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Theorem pgpssslw 17265
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 1006 . . . . . . . . . 10  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  X  e.  Fin )
2 elrabi 3225 . . . . . . . . . . 11  |-  ( x  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  ->  x  e.  (SubGrp `  G
) )
3 pgpssslw.1 . . . . . . . . . . . 12  |-  X  =  ( Base `  G
)
43subgss 16817 . . . . . . . . . . 11  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  X
)
52, 4syl 17 . . . . . . . . . 10  |-  ( x  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  ->  x 
C_  X )
6 ssfi 7801 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  x  C_  X )  ->  x  e.  Fin )
71, 5, 6syl2an 479 . . . . . . . . 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 )
8 hashcl 12544 . . . . . . . . 9  |-  ( x  e.  Fin  ->  ( # `
 x )  e. 
NN0 )
97, 8syl 17 . . . . . . . 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 )
109nn0zd 11045 . . . . . . 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 )
11 pgpssslw.3 . . . . . . 7  |-  F  =  ( x  e.  {
y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) }  |->  ( # `  x ) )
1210, 11fmptd 6061 . . . . . 6  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  F : { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) } --> ZZ )
13 frn 5752 . . . . . 6  |-  ( F : { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) } --> ZZ  ->  ran 
F  C_  ZZ )
1412, 13syl 17 . . . . 5  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ran  F  C_  ZZ )
15 fvex 5891 . . . . . . . 8  |-  ( # `  x )  e.  _V
1615, 11fnmpti 5724 . . . . . . 7  |-  F  Fn  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) }
17 simp1 1005 . . . . . . . 8  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  H  e.  (SubGrp `  G ) )
18 simp3 1007 . . . . . . . 8  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  P pGrp  S )
19 eqimss2 3517 . . . . . . . . . . 11  |-  ( y  =  H  ->  H  C_  y )
2019biantrud 509 . . . . . . . . . 10  |-  ( y  =  H  ->  ( P pGrp  ( Gs  y )  <->  ( P pGrp  ( Gs  y )  /\  H  C_  y ) ) )
21 oveq2 6313 . . . . . . . . . . . 12  |-  ( y  =  H  ->  ( Gs  y )  =  ( Gs  H ) )
22 pgpssslw.2 . . . . . . . . . . . 12  |-  S  =  ( Gs  H )
2321, 22syl6eqr 2481 . . . . . . . . . . 11  |-  ( y  =  H  ->  ( Gs  y )  =  S )
2423breq2d 4435 . . . . . . . . . 10  |-  ( y  =  H  ->  ( P pGrp  ( Gs  y )  <->  P pGrp  S ) )
2520, 24bitr3d 258 . . . . . . . . 9  |-  ( y  =  H  ->  (
( P pGrp  ( Gs  y
)  /\  H  C_  y
)  <->  P pGrp  S )
)
2625elrab 3228 . . . . . . . 8  |-  ( H  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  <->  ( H  e.  (SubGrp `  G )  /\  P pGrp  S )
)
2717, 18, 26sylanbrc 668 . . . . . . 7  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  H  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) } )
28 fnfvelrn 6034 . . . . . . 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
)
2916, 27, 28sylancr 667 . . . . . 6  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ( F `  H )  e.  ran  F )
30 ne0i 3767 . . . . . 6  |-  ( ( F `  H )  e.  ran  F  ->  ran  F  =/=  (/) )
3129, 30syl 17 . . . . 5  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ran  F  =/=  (/) )
32 hashcl 12544 . . . . . . . 8  |-  ( X  e.  Fin  ->  ( # `
 X )  e. 
