14  |-  0  <  pi
6043, 59gt0ne0ii 9488 . . . . . . . . . . . . 13  |-  pi  =/=  0
6130, 44, 58, 60mulne0i 9590 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  =/=  0
6242, 54, 57, 61mulne0i 9590 . . . . . . . . . . 11  |-  ( _i  x.  ( 2  x.  pi ) )  =/=  0
6362a1i 11 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  (
2  x.  pi ) )  =/=  0 )
6441, 56, 63divcan4d 9721 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6553, 64eqtrd 2412 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1
) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6665eleq1d 2446 . . . . . . 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 ) )
6723, 29, 663bitrd 271 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( -u 1  ^ c  ( (
2  /  N )  x.  x ) )  =  1  <->  ( x  /  N )  e.  ZZ ) )
686adantr 452 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( 2  /  N
)  e.  CC )
69 simpr 448 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  NN0 )
7017, 68, 69cxpmul2d 20460 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^ c 
( ( 2  /  N )  x.  x
) )  =  ( ( -u 1  ^ c  ( 2  /  N ) ) ^
x ) )
71 cnfldexp 16650 . . . . . . . . 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 ) )
728, 71sylan 458 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  (mulGrp ` fld ) ) ( -u 1  ^ c  ( 2  /  N ) ) )  =  ( (
-u 1  ^ c 
( 2  /  N
) ) ^ x
) )
73 cnrng 16639 . . . . . . . . . 10  |-fld  e.  Ring
74 cnfldbas 16623 . . . . . . . . . . . 12  |-  CC  =  ( Base ` fld )
75 cnfld0 16641 . . . . . . . . . . . 12  |-  0  =  ( 0g ` fld )
76 cndrng 16646 . . . . . . . . . . . 12  |-fld  e.  DivRing
7774, 75, 76drngui 15761 . . . . . . . . . . 11  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
78 eqid 2380 . . . . . . . . . . 11  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
7977, 78unitsubm 15695 . . . . . . . . . 10  |-  (fld  e.  Ring  -> 
( CC  \  {
0 } )  e.  (SubMnd `  (mulGrp ` fld ) ) )
8073, 79mp1i 12 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( CC  \  {
0 } )  e.  (SubMnd `  (mulGrp ` fld ) ) )
8116adantr 452 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^ c 
( 2  /  N
) )  e.  ( CC  \  { 0 } ) )
82 eqid 2380 . . . . . . . . . 10  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
83 proot1ex.g . . . . . . . . . 10  |-  G  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
84 eqid 2380 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
8582, 83, 84submmulg 14845 . . . . . . . . 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
) ) ) )
8680, 69, 81, 85syl3anc 1184 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  (mulGrp ` fld ) ) ( -u 1  ^ c  ( 2  /  N ) ) )  =  ( x (.g `  G ) (
-u 1  ^ c 
( 2  /  N
) ) ) )
8770, 72, 863eqtr2rd 2419 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  G
) ( -u 1  ^ c  ( 2  /  N ) ) )  =  ( -u
1  ^ c  ( ( 2  /  N
)  x.  x ) ) )
8887eqeq1d 2388 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( x (.g `  G ) ( -u
1  ^ c  ( 2  /  N ) ) )  =  1  <-> 
( -u 1  ^ c 
( ( 2  /  N )  x.  x
) )  =  1 ) )
89 nnz 10228 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  ZZ )
9089adantr 452 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  ZZ )
91 nn0z 10229 . . . . . . . 8  |-  ( x  e.  NN0  ->  x  e.  ZZ )
9291adantl 453 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  ZZ )
93 dvdsval2 12775 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0  /\  x  e.  ZZ )  ->  ( N  ||  x  <->  ( x  /  N )  e.  ZZ ) )
9490, 36, 92, 93syl3anc 1184 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( N  ||  x  <->  ( x  /  N )  e.  ZZ ) )
9567, 88, 943bitr4rd 278 . . . . 5  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( N  ||  x  <->  ( x (.g `  G ) (
-u 1  ^ c 
( 2  /  N
) ) )  =  1 ) )
9695ralrimiva 2725 . . . 4  |-  ( N  e.  NN  ->  A. x  e.  NN0  ( N  ||  x 
<->  ( x (.g `  G
) ( -u 1  ^ c  ( 2  /  N ) ) )  =  1 ) )
9777, 83unitgrp 15692 . . . . . 6  |-  (fld  e.  Ring  ->  G  e.  Grp )
