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Theorem pgpssslw 14941
Description: Every  P-subgroup is contained in a Sylow  P-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypotheses
Ref Expression
pgpssslw.1  |-  X  =  ( Base `  G
)
pgpssslw.2  |-  S  =  ( Gs  H )
pgpssslw.3  |-  F  =  ( x  e.  {
y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) }  |->  ( # `  x ) )
Assertion
Ref Expression
pgpssslw  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  E. k  e.  ( P pSyl  G ) H  C_  k )
Distinct variable groups:    x, k,
y, G    k, H, x, y    P, k, x, y    k, X, x   
k, F    S, k, x, y
Allowed substitution hints:    F( x, y)    X( y)

Proof of Theorem pgpssslw
Dummy variables  m  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 956 . . . . . . . . . 10  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  X  e.  Fin )
2 ssrab2 3271 . . . . . . . . . . . 12  |-  { y  e.  (SubGrp `  G
)  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) } 
C_  (SubGrp `  G )
32sseli 3189 . . . . . . . . . . 11  |-  ( x  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  ->  x  e.  (SubGrp `  G
) )
4 pgpssslw.1 . . . . . . . . . . . 12  |-  X  =  ( Base `  G
)
54subgss 14638 . . . . . . . . . . 11  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  X
)
63, 5syl 15 . . . . . . . . . 10  |-  ( x  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  ->  x 
C_  X )
7 ssfi 7099 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  x  C_  X )  ->  x  e.  Fin )
81, 6, 7syl2an 463 . . . . . . . . 9  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  x  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) } )  ->  x  e.  Fin )
9 hashcl 11366 . . . . . . . . 9  |-  ( x  e.  Fin  ->  ( # `
 x )  e. 
NN0 )
108, 9syl 15 . . . . . . . 8  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  x  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) } )  -> 
( # `  x )  e.  NN0 )
1110nn0zd 10131 . . . . . . 7  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  x  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) } )  -> 
( # `  x )  e.  ZZ )
12 pgpssslw.3 . . . . . . 7  |-  F  =  ( x  e.  {
y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) }  |->  ( # `  x ) )
1311, 12fmptd 5700 . . . . . 6  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  F : { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) } --> ZZ )
14 frn 5411 . . . . . 6  |-  ( F : { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) } --> ZZ  ->  ran 
F  C_  ZZ )
1513, 14syl 15 . . . . 5  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ran  F  C_  ZZ )
16 fvex 5555 . . . . . . . 8  |-  ( # `  x )  e.  _V
1716, 12fnmpti 5388 . . . . . . 7  |-  F  Fn  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) }
18 simp1 955 . . . . . . . 8  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  H  e.  (SubGrp `  G ) )
19 simp3 957 . . . . . . . 8  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  P pGrp  S )
20 eqimss2 3244 . . . . . . . . . . 11  |-  ( y  =  H  ->  H  C_  y )
2120biantrud 493 . . . . . . . . . 10  |-  ( y  =  H  ->  ( P pGrp  ( Gs  y )  <->  ( P pGrp  ( Gs  y )  /\  H  C_  y ) ) )
22 oveq2 5882 . . . . . . . . . . . 12  |-  ( y  =  H  ->  ( Gs  y )  =  ( Gs  H ) )
23 pgpssslw.2 . . . . . . . . . . . 12  |-  S  =  ( Gs  H )
2422, 23syl6eqr 2346 . . . . . . . . . . 11  |-  ( y  =  H  ->  ( Gs  y )  =  S )
2524breq2d 4051 . . . . . . . . . 10  |-  ( y  =  H  ->  ( P pGrp  ( Gs  y )  <->  P pGrp  S ) )
2621, 25bitr3d 246 . . . . . . . . 9  |-  ( y  =  H  ->  (
( P pGrp  ( Gs  y
)  /\  H  C_  y
)  <->  P pGrp  S )
)
2726elrab 2936 . . . . . . . 8  |-  ( H  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  <->  ( H  e.  (SubGrp `  G )  /\  P pGrp  S )
)
2818, 19, 27sylanbrc 645 . . . . . . 7  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  H  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) } )
29 fnfvelrn 5678 . . . . . . 7  |-  ( ( F  Fn  { y  e.  (SubGrp `  G
)  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  /\  H  e.  {
y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) } )  -> 
( F `  H
)  e.  ran  F
)
3017, 28, 29sylancr 644 . . . . . 6  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ( F `  H )  e.  ran  F )
31 ne0i 3474 . . . . . 6  |-  ( ( F `  H )  e.  ran  F  ->  ran  F  =/=  (/) )
3230, 31syl 15 . . . . 5  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ran  F  =/=  (/) )
33 hashcl 11366 . . . . . . . 8  |-  ( X  e.  Fin  ->  ( # `
 X )  e. 