NN0 )
331, 32syl 17 . . . . . . 7  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ( # `  X
)  e.  NN0 )
3433nn0red 10933 . . . . . 6  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ( # `  X
)  e.  RR )
35 fveq2 5881 . . . . . . . . . . 11  |-  ( x  =  m  ->  ( # `
 x )  =  ( # `  m
) )
36 fvex 5891 . . . . . . . . . . 11  |-  ( # `  m )  e.  _V
3735, 11, 36fvmpt 5964 . . . . . . . . . 10  |-  ( m  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  ->  ( F `  m )  =  ( # `  m
) )
3837adantl 467 . . . . . . . . 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 ) )
39 oveq2 6313 . . . . . . . . . . . . 13  |-  ( y  =  m  ->  ( Gs  y )  =  ( Gs  m ) )
4039breq2d 4435 . . . . . . . . . . . 12  |-  ( y  =  m  ->  ( P pGrp  ( Gs  y )  <->  P pGrp  ( Gs  m ) ) )
41 sseq2 3486 . . . . . . . . . . . 12  |-  ( y  =  m  ->  ( H  C_  y  <->  H  C_  m
) )
4240, 41anbi12d 715 . . . . . . . . . . 11  |-  ( y  =  m  ->  (
( P pGrp  ( Gs  y
)  /\  H  C_  y
)  <->  ( P pGrp  ( Gs  m )  /\  H  C_  m ) ) )
4342elrab 3228 . . . . . . . . . 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 ) ) )
441adantr 466 . . . . . . . . . . . 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 )
453subgss 16817 . . . . . . . . . . . . 13  |-  ( m  e.  (SubGrp `  G
)  ->  m  C_  X
)
4645ad2antrl 732 . . . . . . . . . . . 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
)
47 ssdomg 7625 . . . . . . . . . . . 12  |-  ( X  e.  Fin  ->  (
m  C_  X  ->  m  ~<_  X ) )
4844, 46, 47sylc 62 . . . . . . . . . . 11  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( m  e.  (SubGrp `  G )  /\  ( P pGrp  ( Gs  m )  /\  H  C_  m ) ) )  ->  m  ~<_  X )
49 ssfi 7801 . . . . . . . . . . . . 13  |-  ( ( X  e.  Fin  /\  m  C_  X )  ->  m  e.  Fin )
5044, 46, 49syl2anc 665 . . . . . . . . . . . 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 )
51 hashdom 12564 . . . . . . . . . . . 12  |-  ( ( m  e.  Fin  /\  X  e.  Fin )  ->  ( ( # `  m
)  <_  ( # `  X
)  <->  m  ~<_  X )
)
5250, 44, 51syl2anc 665 . . . . . . . . . . 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 ) )
5348, 52mpbird 235 . . . . . . . . . 10  |-  ( ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  /\  ( m  e.  (SubGrp `  G )  /\  ( P pGrp  ( Gs  m )  /\  H  C_  m ) ) )  ->  ( # `  m
)  <_  ( # `  X
) )
5443, 53sylan2b 477 . . . . . . . . 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
) )
5538, 54eqbrtrd 4444 . . . . . . . 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
) )
5655ralrimiva 2836 . . . . . . 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 ) )
57 breq1 4426 . . . . . . . . 9  |-  ( w  =  ( F `  m )  ->  (
w  <_  ( # `  X
)  <->  ( F `  m )  <_  ( # `
 X ) ) )
5857ralrn 6040 . . . . . . . 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 ) ) )
5916, 58ax-mp 5 . . . . . . 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 ) )
6056, 59sylibr 215 . . . . . 6  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  A. w  e.  ran  F  w  <_ 
( # `  X ) )
61 breq2 4427 . . . . . . . 8  |-  ( z  =  ( # `  X
)  ->  ( w  <_  z  <->  w  <_  ( # `  X ) ) )
6261ralbidv 2861 . . . . . . 7  |-  ( z  =  ( # `  X
)  ->  ( A. w  e.  ran  F  w  <_  z  <->  A. w  e.  ran  F  w  <_ 
( # `  X ) ) )
6362rspcev 3182 . . . . . 6  |-  ( ( ( # `  X
)  e.  RR  /\  A. w  e.  ran  F  w  <_  ( # `  X
) )  ->  E. z  e.  RR  A. w  e. 
ran  F  w  <_  z )
6434, 60, 63syl2anc 665 . . . . 5  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  E. z  e.  RR  A. w  e. 
