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Theorem proot1ex 27535
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 10098 . . . 4  |-  -u 1  e.  CC
2 2rp 10648 . . . . . 6  |-  2  e.  RR+
3 nnrp 10652 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  RR+ )
4 rpdivcl 10665 . . . . . 6  |-  ( ( 2  e.  RR+  /\  N  e.  RR+ )  ->  (
2  /  N )  e.  RR+ )
52, 3, 4sylancr 646 . . . . 5  |-  ( N  e.  NN  ->  (
2  /  N )  e.  RR+ )
65rpcnd 10681 . . . 4  |-  ( N  e.  NN  ->  (
2  /  N )  e.  CC )
7 cxpcl 20596 . . . 4  |-  ( (
-u 1  e.  CC  /\  ( 2  /  N
)  e.  CC )  ->  ( -u 1  ^ c  ( 2  /  N ) )  e.  CC )
81, 6, 7sylancr 646 . . 3  |-  ( N  e.  NN  ->  ( -u 1  ^ c  ( 2  /  N ) )  e.  CC )
91a1i 11 . . . 4  |-  ( N  e.  NN  ->  -u 1  e.  CC )
10 ax-1cn 9079 . . . . . 6  |-  1  e.  CC
11 ax-1ne0 9090 . . . . . 6  |-  1  =/=  0
1210, 11negne0i 9406 . . . . 5  |-  -u 1  =/=  0
1312a1i 11 . . . 4  |-  ( N  e.  NN  ->  -u 1  =/=  0 )
149, 13, 6cxpne0d 20635 . . 3  |-  ( N  e.  NN  ->  ( -u 1  ^ c  ( 2  /  N ) )  =/=  0 )
15 eldifsn 3951 . . 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 647 . 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 10262 . . . . . . . . . 10  |-  ( x  e.  NN0  ->  x  e.  CC )
20 mulcl 9105 . . . . . . . . . 10  |-  ( ( ( 2  /  N
)  e.  CC  /\  x  e.  CC )  ->  ( ( 2  /  N )  x.  x
)  e.  CC )
216, 19, 20syl2an 465 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  e.  CC )
2217, 18, 21cxpefd 20634 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^ c 
( ( 2  /  N )  x.  x
) )  =  ( exp `  ( ( ( 2  /  N
)  x.  x )  x.  ( log `  -u 1
) ) ) )
2322eqeq1d 2450 . . . . . . 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 20497 . . . . . . . . . 10  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0
)  ->  ( log `  -u 1 )  e.  CC )
251, 12, 24mp2an 655 . . . . . . . . 9  |-  ( log `  -u 1 )  e.  CC
26 mulcl 9105 . . . . . . . . 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 645 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  e.  CC )
28 efeq1 20462 . . . . . . . 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 10101 . . . . . . . . . . . . . 14  |-  2  e.  CC
3130a1i 11 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
2  e.  CC )
32 nncn 10039 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  CC )
3332adantr 453 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  CC )
3419adantl 454 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  CC )
35 nnne0 10063 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  =/=  0 )
3635adantr 453 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  =/=  0 )
3731, 33, 34, 36div13d 9845 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  =  ( ( x  /  N )  x.  2 ) )
38 logm1 20514 . . . . . . . . . . . . 13  |-  ( log `  -u 1 )  =  ( _i  x.  pi )
3938a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( log `  -u 1
)  =  ( _i  x.  pi ) )
4037, 39oveq12d 6128 . . . . . . . . . . 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 9821 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x  /  N
)  e.  CC )
42 ax-icn 9080 . . . . . . . . . . . . . 14  |-  _i  e.  CC
43 pire 20403 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
4443recni 9133 . . . . . . . . . . . . . 14  |-  pi  e.  CC
4542, 44mulcli 9126 . . . . . . . . . . . . 13  |-  ( _i  x.  pi )  e.  CC
4645a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  pi )  e.  CC )
4741, 31, 46mulassd 9142 . . . . . . . . . . 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 9306 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( 2  x.  (
_i  x.  pi )
)  =  ( _i  x.  ( 2  x.  pi ) ) )
5150oveq2d 6126 . . . . . . . . . . 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 2478 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  =  ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) ) )