NN0 )
341, 33syl 15 . . . . . . 7  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ( # `  X
)  e.  NN0 )
3534nn0red 10035 . . . . . 6  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ( # `  X
)  e.  RR )
36 fveq2 5541 . . . . . . . . . . 11  |-  ( x  =  m  ->  ( # `
 x )  =  ( # `  m
) )
37 fvex 5555 . . . . . . . . . . 11  |-  ( # `  m )  e.  _V
3836, 12, 37fvmpt 5618 . . . . . . . . . 10  |-  ( m  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  ->  ( F `  m )  =  ( # `  m
) )
3938adantl 452 . . . . . . . . 9  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  m  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) } )  -> 
( F `  m
)  =  ( # `  m ) )
40 oveq2 5882 . . . . . . . . . . . . 13  |-  ( y  =  m  ->  ( Gs  y )  =  ( Gs  m ) )
4140breq2d 4051 . . . . . . . . . . . 12  |-  ( y  =  m  ->  ( P pGrp  ( Gs  y )  <->  P pGrp  ( Gs  m ) ) )
42 sseq2 3213 . . . . . . . . . . . 12  |-  ( y  =  m  ->  ( H  C_  y  <->  H  C_  m
) )
4341, 42anbi12d 691 . . . . . . . . . . 11  |-  ( y  =  m  ->  (
( P pGrp  ( Gs  y
)  /\  H  C_  y
)  <->  ( P pGrp  ( Gs  m )  /\  H  C_  m ) ) )
4443elrab 2936 . . . . . . . . . 10  |-  ( m  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  <->  ( m  e.  (SubGrp `  G )  /\  ( P pGrp  ( Gs  m )  /\  H  C_  m ) ) )
451adantr 451 . . . . . . . . . . . 12  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( m  e.  (SubGrp `  G )  /\  ( P pGrp  ( Gs  m )  /\  H  C_  m ) ) )  ->  X  e.  Fin )
464subgss 14638 . . . . . . . . . . . . 13  |-  ( m  e.  (SubGrp `  G
)  ->  m  C_  X
)
4746ad2antrl 708 . . . . . . . . . . . 12  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( m  e.  (SubGrp `  G )  /\  ( P pGrp  ( Gs  m )  /\  H  C_  m ) ) )  ->  m  C_  X
)
48 ssdomg 6923 . . . . . . . . . . . 12  |-  ( X  e.  Fin  ->  (
m  C_  X  ->  m  ~<_  X ) )
4945, 47, 48sylc 56 . . . . . . . . . . 11  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( m  e.  (SubGrp `  G )  /\  ( P pGrp  ( Gs  m )  /\  H  C_  m ) ) )  ->  m  ~<_  X )
50 ssfi 7099 . . . . . . . . . . . . 13  |-  ( ( X  e.  Fin  /\  m  C_  X )  ->  m  e.  Fin )
5145, 47, 50syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( m  e.  (SubGrp `  G )  /\  ( P pGrp  ( Gs  m )  /\  H  C_  m ) ) )  ->  m  e.  Fin )
52 hashdom 11377 . . . . . . . . . . . 12  |-  ( ( m  e.  Fin  /\  X  e.  Fin )  ->  ( ( # `  m
)  <_  ( # `  X
)  <->  m  ~<_  X )
)
5351, 45, 52syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( m  e.  (SubGrp `  G )  /\  ( P pGrp  ( Gs  m )  /\  H  C_  m ) ) )  ->  ( ( # `  m )  <_  ( # `
 X )  <->  m  ~<_  X ) )
5449, 53mpbird 223 . . . . . . . . . 10  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( m  e.  (SubGrp `  G )  /\  ( P pGrp  ( Gs  m )  /\  H  C_  m ) ) )  ->  ( # `  m
)  <_  ( # `  X
) )
5544, 54sylan2b 461 . . . . . . . . 9  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  m  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) } )  -> 
( # `  m )  <_  ( # `  X
) )
5639, 55eqbrtrd 4059 . . . . . . . 8  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  m  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) } )  -> 
( F `  m
)  <_  ( # `  X
) )
5756ralrimiva 2639 . . . . . . 7  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  A. m  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) }  ( F `
 m )  <_ 
( # `  X ) )
58 breq1 4042 . . . . . . . . 9  |-  ( w  =  ( F `  m )  ->  (
w  <_  ( # `  X
)  <->  ( F `  m )  <_  ( # `
 X ) ) )
5958ralrn 5684 . . . . . . . 8  |-  ( F  Fn  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  ->  ( A. w  e.  ran  F  w  <_  ( # `  X
)  <->  A. m  e.  {
y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) }  ( F `
 m )  <_ 
( # `  X ) ) )
