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Theorem odcau 15201
Description: Cauchy's theorem for the order of an element in a group. A finite group whose order divides a prime 
P contains an element of order  P. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypotheses
Ref Expression
odcau.x  |-  X  =  ( Base `  G
)
odcau.o  |-  O  =  ( od `  G
)
Assertion
Ref Expression
odcau  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  E. g  e.  X  ( O `  g )  =  P )
Distinct variable groups:    g, G    P, g    g, X
Allowed substitution hint:    O( g)

Proof of Theorem odcau
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 odcau.x . . 3  |-  X  =  ( Base `  G
)
2 simpl1 960 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  G  e.  Grp )
3 simpl2 961 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  X  e.  Fin )
4 simpl3 962 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  P  e.  Prime )
5 1nn0 10201 . . . 4  |-  1  e.  NN0
65a1i 11 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  1  e.  NN0 )
7 prmnn 13045 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
84, 7syl 16 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  P  e.  NN )
98nncnd 9980 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  P  e.  CC )
109exp1d 11481 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  ( P ^ 1 )  =  P )
11 simpr 448 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  P  ||  ( # `  X
) )
1210, 11eqbrtrd 4200 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  ( P ^ 1 )  ||  ( # `  X ) )
131, 2, 3, 4, 6, 12sylow1 15200 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  E. s  e.  (SubGrp `  G )
( # `  s )  =  ( P ^
1 ) )
1410eqeq2d 2423 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  (
( # `  s )  =  ( P ^
1 )  <->  ( # `  s
)  =  P ) )
1514adantr 452 . . . 4  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  s  e.  (SubGrp `  G ) )  -> 
( ( # `  s
)  =  ( P ^ 1 )  <->  ( # `  s
)  =  P ) )
16 fvex 5709 . . . . . . . . . . . 12  |-  ( 0g
`  G )  e. 
_V
17 hashsng 11610 . . . . . . . . . . . 12  |-  ( ( 0g `  G )  e.  _V  ->  ( # `
 { ( 0g
`  G ) } )  =  1 )
1816, 17ax-mp 8 . . . . . . . . . . 11  |-  ( # `  { ( 0g `  G ) } )  =  1
19 simprr 734 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( # `  s
)  =  P )
204adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  P  e.  Prime )
21 prmuz2 13060 . . . . . . . . . . . . . 14  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
2220, 21syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  P  e.  ( ZZ>= `  2 )
)
2319, 22eqeltrd 2486 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( # `  s
)  e.  ( ZZ>= ` 
2 ) )
24 eluz2b2 10512 . . . . . . . . . . . . 13  |-  ( (
# `  s )  e.  ( ZZ>= `  2 )  <->  ( ( # `  s
)  e.  NN  /\  1  <  ( # `  s
) ) )
2524simprbi 451 . . . . . . . . . . . 12  |-  ( (
# `  s )  e.  ( ZZ>= `  2 )  ->  1  <  ( # `  s ) )
2623, 25syl 16 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  1  <  (
# `  s )
)
2718, 26syl5eqbr 4213 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( # `  {
( 0g `  G
) } )  < 
( # `  s ) )
28 snfi 7154 . . . . . . . . . . 11  |-  { ( 0g `  G ) }  e.  Fin
293adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  X  e.  Fin )
301subgss 14908 . . . . . . . . . . . . 13  |-  ( s  e.  (SubGrp `  G
)  ->  s  C_  X )
3130ad2antrl 709 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  s  C_  X )
32 ssfi 7296 . . . . . . . . . . . 12  |-  ( ( X  e.  Fin  /\  s  C_  X )  -> 
s  e.  Fin )
3329, 31, 32syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  s  e.  Fin )
34 hashsdom 11618 . . . . . . . . . . 11  |-  ( ( { ( 0g `  G ) }  e.  Fin  /\  s  e.  Fin )  ->  ( ( # `  { ( 0g `  G ) } )  <  ( # `  s
)  <->  { ( 0g `  G ) }  ~<  s ) )
3528, 33, 34sylancr 645 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( ( # `
 { ( 0g
`  G ) } )  <  ( # `  s )  <->  { ( 0g `  G ) } 
~<  s ) )
3627, 35mpbid 202 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  { ( 0g `  G ) } 
~<  s )
37 sdomdif 7222 . . . . . . . . 9  |-  ( { ( 0g `  G
) }  ~<  s  ->  ( s  \  {
( 0g `  G
) } )  =/=  (/) )
3836, 37syl 16 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( s  \  { ( 0g `  G ) } )  =/=  (/) )
39 n0 3605 . . . . . . . 8  |-  ( ( s  \  { ( 0g `  G ) } )  =/=  (/)  <->  E. g 
g  e.  ( s 
\  { ( 0g
`  G ) } ) )
4038, 39sylib 189 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  E. g 
g  e.  ( s 
\  { ( 0g
`  G ) } ) )
41 eldifsn 3895 . . . . . . . . 9  |-  ( g  e.  ( s  \  { ( 0g `  G ) } )  <-> 
( g  e.  s  /\  g  =/=  ( 0g `  G ) ) )
4231adantrr 698 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  s  C_  X
)
43 simprrl 741 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  g  e.  s )
4442, 43sseldd 3317 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  g  e.  X
)
45 simprrr 742 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  g  =/=  ( 0g `  G ) )
46 simprll 739 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  s  e.  (SubGrp `  G ) )
4733adantrr 698 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  s  e.  Fin )
48 odcau.o . . . . . . . . . . . . . . . . . . 19  |-  O  =  ( od `  G
