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Theorem odcau 15193
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 10193 . . . 4  |-  1  e.  NN0
65a1i 11 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  1  e.  NN0 )
7 prmnn 13037 . . . . . . 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 9972 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  P  e.  CC )
109exp1d 11473 . . . 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 4192 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  ( P ^ 1 )  ||  ( # `  X ) )
131, 2, 3, 4, 6, 12sylow1 15192 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  E. s  e.  (SubGrp `  G )
( # `  s )  =  ( P ^
1 ) )
1410eqeq2d 2415 . . . . 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 5701 . . . . . . . . . . . 12  |-  ( 0g
`  G )  e. 
_V
17 hashsng 11602 . . . . . . . . . . . 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 13052 . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( # `  s
)  e.  ( ZZ>= ` 
2 ) )
24 eluz2b2 10504 . . . . . . . . . . . . 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 4205 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( # `  {
( 0g `  G
) } )  < 
( # `  s ) )
28 snfi 7146 . . . . . . . . . . 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 14900 . . . . . . . . . . . . 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 7288 . . . . . . . . . . . 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 11610 . . . . . . . . . . 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 7214 . . . . . . . . 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 3597 . . . . . . . 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 3887 . . . . . . . . 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 3309 . . . . . . . . . . 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 15160 . . . . . . . . . . . . . . . . . 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 4196 . . . . . . . . . . . . . . . 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 15156 . . . . . . . . . . . . . . . . . 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 13047 . . . . . . . . . . . . . . . . 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 2404 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  G )  =  ( 0g `  G
)
6348, 62, 1odeq1 15151 . . . . . . . . . . . . . . 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 2634 . . . . . . . . . . . 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 2672 . . . . . 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 2790 . 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 2567   E.wrex 2667   _Vcvv 2916    \ cdif 3277    C_ wss 3280   (/)c0 3588   {csn 3774   class class class wbr 4172   ` cfv 5413  (class class class)co 6040    ~< csdm 7067   Fincfn 7068   1c1 8947    < clt 9076   NNcn 9956   2c2 10005   NN0cn0 10177   ZZ>=cuz 10444   ^cexp 11337   #chash 11573    || cdivides 12807   Primecprime 13034   Basecbs 13424   0gc0g 13678   Grpcgrp 14640  SubGrpcsubg 14893   odcod 15118
This theorem is referenced by:  pgpfi  15194  ablfacrplem  15578
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-eqg 14898  df-ga 15022  df-od 15122
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