6017, 59ax-mp 8 . . . . . . 7  |-  ( A. w  e.  ran  F  w  <_  ( # `  X
)  <->  A. m  e.  {
y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) }  ( F `
 m )  <_ 
( # `  X ) )
6157, 60sylibr 203 . . . . . 6  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  A. w  e.  ran  F  w  <_ 
( # `  X ) )
62 breq2 4043 . . . . . . . 8  |-  ( z  =  ( # `  X
)  ->  ( w  <_  z  <->  w  <_  ( # `  X ) ) )
6362ralbidv 2576 . . . . . . 7  |-  ( z  =  ( # `  X
)  ->  ( A. w  e.  ran  F  w  <_  z  <->  A. w  e.  ran  F  w  <_ 
( # `  X ) ) )
6463rspcev 2897 . . . . . 6  |-  ( ( ( # `  X
)  e.  RR  /\  A. w  e.  ran  F  w  <_  ( # `  X
) )  ->  E. z  e.  RR  A. w  e. 
ran  F  w  <_  z )
6535, 61, 64syl2anc 642 . . . . 5  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  E. z  e.  RR  A. w  e. 
ran  F  w  <_  z )
66 suprzcl 10107 . . . . 5  |-  ( ( ran  F  C_  ZZ  /\ 
ran  F  =/=  (/)  /\  E. z  e.  RR  A. w  e.  ran  F  w  <_ 
z )  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F )
6715, 32, 65, 66syl3anc 1182 . . . 4  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F
)
68 fvelrnb 5586 . . . . 5  |-  ( F  Fn  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  ->  ( sup ( ran  F ,  RR ,  <  )  e.  ran  F  <->  E. k  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) }  ( F `
 k )  =  sup ( ran  F ,  RR ,  <  )
) )
6917, 68ax-mp 8 . . . 4  |-  ( sup ( ran  F ,  RR ,  <  )  e. 
ran  F  <->  E. k  e.  {
y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) }  ( F `
 k )  =  sup ( ran  F ,  RR ,  <  )
)
7067, 69sylib 188 . . 3  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  E. k  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) }  ( F `
 k )  =  sup ( ran  F ,  RR ,  <  )
)
71 oveq2 5882 . . . . . 6  |-  ( y  =  k  ->  ( Gs  y )  =  ( Gs  k ) )
7271breq2d 4051 . . . . 5  |-  ( y  =  k  ->  ( P pGrp  ( Gs  y )  <->  P pGrp  ( Gs  k ) ) )
73 sseq2 3213 . . . . 5  |-  ( y  =  k  ->  ( H  C_  y  <->  H  C_  k
) )
7472, 73anbi12d 691 . . . 4  |-  ( y  =  k  ->  (
( P pGrp  ( Gs  y
)  /\  H  C_  y
)  <->  ( P pGrp  ( Gs  k )  /\  H  C_  k ) ) )
7574rexrab 2942 . . 3  |-  ( E. k  e.  { y  e.  (SubGrp `  G
)  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  ( F `  k
)  =  sup ( ran  F ,  RR ,  <  )  <->  E. k  e.  (SubGrp `  G ) ( ( P pGrp  ( Gs  k )  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) )
7670, 75sylib 188 . 2  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  E. k  e.  (SubGrp `  G )
( ( P pGrp  ( Gs  k )  /\  H  C_  k )  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  )
) )
77 simpl3 960 . . . . . . 7  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( k  e.  (SubGrp `  G )  /\  ( ( P pGrp  ( Gs  k )  /\  H  C_  k )  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  )
) ) )  ->  P pGrp  S )
78 pgpprm 14920 . . . . . . 7  |-  ( P pGrp 
S  ->  P  e.  Prime )
7977, 78syl 15 . . . . . 6  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( k  e.  (SubGrp `  G )  /\  ( ( P pGrp  ( Gs  k )  /\  H  C_  k )  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  )
) ) )  ->  P  e.  Prime )
80 simprl 732 . . . . . 6  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( k  e.  (SubGrp `  G )  /\  ( ( P pGrp  ( Gs  k )  /\  H  C_  k )  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  )
) ) )  -> 
k  e.  (SubGrp `  G ) )
81 zssre 10047 . . . . . . . . . . . . . . . 16  |-  ZZ  C_  RR
8215, 81syl6ss 3204 . . . . . . . . . . . . . . 15  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ran  F  C_  RR )
8382ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ran  F  C_  RR )
8432ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ran  F  =/=  (/) )
8565ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  E. z  e.  RR  A. w  e. 
