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Theorem proot1ex 27623
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 9829 . . . 4  |-  -u 1  e.  CC
2 2rp 10375 . . . . . 6  |-  2  e.  RR+
3 nnrp 10379 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  RR+ )
4 rpdivcl 10392 . . . . . 6  |-  ( ( 2  e.  RR+  /\  N  e.  RR+ )  ->  (
2  /  N )  e.  RR+ )
52, 3, 4sylancr 644 . . . . 5  |-  ( N  e.  NN  ->  (
2  /  N )  e.  RR+ )
65rpcnd 10408 . . . 4  |-  ( N  e.  NN  ->  (
2  /  N )  e.  CC )
7 cxpcl 20037 . . . 4  |-  ( (
-u 1  e.  CC  /\  ( 2  /  N
)  e.  CC )  ->  ( -u 1  ^ c  ( 2  /  N ) )  e.  CC )
81, 6, 7sylancr 644 . . 3  |-  ( N  e.  NN  ->  ( -u 1  ^ c  ( 2  /  N ) )  e.  CC )
91a1i 10 . . . 4  |-  ( N  e.  NN  ->  -u 1  e.  CC )
10 ax-1cn 8811 . . . . . 6  |-  1  e.  CC
11 ax-1ne0 8822 . . . . . 6  |-  1  =/=  0
1210, 11negne0i 9137 . . . . 5  |-  -u 1  =/=  0
1312a1i 10 . . . 4  |-  ( N  e.  NN  ->  -u 1  =/=  0 )
149, 13, 6cxpne0d 20076 . . 3  |-  ( N  e.  NN  ->  ( -u 1  ^ c  ( 2  /  N ) )  =/=  0 )
15 eldifsn 3762 . . 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 645 . 2  |-  ( N  e.  NN  ->  ( -u 1  ^ c  ( 2  /  N ) )  e.  ( CC 
\  { 0 } ) )
171a1i 10 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  -u 1  e.  CC )
1812a1i 10 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  -u 1  =/=  0 )
19 nn0cn 9991 . . . . . . . . . 10  |-  ( x  e.  NN0  ->  x  e.  CC )
20 mulcl 8837 . . . . . . . . . 10  |-  ( ( ( 2  /  N
)  e.  CC  /\  x  e.  CC )  ->  ( ( 2  /  N )  x.  x
)  e.  CC )
216, 19, 20syl2an 463 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  e.  CC )
2217, 18, 21cxpefd 20075 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^ c 
( ( 2  /  N )  x.  x
) )  =  ( exp `  ( ( ( 2  /  N
)  x.  x )  x.  ( log `  -u 1
) ) ) )
2322eqeq1d 2304 . . . . . . 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 19942 . . . . . . . . . 10  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0
)  ->  ( log `  -u 1 )  e.  CC )
251, 12, 24mp2an 653 . . . . . . . . 9  |-  ( log `  -u 1 )  e.  CC
26 mulcl 8837 . . . . . . . . 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 643 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  e.  CC )
28 efeq1 19907 . . . . . . . 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 15 . . . . . . 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 9832 . . . . . . . . . . . . . 14  |-  2  e.  CC
3130a1i 10 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
2  e.  CC )
32 nncn 9770 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  CC )
3332adantr 451 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  CC )
3419adantl 452 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  CC )
35 nnne0 9794 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  =/=  0 )
3635adantr 451 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  =/=  0 )
3731, 33, 34, 36div13d 9576 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  =  ( ( x  /  N )  x.  2 ) )
38 logm1 19958 . . . . . . . . . . . . 13  |-  ( log `  -u 1 )  =  ( _i  x.  pi )
3938a1i 10 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( log `  -u 1
)  =  ( _i  x.  pi ) )
4037, 39oveq12d 5892 . . . . . . . . . . 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 9552 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x  /  N
)  e.  CC )
42 ax-icn 8812 . . . . . . . . . . . . . 14  |-  _i  e.  CC
43 pire 19848 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
4443recni 8865 . . . . . . . . . . . . . 14  |-  pi  e.  CC
4542, 44mulcli 8858 . . . . . . . . . . . . 13  |-  ( _i  x.  pi )  e.  CC
4645a1i 10 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  pi )  e.  CC )
4741, 31, 46mulassd 8874 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( x  /  N )  x.  2 )  x.  (
_i  x.  pi )
)  =  ( ( x  /  N )  x.  ( 2  x.  ( _i  x.  pi ) ) ) )
4842a1i 10 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  _i  e.  CC )
4944a1i 10 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  pi  e.  CC )
5031, 48, 49mul12d 9037 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( 2  x.  (
_i  x.  pi )
)  =  ( _i  x.  ( 2  x.  pi ) ) )
5150oveq2d 5890 . . . . . . . . . . 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 2332 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  =  ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) ) )
5352oveq1d 5889 . . . . . . . . 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 8858 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  e.  CC
5542, 54mulcli 8858 . . . . . . . . . . 11  |-  ( _i  x.  ( 2  x.  pi ) )  e.  CC
5655a1i 10 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  (
2  x.  pi ) )  e.  CC )
57 ine0 9231 . . . . . . . . . . . 12  |-  _i  =/=  0
58 2ne0 9845 . . . . . . . . . . . . 13  |-  2  =/=  0
59 pipos 19849 . . . . . . . . . . . . . 14  |-  0  <  pi
6043, 59gt0ne0ii 9325 . . . . . . . . . . . . 13  |-  pi  =/=  0
6130, 44, 58, 60mulne0i 9427 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  =/=  0
6242, 54, 57, 61mulne0i 9427 . . . . . . . . . . 11  |-  ( _i  x.  ( 2  x.  pi ) )  =/=  0
6362a1i 10 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  (
2  x.  pi ) )  =/=  0 )
