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Theorem proot1ex 27520
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 9813 . . . 4  |-  -u 1  e.  CC
2 2rp 10359 . . . . . 6  |-  2  e.  RR+
3 nnrp 10363 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  RR+ )
4 rpdivcl 10376 . . . . . 6  |-  ( ( 2  e.  RR+  /\  N  e.  RR+ )  ->  (
2  /  N )  e.  RR+ )
52, 3, 4sylancr 644 . . . . 5  |-  ( N  e.  NN  ->  (
2  /  N )  e.  RR+ )
65rpcnd 10392 . . . 4  |-  ( N  e.  NN  ->  (
2  /  N )  e.  CC )
7 cxpcl 20021 . . . 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 8795 . . . . . 6  |-  1  e.  CC
11 ax-1ne0 8806 . . . . . 6  |-  1  =/=  0
1210, 11negne0i 9121 . . . . 5  |-  -u 1  =/=  0
1312a1i 10 . . . 4  |-  ( N  e.  NN  ->  -u 1  =/=  0 )
149, 13, 6cxpne0d 20060 . . 3  |-  ( N  e.  NN  ->  ( -u 1  ^ c  ( 2  /  N ) )  =/=  0 )
15 eldifsn 3749 . . 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 9975 . . . . . . . . . 10  |-  ( x  e.  NN0  ->  x  e.  CC )
20 mulcl 8821 . . . . . . . . . 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 20059 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^ c 
( ( 2  /  N )  x.  x
) )  =  ( exp `  ( ( ( 2  /  N
)  x.  x )  x.  ( log `  -u 1
) ) ) )
2322eqeq1d 2291 . . . . . . 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 19926 . . . . . . . . . 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 8821 . . . . . . . . 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 19891 . . . . . . . 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 9816 . . . . . . . . . . . . . 14  |-  2  e.  CC
3130a1i 10 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
2  e.  CC )
32 nncn 9754 . . . . . . . . . . . . . 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 9778 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  =/=  0 )
3635adantr 451 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  =/=  0 )
3731, 33, 34, 36div13d 9560 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( 2  /  N )  x.  x
)  =  ( ( x  /  N )  x.  2 ) )
38 logm1 19942 . . . . . . . . . . . . 13  |-  ( log `  -u 1 )  =  ( _i  x.  pi )
3938a1i 10 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( log `  -u 1
)  =  ( _i  x.  pi ) )
4037, 39oveq12d 5876 . . . . . . . . . . 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 9536 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x  /  N
)  e.  CC )
42 ax-icn 8796 . . . . . . . . . . . . . 14  |-  _i  e.  CC
43 pire 19832 . . . . . . . . . . . . . . 15  |-  pi  e.  RR
4443recni 8849 . . . . . . . . . . . . . 14  |-  pi  e.  CC
4542, 44mulcli 8842 . . . . . . . . . . . . 13  |-  ( _i  x.  pi )  e.  CC
4645a1i 10 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( _i  x.  pi )  e.  CC )
4741, 31, 46mulassd 8858 . . . . . . . . . . 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 9021 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( 2  x.  (
_i  x.  pi )
)  =  ( _i  x.  ( 2  x.  pi ) ) )
5150oveq2d 5874 . . . . . . . . . . 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 2319 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1 ) )  =  ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) ) )
5352oveq1d 5873 . . . . . . . . 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 8842 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  e.  CC
5542, 54mulcli 8842 . . . . . . . . . . 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 9215 . . . . . . . . . . . 12  |-  _i  =/=  0
58 2ne0 9829 . . . . . . . . . . . . 13  |-  2  =/=  0
59 pipos 19833 . . . . . . . . . . . . . 14  |-  0  <  pi
6043, 59gt0ne0ii 9309 . . . . . . . . . . . . 13  |-  pi  =/=  0
6130, 44, 58, 60mulne0i 9411 . . . . . . . . . . . 12  |-  ( 2  x.  pi )  =/=  0
6242, 54, 57, 61mulne0i 9411 . . . . . . . . . . 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 9542 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( x  /  N )  x.  ( _i  x.  (
