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Theorem odcau 16427
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 999 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  G  e.  Grp )
3 simpl2 1000 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  X  e.  Fin )
4 simpl3 1001 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  P  e.  Prime )
5 1nn0 10810 . . . 4  |-  1  e.  NN0
65a1i 11 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  1  e.  NN0 )
7 prmnn 14078 . . . . . . 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 10551 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  P  e.  CC )
109exp1d 12272 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  ( P ^ 1 )  =  P )
11 simpr 461 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  P  ||  ( # `  X
) )
1210, 11eqbrtrd 4467 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  ( P ^ 1 )  ||  ( # `  X ) )
131, 2, 3, 4, 6, 12sylow1 16426 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  E. s  e.  (SubGrp `  G )
( # `  s )  =  ( P ^
1 ) )
1410eqeq2d 2481 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  (
( # `  s )  =  ( P ^
1 )  <->  ( # `  s
)  =  P ) )
1514adantr 465 . . . 4  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  s  e.  (SubGrp `  G ) )  -> 
( ( # `  s
)  =  ( P ^ 1 )  <->  ( # `  s
)  =  P ) )
16 fvex 5875 . . . . . . . . . . . 12  |-  ( 0g
`  G )  e. 
_V
17 hashsng 12405 . . . . . . . . . . . 12  |-  ( ( 0g `  G )  e.  _V  ->  ( # `
 { ( 0g
`  G ) } )  =  1 )
1816, 17ax-mp 5 . . . . . . . . . . 11  |-  ( # `  { ( 0g `  G ) } )  =  1
19 simprr 756 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( # `  s
)  =  P )
204adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  P  e.  Prime )
21 prmuz2 14093 . . . . . . . . . . . . . 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 2555 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( # `  s
)  e.  ( ZZ>= ` 
2 ) )
24 eluz2b2 11153 . . . . . . . . . . . . 13  |-  ( (
# `  s )  e.  ( ZZ>= `  2 )  <->  ( ( # `  s
)  e.  NN  /\  1  <  ( # `  s
) ) )
2524simprbi 464 . . . . . . . . . . . 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 4480 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( # `  {
( 0g `  G
) } )  < 
( # `  s ) )
28 snfi 7596 . . . . . . . . . . 11  |-  { ( 0g `  G ) }  e.  Fin
293adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  X  e.  Fin )
301subgss 16004 . . . . . . . . . . . . 13  |-  ( s  e.  (SubGrp `  G
)  ->  s  C_  X )
3130ad2antrl 727 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  s  C_  X )
32 ssfi 7740 . . . . . . . . . . . 12  |-  ( ( X  e.  Fin  /\  s  C_  X )  -> 
s  e.  Fin )
3329, 31, 32syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  s  e.  Fin )
34 hashsdom 12416 . . . . . . . . . . 11  |-  ( ( { ( 0g `  G ) }  e.  Fin  /\  s  e.  Fin )  ->  ( ( # `  { ( 0g `  G ) } )  <  ( # `  s
)  <->  { ( 0g `  G ) }  ~<  s ) )
3528, 33, 34sylancr 663 . . . . . . . . . 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 210 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  { ( 0g `  G ) } 
~<  s )
37 sdomdif 7665 . . . . . . . . 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 3794 . . . . . . . 8  |-  ( ( s  \  { ( 0g `  G ) } )  =/=  (/)  <->  E. g 
g  e.  ( s 
\  { ( 0g
`  G ) } ) )
4038, 39sylib 196 . . . . . . 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 4152 . . . . . . . . 9  |-  ( g  e.  ( s  \  { ( 0g `  G ) } )  <-> 
( g  e.  s  /\  g  =/=  ( 0g `  G ) ) )
4231adantrr 716 . . . . . . . . . . . 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 763 . . . . . . . . . . . 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 3505 . . . . . . . . . . 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 764 . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . . . . 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 16394 . . . . . . . . . . . . . . . . . 18  |-  ( ( s  e.  (SubGrp `  G )  /\  s  e.  Fin  /\  g  e.  s )  ->  ( O `  g )  ||  ( # `  s
) )
5046, 47, 43, 49syl3anc 1228 . . . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . . 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 4471 . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . 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 16390 . . . . . . . . . . . . . . . . . 18  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  g  e.  X )  ->  ( O `  g )  e.  NN )
5754, 55, 44, 56syl3anc 1228 . . . . . . . . . . . . . . . . 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 14088 . . . . . . . . . . . . . . . . 17  |-  ( ( P  e.  Prime  /\  ( O `  g )  e.  NN )  ->  (
( O `  g
)  ||  P  <->  ( ( O `  g )  =  P  \/  ( O `  g )  =  1 ) ) )
5953, 57, 58syl2anc 661 . . . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . . . 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 377 . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  G )  =  ( 0g `  G
)
6348, 62, 1odeq1 16385 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  Grp  /\  g  e.  X )  ->  ( ( O `  g )  =  1  <-> 
g  =  ( 0g
`  G ) ) )
6454, 44, 63syl2anc 661 . . . . . . . . . . . . . 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 214 . . . . . . . . . . . . 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 2683 . . . . . . . . . . . 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 532 . . . . . . . . . 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 615 . . . . . . . . 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 217 . . . . . . . 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 1686 . . . . . . 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 2820 . . . . . 6  |-  ( E. g  e.  X  ( O `  g )  =  P  <->  E. g
( g  e.  X  /\  ( O `  g
)  =  P ) )
7472, 73sylibr 212 . . . . 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 615 . . . 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 215 . . 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 2955 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   E.wrex 2815   _Vcvv 3113    \ cdif 3473    C_ wss 3476   (/)c0 3785   {csn 4027   class class class wbr 4447   ` cfv 5587  (class class class)co 6283    ~< csdm 7515   Fincfn 7516   1c1 9492    < clt 9627   NNcn 10535   2c2 10584   NN0cn0 10794   ZZ>=cuz 11081   ^cexp 12133   #chash 12372    || cdivides 13846   Primecprime 14075   Basecbs 14489   0gc0g 14694   Grpcgrp 15726  SubGrpcsubg 15997   odcod 16352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-omul 7135  df-er 7311  df-ec 7313  df-qs 7317  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-oi 7934  df-card 8319  df-acn 8322  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-n0 10795  df-z 10864  df-uz 11082  df-q 11182  df-rp 11220  df-fz 11672  df-fzo 11792  df-fl 11896  df-mod 11964  df-seq 12075  df-exp 12134  df-fac 12321  df-bc 12348  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-sum 13471  df-dvds 13847  df-gcd 14003  df-prm 14076  df-pc 14219  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-0g 14696  df-mnd 15731  df-submnd 15784  df-grp 15864  df-minusg 15865  df-sbg 15866  df-mulg 15867  df-subg 16000  df-eqg 16002  df-ga 16130  df-od 16356
This theorem is referenced by:  pgpfi  16428  ablfacrplem  16915
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