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Theorem odcau 16096
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 986 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  G  e.  Grp )
3 simpl2 987 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  X  e.  Fin )
4 simpl3 988 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  P  e.  Prime )
5 1nn0 10591 . . . 4  |-  1  e.  NN0
65a1i 11 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  1  e.  NN0 )
7 prmnn 13762 . . . . . . 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 10334 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  P  e.  CC )
109exp1d 11999 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  ( P ^ 1 )  =  P )
11 simpr 458 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  P  ||  ( # `  X
) )
1210, 11eqbrtrd 4309 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  ( P ^ 1 )  ||  ( # `  X ) )
131, 2, 3, 4, 6, 12sylow1 16095 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  E. s  e.  (SubGrp `  G )
( # `  s )  =  ( P ^
1 ) )
1410eqeq2d 2452 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  (
( # `  s )  =  ( P ^
1 )  <->  ( # `  s
)  =  P ) )
1514adantr 462 . . . 4  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  s  e.  (SubGrp `  G ) )  -> 
( ( # `  s
)  =  ( P ^ 1 )  <->  ( # `  s
)  =  P ) )
16 fvex 5698 . . . . . . . . . . . 12  |-  ( 0g
`  G )  e. 
_V
17 hashsng 12132 . . . . . . . . . . . 12  |-  ( ( 0g `  G )  e.  _V  ->  ( # `
 { ( 0g
`  G ) } )  =  1 )
1816, 17ax-mp 5 . . . . . . . . . . 11  |-  ( # `  { ( 0g `  G ) } )  =  1
19 simprr 751 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( # `  s
)  =  P )
204adantr 462 . . . . . . . . . . . . . 14  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  P  e.  Prime )
21 prmuz2 13777 . . . . . . . . . . . . . 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 2515 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( # `  s
)  e.  ( ZZ>= ` 
2 ) )
24 eluz2b2 10923 . . . . . . . . . . . . 13  |-  ( (
# `  s )  e.  ( ZZ>= `  2 )  <->  ( ( # `  s
)  e.  NN  /\  1  <  ( # `  s
) ) )
2524simprbi 461 . . . . . . . . . . . 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 4322 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( # `  {
( 0g `  G
) } )  < 
( # `  s ) )
28 snfi 7386 . . . . . . . . . . 11  |-  { ( 0g `  G ) }  e.  Fin
293adantr 462 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  X  e.  Fin )
301subgss 15675 . . . . . . . . . . . . 13  |-  ( s  e.  (SubGrp `  G
)  ->  s  C_  X )
3130ad2antrl 722 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  s  C_  X )
32 ssfi 7529 . . . . . . . . . . . 12  |-  ( ( X  e.  Fin  /\  s  C_  X )  -> 
s  e.  Fin )
3329, 31, 32syl2anc 656 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  s  e.  Fin )
34 hashsdom 12140 . . . . . . . . . . 11  |-  ( ( { ( 0g `  G ) }  e.  Fin  /\  s  e.  Fin )  ->  ( ( # `  { ( 0g `  G ) } )  <  ( # `  s
)  <->  { ( 0g `  G ) }  ~<  s ) )
3528, 33, 34sylancr 658 . . . . . . . . . 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 7455 . . . . . . . . 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 3643 . . . . . . . 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 3997 . . . . . . . . 9  |-  ( g  e.  ( s  \  { ( 0g `  G ) } )  <-> 
( g  e.  s  /\  g  =/=  ( 0g `  G ) ) )
4231adantrr 711 . . . . . . . . . . . 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 758 . . . . . . . . . . . 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 3354 . . . . . . . . . . 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 759 . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . . . . 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 16063 . . . . . . . . . . . . . . . . . 18  |-  ( ( s  e.  (SubGrp `  G )  /\  s  e.  Fin  /\  g  e.  s )  ->  ( O `  g )  ||  ( # `  s
) )
5046, 47, 43, 49syl3anc 1213 . . . . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . . 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 4313 . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . 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 16059 . . . . . . . . . . . . . . . . . 18  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  g  e.  X )  ->  ( O `  g )  e.  NN )
5754, 55, 44, 56syl3anc 1213 . . . . . . . . . . . . . . . . 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 13772 . . . . . . . . . . . . . . . . 17  |-  ( ( P  e.  Prime  /\  ( O `  g )  e.  NN )  ->  (
( O `  g
)  ||  P  <->  ( ( O `  g )  =  P  \/  ( O `  g )  =  1 ) ) )
5953, 57, 58syl2anc 656 . . . . . . . . . . . . . . . 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 2441 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  G )  =  ( 0g `  G
)
6348, 62, 1odeq1 16054 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  Grp  /\  g  e.  X )  ->  ( ( O `  g )  =  1  <-> 
g  =  ( 0g
`  G ) ) )
6454, 44, 63syl2anc 656 . . . . . . . . . . . . . 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 2676 . . . . . . . . . . . 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 529 . . . . . . . . . 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 612 . . . . . . . . 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 1681 . . . . . . 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 2719 . . . . . 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 612 . . . 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 2839 . 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 960    = wceq 1364   E.wex 1591    e. wcel 1761    =/= wne 2604   E.wrex 2714   _Vcvv 2970    \ cdif 3322    C_ wss 3325   (/)c0 3634   {csn 3874   class class class wbr 4289   ` cfv 5415  (class class class)co 6090    ~< csdm 7305   Fincfn 7306   1c1 9279    < clt 9414   NNcn 10318   2c2 10367   NN0cn0 10575   ZZ>=cuz 10857   ^cexp 11861   #chash 12099    || cdivides 13531   Primecprime 13759   Basecbs 14170   0gc0g 14374   Grpcgrp 15406  SubGrpcsubg 15668   odcod 16021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-disj 4260  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-omul 6921  df-er 7097  df-ec 7099  df-qs 7103  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-acn 8108  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-dvds 13532  df-gcd 13687  df-prm 13760  df-pc 13900  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-mnd 15411  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-eqg 15673  df-ga 15801  df-od 16025
This theorem is referenced by:  pgpfi  16097  ablfacrplem  16556
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