ran  F  w  <_  z )
65 suprzcl 11022 . . . . 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 )
6614, 31, 64, 65syl3anc 1264 . . . 4  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F
)
67 fvelrnb 5928 . . . . 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 ,  <  )
) )
6816, 67ax-mp 5 . . . 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 ,  <  )
)
6966, 68sylib 199 . . 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 ,  <  )
)
70 oveq2 6313 . . . . . 6  |-  ( y  =  k  ->  ( Gs  y )  =  ( Gs  k ) )
7170breq2d 4435 . . . . 5  |-  ( y  =  k  ->  ( P pGrp  ( Gs  y )  <->  P pGrp  ( Gs  k ) ) )
72 sseq2 3486 . . . . 5  |-  ( y  =  k  ->  ( H  C_  y  <->  H  C_  k
) )
7371, 72anbi12d 715 . . . 4  |-  ( y  =  k  ->  (
( P pGrp  ( Gs  y
)  /\  H  C_  y
)  <->  ( P pGrp  ( Gs  k )  /\  H  C_  k ) ) )
7473rexrab 3234 . . 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 ,  <  ) ) )
7569, 74sylib 199 . 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 ,  <  )
) )
76 simpl3 1010 . . . . . . 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 )
77 pgpprm 17244 . . . . . . 7  |-  ( P pGrp 
S  ->  P  e.  Prime )
7876, 77syl 17 . . . . . 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 )
79 simprl 762 . . . . . 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 ) )
80 zssre 10951 . . . . . . . . . . . . . . . 16  |-  ZZ  C_  RR
8114, 80syl6ss 3476 . . . . . . . . . . . . . . 15  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ran  F  C_  RR )
8281ad2antrr 730 . . . . . . . . . . . . . 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 )
8331ad2antrr 730 . . . . . . . . . . . . . 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  =/=  (/) )
8464ad2antrr 730 . . . . . . . . . . . . . 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 )
85 simprl 762 . . . . . . . . . . . . . . . . 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 ) )
86 simprrr 773 . . . . . . . . . . . . . . . . . 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 ) )
87 simprrl 772 . . . . . . . . . . . . . . . . . . . . 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
) )
8887adantr 466 . . . . . . . . . . . . . . . . . . . 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 ) )
8988simprd 464 . . . . . . . . . . . . . . . . . . 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
)
90 simprrl 772 . . . . . . . . . . . . . . . . . . 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 )
9189, 90sstrd 3474 . . . . . . . . . . . . . . . . . 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
)
9286, 91jca 534 . . . . . . . . . . . . . . . . 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 ) )
9385, 92, 43sylanbrc 668 . . . . . . . . . . . . . . . 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
) } )
9493, 37syl 17 . . . . . . . . . . . . . . 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 )
)
95 fnfvelrn 6034 . . . . . . . . . . . . . . . 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
)
9616, 93, 95sylancr 667 . . . . . . . . . . . . . . 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 )
9794, 96eqeltrrd 2508 . . . . . . . . . . . . . 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
)
98 suprub 10577 . . . . . . . . . . . . . 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 ,  <  ) )
9982, 83, 84, 97, 98syl31anc 1267 . . . . . . . . . . . . 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 ,  <  ) )
100 simprrr 773 . . . . . . . . . . . . . . 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 ,  <  ) )
101100adantr 466 . . . . . . . . . . . . . 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 ,  <  ) )
10279adantr 466 . . . . . . . . . . . . . . . 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 ) )
10373elrab 3228 . . . . . . . . . . . . . . . 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 ) ) )
104102, 88, 103sylanbrc 668 . . . . . . . . . . . . . . 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
) } )
105 fveq2 5881 . . . . . . . . . . . . . . . 16  |-  ( x  =  k  ->  ( # `
 x )  =  ( # `  k
) )
106 fvex 5891 . . . . . . . . . . . . . . . 16  |-  ( # `  k )  e.  _V
107105, 11, 106fvmpt 5964 . . . . . . . . . . . . . . 15  |-  ( k  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  ->  ( F `  k )  =  ( # `  k
) )
108104, 107syl 17 . . . . . . . . . . . . . 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 )
)
109101, 108eqtr3d 2465 . . . . . . . . . . . . 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 ) )
11099, 109breqtrd 4448 . . . . . . . . . . . 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
) )
111 simpll2 1045 . . . . . . . . . . . . . 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 )
11245ad2antrl 732 . . . . . . . . . . . . . 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
)
113111, 112, 49syl2anc 665 . . . . . . . . . . . . 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 )
114 ssfi 7801 . . . . . . . . . . . . . 14  |-  ( ( m  e.  Fin  /\  k  C_  m )  -> 
k  e.  Fin )
115113, 90, 114syl2anc 665 . . . . . . . . . . . . 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 )
116 hashcl 12544 . . . . . . . . . . . . . 14  |-  ( m  e.  Fin  ->  ( # `
 m )  e. 
NN0 )
117 hashcl 12544 . . . . . . . . . . . . . 14  |-  ( k  e.  Fin  ->  ( # `
 k )  e. 