9873, 97mp1i 12 . . . . 5  |-  ( N  e.  NN  ->  G  e.  Grp )
99 nnnn0 10153 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
10077, 83unitgrpbas 15691 . . . . . 6  |-  ( CC 
\  { 0 } )  =  ( Base `  G )
101 proot1ex.o . . . . . 6  |-  O  =  ( od `  G
)
102 cnfld1 16642 . . . . . . . 8  |-  1  =  ( 1r ` fld )
10377, 83, 102unitgrpid 15694 . . . . . . 7  |-  (fld  e.  Ring  -> 
1  =  ( 0g
`  G ) )
10473, 103ax-mp 8 . . . . . 6  |-  1  =  ( 0g `  G )
105100, 101, 84, 104odeq 15108 . . . . 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 ) ) )
10698, 16, 99, 105syl3anc 1184 . . . 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 ) ) )
10796, 106mpbird 224 . . 3  |-  ( N  e.  NN  ->  N  =  ( O `  ( -u 1  ^ c 
( 2  /  N
) ) ) )
108107eqcomd 2385 . 2  |-  ( N  e.  NN  ->  ( O `  ( -u 1  ^ c  ( 2  /  N ) ) )  =  N )
109100, 101odf 15095 . . . 4  |-  O :
( CC  \  {
0 } ) --> NN0
110 ffn 5524 . . . 4  |-  ( O : ( CC  \  { 0 } ) --> NN0  ->  O  Fn  ( CC  \  { 0 } ) )
111109, 110ax-mp 8 . . 3  |-  O  Fn  ( CC  \  { 0 } )
112 fniniseg 5783 . . 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 ) ) )
113111, 112mp1i 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 ) ) )
11416, 108, 113mpbir2and 889 1  |-  ( N  e.  NN  ->  ( -u 1  ^ c  ( 2  /  N ) )  e.  ( `' O " { N } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543   A.wral 2642    \ cdif 3253   {csn 3750   class class class wbr 4146   `'ccnv 4810   "cima 4814    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013   CCcc 8914   0cc0 8916   1c1 8917   _ici 8918    x. cmul 8921   -ucneg 9217    / cdiv 9602   NNcn 9925   2c2 9974   NN0cn0 10146   ZZcz 10207   RR+crp 10537   ^cexp 11302   expce 12584   picpi 12589    || cdivides 12772   ↾s cress 13390   0gc0g 13643   Grpcgrp 14605  .gcmg 14609  SubMndcsubmnd 14657   odcod 15083  mulGrpcmgp 15568   Ringcrg 15580  ℂfldccnfld 16619   logclog 20312    ^ c ccxp 20313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994  ax-addf 8995  ax-mulf 8996
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-tpos 6408  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-map 6949  df-pm 6950  df-ixp 6993  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-fi 7344  df-sup 7374  df-oi 7405  df-card 7752  df-cda 7974  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-q 10500  df-rp 10538  df-xneg 10635  df-xadd 10636  df-xmul 10637  df-ioo 10845  df-ioc 10846  df-ico 10847  df-icc 10848  df-fz 10969  df-fzo 11059  df-fl 11122  df-mod 11171  df-seq 11244  df-exp 11303  df-fac 11487  df-bc 11514  df-hash 11539  df-shft 11802  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-limsup 12185  df-clim 12202  df-rlim 12203  df-sum 12400  df-ef 12590  df-sin 12592  df-cos 12593  df-pi 12595  df-dvds 12773  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-starv 13464  df-sca 13465  df-vsca 13466  df-tset 13468  df-ple 13469  df-ds 13471  df-unif 13472  df-hom 13473  df-cco 13474  df-rest 13570  df-topn 13571  df-topgen 13587  df-pt 13588  df-prds 13591  df-xrs 13646  df-0g 13647  df-gsum 13648  df-qtop 13653  df-imas 13654  df-xps 13656  df-mre 13731  df-mrc 13732  df-acs 13734  df-mnd 14610  df-submnd 14659  df-grp 14732  df-minusg 14733  df-sbg 14734  df-mulg 14735  df-cntz 15036  df-od 15087  df-cmn 15334  df-mgp 15569  df-rng 15583  df-cring 15584  df-ur 15585  df-oppr 15648  df-dvdsr 15666  df-unit 15667  df-invr 15697  df-dvr 15708  df-drng 15757  df-xmet 16612  df-met 16613  df-bl 16614  df-mopn 16615  df-fbas 16616  df-fg 16617  df-cnfld 16620  df-top 16879  df-bases 16881  df-topon 16882  df-topsp 16883  df-cld 16999  df-ntr 17000  df-cls 17001  df-nei 17078  df-lp 17116  df-perf 17117  df-cn 17206  df-cnp 17207  df-haus 17294  df-tx 17508  df-hmeo 17701  df-fil 17792  df-fm 17884  df-flim 17885  df-flf 17886  df-xms 18252  df-ms 18253  df-tms 18254  df-cncf 18772  df-limc 19613  df-dv 19614  df-log 20314  df-cxp 20315
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