)
4948odsubdvds 15168 . . . . . . . . . . . . . . . . . 18  |-  ( ( s  e.  (SubGrp `  G )  /\  s  e.  Fin  /\  g  e.  s )  ->  ( O `  g )  ||  ( # `  s
) )
5046, 47, 43, 49syl3anc 1184 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  ( O `  g )  ||  ( # `
 s ) )
51 simprlr 740 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  ( # `  s
)  =  P )
5250, 51breqtrd 4204 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  ( O `  g )  ||  P
)
534adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  P  e.  Prime )
542adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  G  e.  Grp )
553adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  X  e.  Fin )
561, 48odcl2 15164 . . . . . . . . . . . . . . . . . 18  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  g  e.  X )  ->  ( O `  g )  e.  NN )
5754, 55, 44, 56syl3anc 1184 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  ( O `  g )  e.  NN )
58 dvdsprime 13055 . . . . . . . . . . . . . . . . 17  |-  ( ( P  e.  Prime  /\  ( O `  g )  e.  NN )  ->  (
( O `  g
)  ||  P  <->  ( ( O `  g )  =  P  \/  ( O `  g )  =  1 ) ) )
5953, 57, 58syl2anc 643 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  ( ( O `
 g )  ||  P 
<->  ( ( O `  g )  =  P  \/  ( O `  g )  =  1 ) ) )
6052, 59mpbid 202 . . . . . . . . . . . . . . 15  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  ( ( O `
 g )  =  P  \/  ( O `
 g )  =  1 ) )
6160ord 367 . . . . . . . . . . . . . 14  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  ( -.  ( O `  g )  =  P  ->  ( O `
 g )  =  1 ) )
62 eqid 2412 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  G )  =  ( 0g `  G
)
6348, 62, 1odeq1 15159 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  Grp  /\  g  e.  X )  ->  ( ( O `  g )  =  1  <-> 
g  =  ( 0g
`  G ) ) )
6454, 44, 63syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  ( ( O `
 g )  =  1  <->  g  =  ( 0g `  G ) ) )
6561, 64sylibd 206 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  ( -.  ( O `  g )  =  P  ->  g  =  ( 0g `  G
) ) )
6665necon1ad 2642 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  ( g  =/=  ( 0g `  G
)  ->  ( O `  g )  =  P ) )
6745, 66mpd 15 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  ( O `  g )  =  P )
6844, 67jca 519 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( ( s  e.  (SubGrp `  G
)  /\  ( # `  s
)  =  P )  /\  ( g  e.  s  /\  g  =/=  ( 0g `  G
) ) ) )  ->  ( g  e.  X  /\  ( O `
 g )  =  P ) )
6968expr 599 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( (
g  e.  s  /\  g  =/=  ( 0g `  G ) )  -> 
( g  e.  X  /\  ( O `  g
)  =  P ) ) )
7041, 69syl5bi 209 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( g  e.  ( s  \  {
( 0g `  G
) } )  -> 
( g  e.  X  /\  ( O `  g
)  =  P ) ) )
7170eximdv 1629 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( E. g  g  e.  (
s  \  { ( 0g `  G ) } )  ->  E. g
( g  e.  X  /\  ( O `  g
)  =  P ) ) )
7240, 71mpd 15 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  E. g
( g  e.  X  /\  ( O `  g
)  =  P ) )
73 df-rex 2680 . . . . . 6  |-  ( E. g  e.  X  ( O `  g )  =  P  <->  E. g
( g  e.  X  /\  ( O `  g
)  =  P ) )
7472, 73sylibr 204 . . . . 5  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  E. g  e.  X  ( O `  g )  =  P )
7574expr 599 . . . 4  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  s  e.  (SubGrp `  G ) )  -> 
( ( # `  s
)  =  P  ->  E. g  e.  X  ( O `  g )  =  P ) )
7615, 75sylbid 207 . . 3  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  s  e.  (SubGrp `  G ) )  -> 
( ( # `  s
)  =  ( P ^ 1 )  ->  E. g  e.  X  ( O `  g )  =  P ) )
7776rexlimdva 2798 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  ( E. s  e.  (SubGrp `  G ) ( # `  s )  =  ( P ^ 1 )  ->  E. g  e.  X  ( O `  g )  =  P ) )
7813, 77mpd 15 1  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  E. g  e.  X  ( O `  g )  =  P )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675   _Vcvv 2924    \ cdif 3285    C_ wss 3288   (/)c0 3596   {csn 3782   class class class wbr 4180   ` cfv 5421  (class class class)co 6048    ~< csdm 7075   Fincfn 7076   1c1 8955    < clt 9084   NNcn 9964   2c2 10013   NN0cn0 10185   ZZ>=cuz 10452   ^cexp 11345   #chash 11581    || cdivides 12815   Primecprime 13042   Basecbs 13432   0gc0g 13686   Grpcgrp 14648  SubGrpcsubg 14901   odcod 15126
This theorem is referenced by:  pgpfi  15202  ablfacrplem  15586
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-disj 4151  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6872  df-ec 6874  df-qs 6878  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-acn 7793  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-q 10539  df-rp 10577  df-fz 11008  df-fzo 11099  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-fac 11530  df-bc 11557  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-sum 12443  df-dvds 12816  df-gcd 12970  df-prm 13043  df-pc 13174  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-0g 13690  df-mnd 14653  df-submnd 14702  df-grp 14775  df-minusg 14776  df-sbg 14777  df-mulg 14778  df-subg 14904  df-eqg 14906  df-ga 15030  df-od 15130
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