ran  F  w  <_  z )
86 simprl 732 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  m  e.  (SubGrp `  G ) )
87 simprrr 741 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  P pGrp  ( Gs  m ) )
88 simprrl 740 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( k  e.  (SubGrp `  G )  /\  ( ( P pGrp  ( Gs  k )  /\  H  C_  k )  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  )
) ) )  -> 
( P pGrp  ( Gs  k
)  /\  H  C_  k
) )
8988adantr 451 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ( P pGrp  ( Gs  k )  /\  H  C_  k ) )
9089simprd 449 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  H  C_  k
)
91 simprrl 740 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  k  C_  m )
9290, 91sstrd 3202 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  H  C_  m
)
9387, 92jca 518 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ( P pGrp  ( Gs  m )  /\  H  C_  m ) )
9486, 93, 44sylanbrc 645 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  m  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) } )
9594, 38syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ( F `  m )  =  (
# `  m )
)
96 fnfvelrn 5678 . . . . . . . . . . . . . . . 16  |-  ( ( F  Fn  { y  e.  (SubGrp `  G
)  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  /\  m  e.  {
y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) } )  -> 
( F `  m
)  e.  ran  F
)
9717, 94, 96sylancr 644 . . . . . . . . . . . . . . 15  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ( F `  m )  e.  ran  F )
9895, 97eqeltrrd 2371 . . . . . . . . . . . . . 14  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ( # `  m
)  e.  ran  F
)
99 suprub 9731 . . . . . . . . . . . . . 14  |-  ( ( ( ran  F  C_  RR  /\  ran  F  =/=  (/)  /\  E. z  e.  RR  A. w  e. 
ran  F  w  <_  z )  /\  ( # `  m )  e.  ran  F )  ->  ( # `  m
)  <_  sup ( ran  F ,  RR ,  <  ) )
10083, 84, 85, 98, 99syl31anc 1185 . . . . . . . . . . . . 13  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ( # `  m
)  <_  sup ( ran  F ,  RR ,  <  ) )
101 simprrr 741 . . . . . . . . . . . . . . 15  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( k  e.  (SubGrp `  G )  /\  ( ( P pGrp  ( Gs  k )  /\  H  C_  k )  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  )
) ) )  -> 
( F `  k
)  =  sup ( ran  F ,  RR ,  <  ) )
102101adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) )
10380adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  k  e.  (SubGrp `  G ) )
10474elrab 2936 . . . . . . . . . . . . . . . 16  |-  ( k  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  <->  ( k  e.  (SubGrp `  G )  /\  ( P pGrp  ( Gs  k )  /\  H  C_  k ) ) )
105103, 89, 104sylanbrc 645 . . . . . . . . . . . . . . 15  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  k  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) } )
106 fveq2 5541 . . . . . . . . . . . . . . . 16  |-  ( x  =  k  ->  ( # `
 x )  =  ( # `  k
) )
107 fvex 5555 . . . . . . . . . . . . . . . 16  |-  ( # `  k )  e.  _V
108106, 12, 107fvmpt 5618 . . . . . . . . . . . . . . 15  |-  ( k  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  ->  ( F `  k )  =  ( # `  k
) )
109105, 108syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ( F `  k )  =  (
# `  k )
)
110102, 109eqtr3d 2330 . . . . . . . . . . . . 13  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  sup ( ran  F ,  RR ,  <  )  =  ( # `  k ) )
111100, 110breqtrd 4063 . . . . . . . . . . . 12  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ( # `  m
)  <_  ( # `  k
) )
112 simpll2 995 . . . . . . . . . . . . . 14  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  X  e.  Fin )
11346ad2antrl 708 . . . . . . . . . . . . . 14  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  m  C_  X
)
114112, 113, 50syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  m  e.  Fin )
115 ssfi 7099 . . . . . . . . . . . . . 14  |-  ( ( m  e.  Fin  /\  k  C_  m )  -> 
k  e.  Fin )
116114, 91, 115syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  k  e.  Fin )
117 hashcl 11366 . . . . . . . . . . . . . 14  |-  ( m  e.  Fin  ->  ( # `
 m )  e. 
