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Theorem proot1ex 27430
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 10051 . . . 4  |-  -u 1  e.  CC
2 2rp 10601 . . . . . 6  |-  2  e.  RR+
3 nnrp 10605 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  RR+ )
4 rpdivcl 10618 . . . . . 6  |-  ( ( 2  e.  RR+  /\  N  e.  RR+ )  ->  (
2  /  N )  e.  RR+ )
52, 3, 4sylancr 645 . . . . 5  |-  ( N  e.  NN  ->  (
2  /  N )  e.  RR+ )
65rpcnd 10634 . . . 4  |-  ( N  e.  NN  ->  (
2  /  N )  e.  CC )
7 cxpcl 20548 . . . 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 9032 . . . . . 6  |-  1  e.  CC
11 ax-1ne0 9043 . . . . . 6  |-  1  =/=  0
1210, 11negne0i 9359 . . . . 5  |-  -u 1  =/=  0
1312a1i 11 . . . 4  |-  ( N  e.  NN  ->  -u 1  =/=  0 )
149, 13, 6cxpne0d 20587 . . 3  |-  ( N  e.  NN  ->  ( -u 1  ^ c  ( 2  /  N ) )  =/=  0 )
15 eldifsn 3914 . . 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 10215 . . . . . . . . . 10  |-  ( x  e.  NN0  ->  x  e.  CC )
20 mulcl 9058 . . . . . . . . . 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 20586 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^ c 
( ( 2  /  N )  x.  x
) )  =  ( exp `  ( ( ( 2  /  N
)  x.  x )  x.  ( log `  -u 1
) ) ) )
2322eqeq1d 2438 . . . . . . 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 20449 . . . . . . . . . 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 9058 . . . . . . . . 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 20414 . . . . . . . 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 10054 . . . . . . . . . . . . . 14  |-  2  e.  CC
3130a1i 11 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
2  e.  CC )
32 nncn 9992 . . . . . . . . . . . . . 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 10016 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  =/=  0 )
3635adantr 452 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  =/=  0 )
3731, 33, 34, 36div13d 9798 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  =  ( ( x  /  N )  x.  2 ) )
38 logm1 20466 . . . . . . . . . . . . 13  |-  ( log `  -u 1 )  =  ( _i  x.  pi )
3938a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( log `  -u 1
)  =  ( _i  x.  pi ) )
4037, 39oveq12d 6085 . . . . . . . . . . 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 9774 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x  /  N
)  e.  CC )
42 ax-icn 9033 . . . . . . . . . . . . . 14  |-  _i  e.  CC
43 pire 20355 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
4443recni 9086 . . . . . . . . . . . . . 14  |-  pi  e.  CC
4542, 44mulcli 9079 . . . . . . . . . . . . 13  |-  ( _i  x.  pi )  e.  CC
4645a1i 11 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  pi )  e.  CC )
4741, 31, 46mulassd 9095 . . . . . . . . . . 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 9259 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( 2  x.  (
_i  x.  pi )
)  =  ( _i  x.  ( 2  x.  pi ) ) )
5150oveq2d 6083 . . . . . . . . . . 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 2466 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  =  ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) ) )
5352oveq1d 6082 . . . . . . . . 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 9079 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  e.  CC
5542, 54mulcli 9079 . . . . . . . . . . 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 9453 . . . . . . . . . . . 12  |-  _i  =/=  0
58 2ne0 10067 . . . . . . . . . . . . 13  |-  2  =/=  0
59 pipos 20356 . . . . . . . . . . . . . 14  |-  0  <  pi
6043, 59gt0ne0ii 9547 . . . . . . . . . . . . 13  |-  pi  =/=  0
6130, 44, 58, 60mulne0i 9649 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  =/=  0
6242, 54, 57, 61mulne0i 9649 . . . . . . . . . . 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 9780 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6553, 64eqtrd 2462 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1
) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6665eleq1d 2496 . . . . . . 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 20583 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^ c 
( ( 2  /  N )  x.  x
) )  =  ( ( -u 1  ^ c  ( 2  /  N ) ) ^
x ) )
71 cnfldexp 16717 . . . . . . . . 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 16706 . . . . . . . . . 10  |-fld  e.  Ring
74 cnfldbas 16690 . . . . . . . . . . . 12  |-  CC  =  ( Base ` fld )
75 cnfld0 16708 . . . . . . . . . . . 12  |-  0  =  ( 0g ` fld )
76 cndrng 16713 . . . . . . . . . . . 12  |-fld  e.  DivRing
7774, 75, 76drngui 15824 . . . . . . . . . . 11  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
78 eqid 2430 . . . . . . . . . . 11  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
7977, 78unitsubm 15758 . . . . . . . . . 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 2430 . . . . . . . . . 10  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
83 proot1ex.g . . . . . . . . . 10  |-  G  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
84 eqid 2430 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