2  x.  pi ) ) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6553, 64eqtrd 2315 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( ( ( ( 2  /  N )  x.  x )  x.  ( log `  -u 1
) )  /  (
_i  x.  ( 2  x.  pi ) ) )  =  ( x  /  N ) )
6665eleq1d 2349 . . . . . . 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 20056 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( -u 1  ^ c 
( ( 2  /  N )  x.  x
) )  =  ( ( -u 1  ^ c  ( 2  /  N ) ) ^
x ) )
71 cnfldexp 16407 . . . . . . . . 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 16396 . . . . . . . . . 10  |-fld  e.  Ring
74 cnfldbas 16383 . . . . . . . . . . . 12  |-  CC  =  ( Base ` fld )
75 cnfld0 16398 . . . . . . . . . . . 12  |-  0  =  ( 0g ` fld )
76 cndrng 16403 . . . . . . . . . . . 12  |-fld  e.  DivRing
7774, 75, 76drngui 15518 . . . . . . . . . . 11  |-  ( CC 
\  { 0 } )  =  (Unit ` fld )
78 eqid 2283 . . . . . . . . . . 11  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
7977, 78unitsubm 15452 . . . . . . . . . 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 2283 . . . . . . . . . 10  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
83 proot1ex.g . . . . . . . . . 10  |-  G  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )
84 eqid 2283 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
8582, 83, 84submmulg 14602 . . . . . . . . 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 2322 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  -> 
( x (.g `  G
) ( -u 1  ^ c  ( 2  /  N ) ) )  =  ( -u
1  ^ c  ( ( 2  /  N
)  x.  x ) ) )
8887eqeq1d 2291 . . . . . 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 10045 . . . . . . . 8  |-  ( N  e.  NN  ->  N  e.  ZZ )
9089adantr 451 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  N  e.  ZZ )
91 nn0z 10046 . . . . . . . 8  |-  ( x  e.  NN0  ->  x  e.  ZZ )
9291adantl 452 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  NN0 )  ->  x  e.  ZZ )
93 dvdsval2 12534 . . . . . . 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 2626 . . . 4  |-  ( N  e.  NN  ->  A. x  e.  NN0  ( N  ||  x 
<->  ( x (.g `  G
) ( -u 1  ^ c  ( 2  /  N ) ) )  =  1 ) )
9777, 83unitgrp 15449 . . . . . 6  |-  (fld  e.  Ring  ->  G  e.  Grp )
9873, 97mp1i 11 . . . . 5  |-  ( N  e.  NN  ->  G  e.  Grp )
99 nnnn0 9972 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN0 )
10077, 83unitgrpbas 15448 . . . . . 6  |-  ( CC 
\  { 0 } )  =  ( Base `  G )
101 proot1ex.o . . . . . 6  |-  O  =  ( od `  G
)
102 cnfld1 16399 . . . . . . . 8  |-  1  =  ( 1r ` fld )
10377, 83, 102unitgrpid 15451 . . . . . . 7  |-  (fld  e.  Ring  -> 
1  =  ( 0g
`  G ) )
10473, 103ax-mp 8 . . . . . 6  |-  1  =  ( 0g `  G )
105100, 101, 84, 104odeq 14865 . . . . 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 2288 . 2  |-  ( N  e.  NN  ->  ( O `  ( -u 1  ^ c  ( 2  /  N ) ) )  =  N )
109100, 101odf 14852 . . . 4  |-  O :
( CC  \  {
0 } ) --> NN0
110 ffn 5389 . . . 4  |-  ( O : ( CC  \  { 0 } ) --> NN0  ->  O  Fn  ( CC  \  { 0 } ) )
111109, 110ax-mp 8 . . 3  |-  O  Fn  ( CC  \  { 0 } )
112 fniniseg 5646 . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149   {csn 3640   class class class wbr 4023   `'ccnv 4688   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738   _ici 8739    x. cmul 8742   -ucneg 9038    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   RR+crp 10354   ^cexp 11104   expce 12343   picpi 12348    || cdivides 12531   ↾s cress 13149   0gc0g 13400   Grpcgrp 14362  .gcmg 14366  SubMndcsubmnd 14414   odcod 14840  mulGrpcmgp 15325   Ringcrg 15337  ℂfldccnfld 16377   logclog 19912    ^ c ccxp 19913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-cntz 14793  df-od 14844  df-cmn 15091  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915
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