NN0 )
118 nn0re 10885 . . . . . . . . . . . . . . 15  |-  ( (
# `  m )  e.  NN0  ->  ( # `  m
)  e.  RR )
119 nn0re 10885 . . . . . . . . . . . . . . 15  |-  ( (
# `  k )  e.  NN0  ->  ( # `  k
)  e.  RR )
120 lenlt 9719 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  m
)  e.  RR  /\  ( # `  k )  e.  RR )  -> 
( ( # `  m
)  <_  ( # `  k
)  <->  -.  ( # `  k
)  <  ( # `  m
) ) )
121118, 119, 120syl2an 479 . . . . . . . . . . . . . 14  |-  ( ( ( # `  m
)  e.  NN0  /\  ( # `  k )  e.  NN0 )  -> 
( ( # `  m
)  <_  ( # `  k
)  <->  -.  ( # `  k
)  <  ( # `  m
) ) )
122116, 117, 121syl2an 479 . . . . . . . . . . . . 13  |-  ( ( m  e.  Fin  /\  k  e.  Fin )  ->  ( ( # `  m
)  <_  ( # `  k
)  <->  -.  ( # `  k
)  <  ( # `  m
) ) )
123113, 115, 122syl2anc 665 . . . . . . . . . . . 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
) ) )
124110, 123mpbid 213 . . . . . . . . . . 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 ) )
125 php3 7767 . . . . . . . . . . . . . 14  |-  ( ( m  e.  Fin  /\  k  C.  m )  -> 
k  ~<  m )
126125ex 435 . . . . . . . . . . . . 13  |-  ( m  e.  Fin  ->  (
k  C.  m  ->  k 
~<  m ) )
127113, 126syl 17 . . . . . . . . . . . 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 ) )
128 hashsdom 12566 . . . . . . . . . . . . 13  |-  ( ( k  e.  Fin  /\  m  e.  Fin )  ->  ( ( # `  k
)  <  ( # `  m
)  <->  k  ~<  m
) )
129115, 113, 128syl2anc 665 . . . . . . . . . . . 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 ) )
130127, 129sylibrd 237 . . . . . . . . . . 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
) ) )
131124, 130mtod 180 . . . . . . . . . 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 )
132 sspss 3564 . . . . . . . . . . . 12  |-  ( k 
C_  m  <->  ( k  C.  m  \/  k  =  m ) )
13390, 132sylib 199 . . . . . . . . . . 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 ) )
134133ord 378 . . . . . . . . . 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 ) )
135131, 134mpd 15 . . . . . . . . 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 )
136135expr 618 . . . . . . . 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 ) )
13787simpld 460 . . . . . . . . . 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 ) )
138137adantr 466 . . . . . . . . 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 ) )
139 oveq2 6313 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( Gs  k )  =  ( Gs  m ) )
140139breq2d 4435 . . . . . . . . . 10  |-  ( k  =  m  ->  ( P pGrp  ( Gs  k )  <->  P pGrp  ( Gs  m ) ) )
141 eqimss 3516 . . . . . . . . . . 11  |-  ( k  =  m  ->  k  C_  m )
142141biantrurd 510 . . . . . . . . . 10  |-  ( k  =  m  ->  ( P pGrp  ( Gs  m )  <->  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )
143140, 142bitrd 256 . . . . . . . . 9  |-  ( k  =  m  ->  ( P pGrp  ( Gs  k )  <->  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )
144138, 143syl5ibcom 223 . . . . . . . 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 ) ) ) )
145136, 144impbid 193 . . . . . . 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 ) )
146145ralrimiva 2836 . . . . . 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 ) )
147 isslw 17259 . . . . . 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 ) ) )
14878, 79, 146, 147syl3anbrc 1189 . . . . 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 ) )
14987simprd 464 . . . . 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 )
150148, 149jca 534 . . . 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 ) )
151150ex 435 . . 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
) ) )
152151reximdv2 2893 . 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 )
)
15375, 152mpd 15 1  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  E. k  e.  ( P pSyl  G ) H  C_  k )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772   {crab 2775    C_ wss 3436    C. wpss 3437   (/)c0 3761   class class class wbr 4423    |-> cmpt 4482   ran crn 4854    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305    ~<_ cdom 7578    ~< csdm 7579   Fincfn 7580   supcsup 7963   RRcr 9545    < clt 9682    <_ cle 9683   NN0cn0 10876   ZZcz 10944   #chash 12521   Primecprime 14621   Basecbs 15120   ↾s cress 15121  SubGrpcsubg 16810   pGrp cpgp 17168   pSyl cslw 17170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-sup 7965  df-card 8381  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-hash 12522  df-subg 16813  df-pgp 17175  df-slw 17177
This theorem is referenced by:  slwn0  17266
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