8582, 83, 84submmulg 14908 . . . . . . . . 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 2469 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  G
) ( -u 1  ^ c  ( 2  /  N ) ) )  =  ( -u
1  ^ c  ( ( 2  /  N
)  x.  x ) ) )
8887eqeq1d 2438 . . . . . 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 10287 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  ZZ )
9089adantr 452 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  ZZ )
91 nn0z 10288 . . . . . . . 8  |-  ( x  e.  NN0  ->  x  e.  ZZ )
9291adantl 453 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  ZZ )
93 dvdsval2 12838 . . . . . . 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 2776 . . . 4  |-  ( N  e.  NN  ->  A. x  e.  NN0  ( N  ||  x 
<->  ( x (.g `  G
) ( -u 1  ^ c  ( 2  /  N ) ) )  =  1 ) )
9777, 83unitgrp 15755 . . . . . 6  |-  (fld  e.  Ring  ->  G  e.  Grp )
9873, 97mp1i 12 . . . . 5  |-  ( N  e.  NN  ->  G  e.  Grp )
99 nnnn0 10212 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
10077, 83unitgrpbas 15754 . . . . . 6  |-  ( CC 
\  { 0 } )  =  ( Base `  G )
101 proot1ex.o . . . . . 6  |-  O  =  ( od `  G
)
102 cnfld1 16709 . . . . . . . 8  |-  1  =  ( 1r ` fld )
10377, 83, 102unitgrpid 15757 . . . . . . 7  |-  (fld  e.  Ring  -> 
1  =  ( 0g
`  G ) )
10473, 103ax-mp 8 . . . . . 6  |-  1  =  ( 0g `  G )
105100, 101, 84, 104odeq 15171 . . . . 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 2435 . 2  |-  ( N  e.  NN  ->  ( O `  ( -u 1  ^ c  ( 2  /  N ) ) )  =  N )
109100, 101odf 15158 . . . 4  |-  O :
( CC  \  {
0 } ) --> NN0
110 ffn 5577 . . . 4  |-  ( O : ( CC  \  { 0 } ) --> NN0  ->  O  Fn  ( CC  \  { 0 } ) )
111109, 110ax-mp 8 . . 3  |-  O  Fn  ( CC  \  { 0 } )
112 fniniseg 5837 . . 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 1652    e. wcel 1725    =/= wne 2593   A.wral 2692    \ cdif 3304   {csn 3801   class class class wbr 4199   `'ccnv 4863   "cima 4867    Fn wfn 5435   -->wf 5436   ` cfv 5440  (class class class)co 6067   CCcc 8972   0cc0 8974   1c1 8975   _ici 8976    x. cmul 8979   -ucneg 9276    / cdiv 9661   NNcn 9984   2c2 10033   NN0cn0 10205   ZZcz 10266   RR+crp 10596   ^cexp 11365   expce 12647   picpi 12652    || cdivides 12835   ↾s cress 13453   0gc0g 13706   Grpcgrp 14668  .gcmg 14672  SubMndcsubmnd 14720   odcod 15146  mulGrpcmgp 15631   Ringcrg 15643  ℂfldccnfld 16686   logclog 20435    ^ c ccxp 20436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-inf2 7580  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051  ax-pre-sup 9052  ax-addf 9053  ax-mulf 9054
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-iin 4083  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-se 4529  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-isom 5449  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-of 6291  df-1st 6335  df-2nd 6336  df-tpos 6465  df-riota 6535  df-recs 6619  df-rdg 6654  df-1o 6710  df-2o 6711  df-oadd 6714  df-er 6891  df-map 7006  df-pm 7007  df-ixp 7050  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-fi 7402  df-sup 7432  df-oi 7463  df-card 7810  df-cda 8032  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-div 9662  df-nn 9985  df-2 10042  df-3 10043  df-4 10044  df-5 10045  df-6 10046  df-7 10047  df-8 10048  df-9 10049  df-10 10050  df-n0 10206  df-z 10267  df-dec 10367  df-uz 10473  df-q 10559  df-rp 10597  df-xneg 10694  df-xadd 10695  df-xmul 10696  df-ioo 10904  df-ioc 10905  df-ico 10906  df-icc 10907  df-fz 11028  df-fzo 11119  df-fl 11185  df-mod 11234  df-seq 11307  df-exp 11366  df-fac 11550  df-bc 11577  df-hash 11602  df-shft 11865  df-cj 11887  df-re 11888  df-im 11889  df-sqr 12023  df-abs 12024  df-limsup 12248  df-clim 12265  df-rlim 12266  df-sum 12463  df-ef 12653  df-sin 12655  df-cos 12656  df-pi 12658  df-dvds 12836  df-struct 13454  df-ndx 13455  df-slot 13456  df-base 13457  df-sets 13458  df-ress 13459  df-plusg 13525  df-mulr 13526  df-starv 13527  df-sca 13528  df-vsca 13529  df-tset 13531  df-ple 13532  df-ds 13534  df-unif 13535  df-hom 13536  df-cco 13537  df-rest 13633  df-topn 13634  df-topgen 13650  df-pt 13651  df-prds 13654  df-xrs 13709  df-0g 13710  df-gsum 13711  df-qtop 13716  df-imas 13717  df-xps 13719  df-mre 13794  df-mrc 13795  df-acs 13797  df-mnd 14673  df-submnd 14722  df-grp 14795  df-minusg 14796  df-sbg 14797  df-mulg 14798  df-cntz 15099  df-od 15150  df-cmn 15397  df-mgp 15632  df-rng 15646  df-cring 15647  df-ur 15648  df-oppr 15711  df-dvdsr 15729  df-unit 15730  df-invr 15760  df-dvr 15771  df-drng 15820  df-psmet 16677  df-xmet 16678  df-met 16679  df-bl 16680  df-mopn 16681  df-fbas 16682  df-fg 16683  df-cnfld 16687  df-top 16946  df-bases 16948  df-topon 16949  df-topsp 16950  df-cld 17066  df-ntr 17067  df-cls 17068  df-nei 17145  df-lp 17183  df-perf 17184  df-cn 17274  df-cnp 17275  df-haus 17362  df-tx 17577  df-hmeo 17770  df-fil 17861  df-fm 17953  df-flim 17954  df-flf 17955  df-xms 18333  df-ms 18334  df-tms 18335  df-cncf 18891  df-limc 19736  df-dv 19737  df-log 20437  df-cxp 20438
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