5352oveq1d 6125 . . . . . . . . 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 9126 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  e.  CC
5542, 54mulcli 9126 . . . . . . . . . . 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 9500 . . . . . . . . . . . 12  |-  _i  =/=  0
58 2ne0 10114 . . . . . . . . . . . . 13  |-  2  =/=  0
59 pipos 20404 . . . . . . . . . . . . . 14  |-  0  <  pi
6043, 59gt0ne0ii 9594 . . . . . . . . . . . . 13  |-  pi  =/=  0
6130, 44, 58, 60mulne0i 9696 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  =/=  0
6242, 54, 57, 61mulne0i 9696 . . . . . . . . . . 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 9827 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6553, 64eqtrd 2474 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1
) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6665eleq1d 2508 . . . . . . 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 272 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( -u 1  ^ c  ( (
2  /  N )  x.  x ) )  =  1  <->  ( x  /  N )  e.  ZZ ) )
686adantr 453 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( 2  /  N
)  e.  CC )
69 simpr 449 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  NN0 )
7017, 68, 69cxpmul2d 20631 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^ c 
( ( 2  /  N )  x.  x
) )  =  ( ( -u 1  ^ c  ( 2  /  N ) ) ^
x ) )
71 cnfldexp 16765 . . . . . . . . 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 459 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  (mulGrp ` fld ) ) ( -u 1  ^ c  ( 2  /  N ) ) )  =  ( (
-u 1  ^ c 
( 2  /  N
) ) ^ x
) )
73 cnrng 16754 . . . . . . . . . 10  |-fld  e.  Ring
74 cnfldbas 16738 . . . . . . . . . . . 12  |-  CC  =  ( Base ` fld )
75 cnfld0 16756 . . . . . . . . . . . 12  |-  0  =  ( 0g ` fld )
76 cndrng 16761 . . . . . . . . . . . 12  |-fld  e.  DivRing
7774, 75, 76drngui 15872 . . . . . . . . . . 11  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
78 eqid 2442 . . . . . . . . . . 11  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
7977, 78unitsubm 15806 . . . . . . . . . 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 453 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^ c 
( 2  /  N
) )  e.  ( CC  \  { 0 } ) )
82 eqid 2442 . . . . . . . . . 10  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
83 proot1ex.g . . . . . . . . . 10  |-  G  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
84 eqid 2442 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
8582, 83, 84submmulg 14956 . . . . . . . . 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 1185 . . . . . . . 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 2481 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  G
) ( -u 1  ^ c  ( 2  /  N ) ) )  =  ( -u
1  ^ c  ( ( 2  /  N
)  x.  x ) ) )
8887eqeq1d 2450 . . . . . 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 10334 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  ZZ )
9089adantr 453 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  ZZ )
91 nn0z 10335 . . . . . . . 8  |-  ( x  e.  NN0  ->  x  e.  ZZ )
9291adantl 454 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  ZZ )
93 dvdsval2 12886 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  N  =/=  0  /\  x  e.  ZZ )  ->  ( N  ||  x  <->  ( x  /  N )  e.  ZZ ) )
9490, 36, 92, 93syl3anc 1185 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( N  ||  x  <->  ( x  /  N )  e.  ZZ ) )
9567, 88, 943bitr4rd 279 . . . . 5  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( N  ||  x  <->  ( x (.g `  G ) (
-u 1  ^ c 
( 2  /  N
) ) )  =  1 ) )
9695ralrimiva 2795 . . . 4  |-  ( N  e.  NN  ->  A. x  e.  NN0  ( N  ||  x 
<->  ( x (.g `  G
) ( -u 1  ^ c  ( 2  /  N ) ) )  =  1 ) )
9777, 83unitgrp 15803 . . . . . 6  |-  (fld  e.  Ring  ->  G  e.  Grp )
9873, 97mp1i 12 . . . . 5  |-  ( N  e.  NN  ->  G  e.  Grp )
99 nnnn0 10259 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
