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Theorem odcau 15125
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 959 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  G  e.  Grp )
3 simpl2 960 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  X  e.  Fin )
4 simpl3 961 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  P  e.  Prime )
5 1nn0 10130 . . . 4  |-  1  e.  NN0
65a1i 10 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  1  e.  NN0 )
7 prmnn 12969 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  NN )
84, 7syl 15 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  P  e.  NN )
98nncnd 9909 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  P  e.  CC )
109exp1d 11405 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  ( P ^ 1 )  =  P )
11 simpr 447 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  P  ||  ( # `  X
) )
1210, 11eqbrtrd 4145 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  ( P ^ 1 )  ||  ( # `  X ) )
131, 2, 3, 4, 6, 12sylow1 15124 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  E. s  e.  (SubGrp `  G )
( # `  s )  =  ( P ^
1 ) )
1410eqeq2d 2377 . . . . 5  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X
) )  ->  (
( # `  s )  =  ( P ^
1 )  <->  ( # `  s
)  =  P ) )
1514adantr 451 . . . 4  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  s  e.  (SubGrp `  G ) )  -> 
( ( # `  s
)  =  ( P ^ 1 )  <->  ( # `  s
)  =  P ) )
16 fvex 5646 . . . . . . . . . . . 12  |-  ( 0g
`  G )  e. 
_V
17 hashsng 11534 . . . . . . . . . . . 12  |-  ( ( 0g `  G )  e.  _V  ->  ( # `
 { ( 0g
`  G ) } )  =  1 )
1816, 17ax-mp 8 . . . . . . . . . . 11  |-  ( # `  { ( 0g `  G ) } )  =  1
19 simprr 733 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( # `  s
)  =  P )
204adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  P  e.  Prime )
21 prmuz2 12984 . . . . . . . . . . . . . 14  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
2220, 21syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  P  e.  ( ZZ>= `  2 )
)
2319, 22eqeltrd 2440 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( # `  s
)  e.  ( ZZ>= ` 
2 ) )
24 eluz2b2 10441 . . . . . . . . . . . . 13  |-  ( (
# `  s )  e.  ( ZZ>= `  2 )  <->  ( ( # `  s
)  e.  NN  /\  1  <  ( # `  s
) ) )
2524simprbi 450 . . . . . . . . . . . 12  |-  ( (
# `  s )  e.  ( ZZ>= `  2 )  ->  1  <  ( # `  s ) )
2623, 25syl 15 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  1  <  (
# `  s )
)
2718, 26syl5eqbr 4158 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( # `  {
( 0g `  G
) } )  < 
( # `  s ) )
28 snfi 7084 . . . . . . . . . . 11  |-  { ( 0g `  G ) }  e.  Fin
293adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  X  e.  Fin )
301subgss 14832 . . . . . . . . . . . . 13  |-  ( s  e.  (SubGrp `  G
)  ->  s  C_  X )
3130ad2antrl 708 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  s  C_  X )
32 ssfi 7226 . . . . . . . . . . . 12  |-  ( ( X  e.  Fin  /\  s  C_  X )  -> 
s  e.  Fin )
3329, 31, 32syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  s  e.  Fin )
34 hashsdom 11542 . . . . . . . . . . 11  |-  ( ( { ( 0g `  G ) }  e.  Fin  /\  s  e.  Fin )  ->  ( ( # `  { ( 0g `  G ) } )  <  ( # `  s
)  <->  { ( 0g `  G ) }  ~<  s ) )
3528, 33, 34sylancr 644 . . . . . . . . . 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 201 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  { ( 0g `  G ) } 
~<  s )
37 sdomdif 7152 . . . . . . . . 9  |-  ( { ( 0g `  G
) }  ~<  s  ->  ( s  \  {
( 0g `  G
) } )  =/=  (/) )
3836, 37syl 15 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `
 X ) )  /\  ( s  e.  (SubGrp `  G )  /\  ( # `  s
)  =  P ) )  ->  ( s  \  { ( 0g `  G ) } )  =/=  (/) )
39 n0 3552 . . . . . . . 8  |-  ( ( s  \  { ( 0g `  G ) } )  =/=  (/)  <->  E. g 
g  e.  ( s 
\  { ( 0g
`  G ) } ) )
4038, 39sylib 188 . . . . . . 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 3842 . . . . . . . . 9  |-  ( g  e.  ( s  \  { ( 0g `  G ) } )  <-> 
( g  e.  s  /\  g  =/=  ( 0g `  G ) ) )
4231adantrr 697 . . . . . . . . . . . 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 740 . . . . . . . . . . . 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 3267 . . . . . . . . . . 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 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  =/=  ( 0g `  G ) )