NN0 )
118 hashcl 11366 . . . . . . . . . . . . . 14  |-  ( k  e.  Fin  ->  ( # `
 k )  e. 
NN0 )
119 nn0re 9990 . . . . . . . . . . . . . . 15  |-  ( (
# `  m )  e.  NN0  ->  ( # `  m
)  e.  RR )
120 nn0re 9990 . . . . . . . . . . . . . . 15  |-  ( (
# `  k )  e.  NN0  ->  ( # `  k
)  e.  RR )
121 lenlt 8917 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  m
)  e.  RR  /\  ( # `  k )  e.  RR )  -> 
( ( # `  m
)  <_  ( # `  k
)  <->  -.  ( # `  k
)  <  ( # `  m
) ) )
122119, 120, 121syl2an 463 . . . . . . . . . . . . . 14  |-  ( ( ( # `  m
)  e.  NN0  /\  ( # `  k )  e.  NN0 )  -> 
( ( # `  m
)  <_  ( # `  k
)  <->  -.  ( # `  k
)  <  ( # `  m
) ) )
123117, 118, 122syl2an 463 . . . . . . . . . . . . 13  |-  ( ( m  e.  Fin  /\  k  e.  Fin )  ->  ( ( # `  m
)  <_  ( # `  k
)  <->  -.  ( # `  k
)  <  ( # `  m
) ) )
124114, 116, 123syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ( ( # `
 m )  <_ 
( # `  k )  <->  -.  ( # `  k
)  <  ( # `  m
) ) )
125111, 124mpbid 201 . . . . . . . . . . 11  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  -.  ( # `
 k )  < 
( # `  m ) )
126 php3 7063 . . . . . . . . . . . . . 14  |-  ( ( m  e.  Fin  /\  k  C.  m )  -> 
k  ~<  m )
127126ex 423 . . . . . . . . . . . . 13  |-  ( m  e.  Fin  ->  (
k  C.  m  ->  k 
~<  m ) )
128114, 127syl 15 . . . . . . . . . . . 12  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ( k  C.  m  ->  k  ~<  m ) )
129 hashsdom 11379 . . . . . . . . . . . . 13  |-  ( ( k  e.  Fin  /\  m  e.  Fin )  ->  ( ( # `  k
)  <  ( # `  m
)  <->  k  ~<  m
) )
130116, 114, 129syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ( ( # `
 k )  < 
( # `  m )  <-> 
k  ~<  m ) )
131128, 130sylibrd 225 . . . . . . . . . . 11  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ( k  C.  m  ->  ( # `  k )  <  ( # `
 m ) ) )
132125, 131mtod 168 . . . . . . . . . 10  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  -.  k  C.  m )
133 sspss 3288 . . . . . . . . . . . 12  |-  ( k 
C_  m  <->  ( k  C.  m  \/  k  =  m ) )
13491, 133sylib 188 . . . . . . . . . . 11  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ( k  C.  m  \/  k  =  m ) )
135134ord 366 . . . . . . . . . 10  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  ( -.  k  C.  m  ->  k  =  m ) )
136132, 135mpd 14 . . . . . . . . 9  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  ( m  e.  (SubGrp `  G
)  /\  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )  ->  k  =  m )
137136expr 598 . . . . . . . 8  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  m  e.  (SubGrp `  G )
)  ->  ( (
k  C_  m  /\  P pGrp  ( Gs  m ) )  -> 
k  =  m ) )
13888simpld 445 . . . . . . . . . 10  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( k  e.  (SubGrp `  G )  /\  ( ( P pGrp  ( Gs  k )  /\  H  C_  k )  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  )
) ) )  ->  P pGrp  ( Gs  k ) )
139138adantr 451 . . . . . . . . 9  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  m  e.  (SubGrp `  G )
)  ->  P pGrp  ( Gs  k ) )
140 oveq2 5882 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( Gs  k )  =  ( Gs  m ) )
141140breq2d 4051 . . . . . . . . . 10  |-  ( k  =  m  ->  ( P pGrp  ( Gs  k )  <->  P pGrp  ( Gs  m ) ) )