10077, 83unitgrpbas 15802 . . . . . 6  |-  ( CC 
\  { 0 } )  =  ( Base `  G )
101 proot1ex.o . . . . . 6  |-  O  =  ( od `  G
)
102 cnfld1 16757 . . . . . . . 8  |-  1  =  ( 1r ` fld )
10377, 83, 102unitgrpid 15805 . . . . . . 7  |-  (fld  e.  Ring  -> 
1  =  ( 0g
`  G ) )
10473, 103ax-mp 5 . . . . . 6  |-  1  =  ( 0g `  G )
105100, 101, 84, 104odeq 15219 . . . . 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 1185 . . . 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 225 . . 3  |-  ( N  e.  NN  ->  N  =  ( O `  ( -u 1  ^ c 
( 2  /  N
) ) ) )
108107eqcomd 2447 . 2  |-  ( N  e.  NN  ->  ( O `  ( -u 1  ^ c  ( 2  /  N ) ) )  =  N )
109100, 101odf 15206 . . . 4  |-  O :
( CC  \  {
0 } ) --> NN0
110 ffn 5620 . . . 4  |-  ( O : ( CC  \  { 0 } ) --> NN0  ->  O  Fn  ( CC  \  { 0 } ) )
111109, 110ax-mp 5 . . 3  |-  O  Fn  ( CC  \  { 0 } )
112 fniniseg 5880 . . 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 890 1  |-  ( N  e.  NN  ->  ( -u 1  ^ c  ( 2  /  N ) )  e.  ( `' O " { N } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605   A.wral 2711    \ cdif 3303   {csn 3838   class class class wbr 4237   `'ccnv 4906   "cima 4910    Fn wfn 5478   -->wf 5479   ` cfv 5483  (class class class)co 6110   CCcc 9019   0cc0 9021   1c1 9022   _ici 9023    x. cmul 9026   -ucneg 9323    / cdiv 9708   NNcn 10031   2c2 10080   NN0cn0 10252   ZZcz 10313   RR+crp 10643   ^cexp 11413   expce 12695   picpi 12700    || cdivides 12883   ↾s cress 13501   0gc0g 13754   Grpcgrp 14716  .gcmg 14720  SubMndcsubmnd 14768   odcod 15194  mulGrpcmgp 15679   Ringcrg 15691  ℂfldccnfld 16734   logclog 20483    ^ c ccxp 20484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099  ax-addf 9100  ax-mulf 9101
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334  df-1st 6378  df-2nd 6379  df-tpos 6508  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-2o 6754  df-oadd 6757  df-er 6934  df-map 7049  df-pm 7050  df-ixp 7093  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-fi 7445  df-sup 7475  df-oi 7508  df-card 7857  df-cda 8079  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-10 10097  df-n0 10253  df-z 10314  df-dec 10414  df-uz 10520  df-q 10606  df-rp 10644  df-xneg 10741  df-xadd 10742  df-xmul 10743  df-ioo 10951  df-ioc 10952  df-ico 10953  df-icc 10954  df-fz 11075  df-fzo 11167  df-fl 11233  df-mod 11282  df-seq 11355  df-exp 11414  df-fac 11598  df-bc 11625  df-hash 11650  df-shft 11913  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072  df-limsup 12296  df-clim 12313  df-rlim 12314  df-sum 12511  df-ef 12701  df-sin 12703  df-cos 12704  df-pi 12706  df-dvds 12884  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-plusg 13573  df-mulr 13574  df-starv 13575  df-sca 13576  df-vsca 13577  df-tset 13579  df-ple 13580  df-ds 13582  df-unif 13583  df-hom 13584  df-cco 13585  df-rest 13681  df-topn 13682  df-topgen 13698  df-pt 13699  df-prds 13702  df-xrs 13757  df-0g 13758  df-gsum 13759  df-qtop 13764  df-imas 13765  df-xps 13767  df-mre 13842  df-mrc 13843  df-acs 13845  df-mnd 14721  df-submnd 14770  df-grp 14843  df-minusg 14844  df-sbg 14845  df-mulg 14846  df-cntz 15147  df-od 15198  df-cmn 15445  df-mgp 15680  df-rng 15694  df-cring 15695  df-ur 15696  df-oppr 15759  df-dvdsr 15777  df-unit 15778  df-invr 15808  df-dvr 15819  df-drng 15868  df-psmet 16725  df-xmet 16726  df-met 16727  df-bl 16728  df-mopn 16729  df-fbas 16730  df-fg 16731  df-cnfld 16735  df-top 16994  df-bases 16996  df-topon 16997  df-topsp 16998  df-cld 17114  df-ntr 17115  df-cls 17116  df-nei 17193  df-lp 17231  df-perf 17232  df-cn 17322  df-cnp 17323  df-haus 17410  df-tx 17625  df-hmeo 17818  df-fil 17909  df-fm 18001  df-flim 18002  df-flf 18003  df-xms 18381  df-ms 18382  df-tms 18383  df-cncf 18939  df-limc 19784  df-dv 19785  df-log 20485  df-cxp 20486
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