46 simprll 738 . . . . . . . . . . . . . . . . . 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 697 . . . . . . . . . . . . . . . . . 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 15092 . . . . . . . . . . . . . . . . . 18  |-  ( ( s  e.  (SubGrp `  G )  /\  s  e.  Fin  /\  g  e.  s )  ->  ( O `  g )  ||  ( # `  s
) )
5046, 47, 43, 49syl3anc 1183 . . . . . . . . . . . . . . . . 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 739 . . . . . . . . . . . . . . . . 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 4149 . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . . . 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 15088 . . . . . . . . . . . . . . . . . 18  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  g  e.  X )  ->  ( O `  g )  e.  NN )
5754, 55, 44, 56syl3anc 1183 . . . . . . . . . . . . . . . . 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 12979 . . . . . . . . . . . . . . . . 17  |-  ( ( P  e.  Prime  /\  ( O `  g )  e.  NN )  ->  (
( O `  g
)  ||  P  <->  ( ( O `  g )  =  P  \/  ( O `  g )  =  1 ) ) )
5953, 57, 58syl2anc 642 . . . . . . . . . . . . . . . 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 201 . . . . . . . . . . . . . . 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 366 . . . . . . . . . . . . . 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 2366 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  G )  =  ( 0g `  G
)
6348, 62, 1odeq1 15083 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  Grp  /\  g  e.  X )  ->  ( ( O `  g )  =  1  <-> 
g  =  ( 0g
`  G ) ) )
6454, 44, 63syl2anc 642 . . . . . . . . . . . . . 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 205 . . . . . . . . . . . . 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 2596 . . . . . . . . . . . 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 14 . . . . . . . . . . 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 518 . . . . . . . . . 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 598 . . . . . . . . 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 208 . . . . . . . 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 1627 . . . . . . 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 14 . . . . . 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 2634 . . . . . 6  |-  ( E. g  e.  X  ( O `  g )  =  P  <->  E. g
( g  e.  X  /\  ( O `  g
)  =  P ) )
7472, 73sylibr 203 . . . . 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 598 . . . 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 206 . . 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 2752 . 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 14 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 176    \/ wo 357    /\ wa 358    /\ w3a 935   E.wex 1546    = wceq 1647    e. wcel 1715    =/= wne 2529   E.wrex 2629   _Vcvv 2873    \ cdif 3235    C_ wss 3238   (/)c0 3543   {csn 3729   class class class wbr 4125   ` cfv 5358  (class class class)co 5981    ~< csdm 7005   Fincfn 7006   1c1 8885    < clt 9014   NNcn 9893   2c2 9942   NN0cn0 10114   ZZ>=cuz 10381   ^cexp 11269   #chash 11505    || cdivides 12739   Primecprime 12966   Basecbs 13356   0gc0g 13610   Grpcgrp 14572  SubGrpcsubg 14825   odcod 15050
This theorem is referenced by:  pgpfi  15126  ablfacrplem  15510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-disj 4096  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-omul 6626  df-er 6802  df-ec 6804  df-qs 6808  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-oi 7372  df-card 7719  df-acn 7722  df-cda 7941  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-n0 10115  df-z 10176  df-uz 10382  df-q 10468  df-rp 10506  df-fz 10936  df-fzo 11026  df-fl 11089  df-mod 11138  df-seq 11211  df-exp 11270  df-fac 11454  df-bc 11481  df-hash 11506  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-clim 12169  df-sum 12367  df-dvds 12740  df-gcd 12894  df-prm 12967  df-pc 13098  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-0g 13614  df-mnd 14577  df-submnd 14626  df-grp 14699  df-minusg 14700  df-sbg 14701  df-mulg 14702  df-subg 14828  df-eqg 14830  df-ga 14954  df-od 15054
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