142 eqimss 3243 . . . . . . . . . . 11  |-  ( k  =  m  ->  k  C_  m )
143142biantrurd 494 . . . . . . . . . 10  |-  ( k  =  m  ->  ( P pGrp  ( Gs  m )  <->  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )
144141, 143bitrd 244 . . . . . . . . 9  |-  ( k  =  m  ->  ( P pGrp  ( Gs  k )  <->  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )
145139, 144syl5ibcom 211 . . . . . . . 8  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  m  e.  (SubGrp `  G )
)  ->  ( k  =  m  ->  ( k 
C_  m  /\  P pGrp  ( Gs  m ) ) ) )
146137, 145impbid 183 . . . . . . 7  |-  ( ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  /\  ( k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) ) )  /\  m  e.  (SubGrp `  G )
)  ->  ( (
k  C_  m  /\  P pGrp  ( Gs  m ) )  <->  k  =  m ) )
147146ralrimiva 2639 . . . . . 6  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( k  e.  (SubGrp `  G )  /\  ( ( P pGrp  ( Gs  k )  /\  H  C_  k )  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  )
) ) )  ->  A. m  e.  (SubGrp `  G ) ( ( k  C_  m  /\  P pGrp  ( Gs  m ) )  <->  k  =  m ) )
148 isslw 14935 . . . . . 6  |-  ( k  e.  ( P pSyl  G
)  <->  ( P  e. 
Prime  /\  k  e.  (SubGrp `  G )  /\  A. m  e.  (SubGrp `  G
) ( ( k 
C_  m  /\  P pGrp  ( Gs  m ) )  <->  k  =  m ) ) )
14979, 80, 147, 148syl3anbrc 1136 . . . . 5  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( k  e.  (SubGrp `  G )  /\  ( ( P pGrp  ( Gs  k )  /\  H  C_  k )  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  )
) ) )  -> 
k  e.  ( P pSyl 
G ) )
15088simprd 449 . . . . 5  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( k  e.  (SubGrp `  G )  /\  ( ( P pGrp  ( Gs  k )  /\  H  C_  k )  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  )
) ) )  ->  H  C_  k )
151149, 150jca 518 . . . 4  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( k  e.  (SubGrp `  G )  /\  ( ( P pGrp  ( Gs  k )  /\  H  C_  k )  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  )
) ) )  -> 
( k  e.  ( P pSyl  G )  /\  H  C_  k ) )
152151ex 423 . . 3  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ( (
k  e.  (SubGrp `  G )  /\  (
( P pGrp  ( Gs  k
)  /\  H  C_  k
)  /\  ( F `  k )  =  sup ( ran  F ,  RR ,  <  ) ) )  ->  ( k  e.  ( P pSyl  G )  /\  H  C_  k
) ) )
153152reximdv2 2665 . 2  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ( E. k  e.  (SubGrp `  G
) ( ( P pGrp  ( Gs  k )  /\  H  C_  k )  /\  ( F `  k )  =  sup ( ran 
F ,  RR ,  <  ) )  ->  E. k  e.  ( P pSyl  G ) H  C_  k )
)
15476, 153mpd 14 1  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  E. k  e.  ( P pSyl  G ) H  C_  k )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    C_ wss 3165    C. wpss 3166   (/)c0 3468   class class class wbr 4039    e. cmpt 4093   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    ~<_ cdom 6877    ~< csdm 6878   Fincfn 6879   supcsup 7209   RRcr 8752    < clt 8883    <_ cle 8884   NN0cn0 9981   ZZcz 10040   #chash 11353   Primecprime 12774   Basecbs 13164   ↾s cress 13165  SubGrpcsubg 14631   pGrp cpgp 14858   pSyl cslw 14859
This theorem is referenced by:  slwn0  14942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354  df-subg 14634  df-pgp 14862  df-slw 14863
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