6441, 56, 63divcan4d 9558 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6553, 64eqtrd 2328 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1
) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6665eleq1d 2362 . . . . . . 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 270 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( -u 1  ^ c  ( (
2  /  N )  x.  x ) )  =  1  <->  ( x  /  N )  e.  ZZ ) )
686adantr 451 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( 2  /  N
)  e.  CC )
69 simpr 447 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  NN0 )
7017, 68, 69cxpmul2d 20072 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^ c 
( ( 2  /  N )  x.  x
) )  =  ( ( -u 1  ^ c  ( 2  /  N ) ) ^
x ) )
71 cnfldexp 16423 . . . . . . . . 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 457 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  (mulGrp ` fld ) ) ( -u 1  ^ c  ( 2  /  N ) ) )  =  ( (
-u 1  ^ c 
( 2  /  N
) ) ^ x
) )
73 cnrng 16412 . . . . . . . . . 10  |-fld  e.  Ring
74 cnfldbas 16399 . . . . . . . . . . . 12  |-  CC  =  ( Base ` fld )
75 cnfld0 16414 . . . . . . . . . . . 12  |-  0  =  ( 0g ` fld )
76 cndrng 16419 . . . . . . . . . . . 12  |-fld  e.  DivRing
7774, 75, 76drngui 15534 . . . . . . . . . . 11  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
78 eqid 2296 . . . . . . . . . . 11  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
7977, 78unitsubm 15468 . . . . . . . . . 10  |-  (fld  e.  Ring  -> 
( CC  \  {
0 } )  e.  (SubMnd `  (mulGrp ` fld ) ) )
8073, 79mp1i 11 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( CC  \  {
0 } )  e.  (SubMnd `  (mulGrp ` fld ) ) )
8116adantr 451 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^ c 
( 2  /  N
) )  e.  ( CC  \  { 0 } ) )
82 eqid 2296 . . . . . . . . . 10  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
83 proot1ex.g . . . . . . . . . 10  |-  G  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
84 eqid 2296 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
8582, 83, 84submmulg 14618 . . . . . . . . 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 1182 . . . . . . . 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 2335 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  G
) ( -u 1  ^ c  ( 2  /  N ) ) )  =  ( -u
1  ^ c  ( ( 2  /  N
)  x.  x ) ) )
8887eqeq1d 2304 . . . . . 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 10061 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  ZZ )
9089adantr 451 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  ZZ )
91 nn0z 10062 . . . . . . . 8  |-  ( x  e.  NN0  ->  x  e.  ZZ )
9291adantl 452 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  ZZ )
93 dvdsval2 12550 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0  /\  x  e.  ZZ )  ->  ( N  ||  x  <->  ( x  /  N )  e.  ZZ ) )
9490, 36, 92, 93syl3anc 1182 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( N  ||  x  <->  ( x  /  N )  e.  ZZ ) )
9567, 88, 943bitr4rd 277 . . . . 5  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( N  ||  x  <->  ( x (.g `  G ) (
-u 1  ^ c 
( 2  /  N
) ) )  =  1 ) )
9695ralrimiva 2639 . . . 4  |-  ( N  e.  NN  ->  A. x  e.  NN0  ( N  ||  x 
<->  ( x (.g `  G
) ( -u 1  ^ c  ( 2  /  N ) ) )  =  1 ) )
9777, 83unitgrp 15465 . . . . . 6  |-  (fld  e.  Ring  ->  G  e.  Grp )
9873, 97mp1i 11 . . . . 5  |-  ( N  e.  NN  ->  G  e.  Grp )
99 nnnn0 9988 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
10077, 83unitgrpbas 15464 . . . . . 6  |-  ( CC 
\  { 0 } )  =  ( Base `  G )
101 proot1ex.o . . . . . 6  |-  O  =  ( od `  G
)
102 cnfld1 16415 . . . . . . . 8  |-  1  =  ( 1r ` fld )
10377, 83, 102unitgrpid 15467 . . . . . . 7  |-  (fld  e.  Ring  -> 
1  =  ( 0g
`  G ) )
10473, 103ax-mp 8 . . . . . 6  |-  1  =  ( 0g `  G )
105100, 101, 84, 104odeq 14881 . . . . 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 1182 . . . 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 223 . . 3  |-  ( N  e.  NN  ->  N  =  ( O `  ( -u 1  ^ c 
( 2  /  N
) ) ) )
108107eqcomd 2301 . 2  |-  ( N  e.  NN  ->  ( O `  ( -u 1  ^ c  ( 2  /  N ) ) )  =  N )
109100, 101odf 14868 . . . 4  |-  O :
( CC  \  {
0 } ) --> NN0
110 ffn 5405 . . . 4  |-  ( O : ( CC  \  { 0 } ) --> NN0  ->  O  Fn  ( CC  \  { 0 } ) )
111109, 110ax-mp 8 . . 3  |-  O  Fn  ( CC  \  { 0 } )
112 fniniseg 5662 . . 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 11 . 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 888 1  |-  ( N  e.  NN  ->  ( -u 1  ^ c  ( 2  /  N ) )  e.  ( `' O " { N } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    \ cdif 3162   {csn 3653   class class class wbr 4039   `'ccnv 4704   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754   _ici 8755    x. cmul 8758   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   RR+crp 10370   ^cexp 11120   expce 12359   picpi 12364    || cdivides 12547   ↾s cress 13165   0gc0g 13416   Grpcgrp 14378  .gcmg 14382  SubMndcsubmnd 14430   odcod 14856  mulGrpcmgp 15341   Ringcrg 15353  ℂfldccnfld 16393   logclog 19928    ^ c ccxp 19929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-dvds 12548  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-cntz 14809  df-od 14860  df-cmn 15107  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cxp 19931
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