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Theorem pgpssslw 15248
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 958 . . . . . . . . . 10  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  X  e.  Fin )
2 elrabi 3090 . . . . . . . . . . 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 14945 . . . . . . . . . . 11  |-  ( x  e.  (SubGrp `  G
)  ->  x  C_  X
)
52, 4syl 16 . . . . . . . . . 10  |-  ( x  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  ->  x 
C_  X )
6 ssfi 7329 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  x  C_  X )  ->  x  e.  Fin )
71, 5, 6syl2an 464 . . . . . . . . 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 11639 . . . . . . . . 9  |-  ( x  e.  Fin  ->  ( # `
 x )  e. 
NN0 )
97, 8syl 16 . . . . . . . 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 10373 . . . . . . 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 5893 . . . . . 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 5597 . . . . . 6  |-  ( F : { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) } --> ZZ  ->  ran 
F  C_  ZZ )
1412, 13syl 16 . . . . 5  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ran  F  C_  ZZ )
15 fvex 5742 . . . . . . . 8  |-  ( # `  x )  e.  _V
1615, 11fnmpti 5573 . . . . . . 7  |-  F  Fn  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) }
17 simp1 957 . . . . . . . 8  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  H  e.  (SubGrp `  G ) )
18 simp3 959 . . . . . . . 8  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  P pGrp  S )
19 eqimss2 3401 . . . . . . . . . . 11  |-  ( y  =  H  ->  H  C_  y )
2019biantrud 494 . . . . . . . . . 10  |-  ( y  =  H  ->  ( P pGrp  ( Gs  y )  <->  ( P pGrp  ( Gs  y )  /\  H  C_  y ) ) )
21 oveq2 6089 . . . . . . . . . . . 12  |-  ( y  =  H  ->  ( Gs  y )  =  ( Gs  H ) )
22 pgpssslw.2 . . . . . . . . . . . 12  |-  S  =  ( Gs  H )
2321, 22syl6eqr 2486 . . . . . . . . . . 11  |-  ( y  =  H  ->  ( Gs  y )  =  S )
2423breq2d 4224 . . . . . . . . . 10  |-  ( y  =  H  ->  ( P pGrp  ( Gs  y )  <->  P pGrp  S ) )
2520, 24bitr3d 247 . . . . . . . . 9  |-  ( y  =  H  ->  (
( P pGrp  ( Gs  y
)  /\  H  C_  y
)  <->  P pGrp  S )
)
2625elrab 3092 . . . . . . . 8  |-  ( H  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  <->  ( H  e.  (SubGrp `  G )  /\  P pGrp  S )
)
2717, 18, 26sylanbrc 646 . . . . . . 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 5867 . . . . . . 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 645 . . . . . 6  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ( F `  H )  e.  ran  F )
30 ne0i 3634 . . . . . 6  |-  ( ( F `  H )  e.  ran  F  ->  ran  F  =/=  (/) )
3129, 30syl 16 . . . . 5  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ran  F  =/=  (/) )
32 hashcl 11639 . . . . . . . 8  |-  ( X  e.  Fin  ->  ( # `
 X )  e. 
NN0 )
331, 32syl 16 . . . . . . 7  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ( # `  X
)  e.  NN0 )
3433nn0red 10275 . . . . . 6  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ( # `  X
)  e.  RR )
35 fveq2 5728 . . . . . . . . . . 11  |-  ( x  =  m  ->  ( # `
 x )  =  ( # `  m
) )
36 fvex 5742 . . . . . . . . . . 11  |-  ( # `  m )  e.  _V
3735, 11, 36fvmpt 5806 . . . . . . . . . 10  |-  ( m  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  ->  ( F `  m )  =  ( # `  m
) )
3837adantl 453 . . . . . . . . 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 6089 . . . . . . . . . . . . 13  |-  ( y  =  m  ->  ( Gs  y )  =  ( Gs  m ) )
4039breq2d 4224 . . . . . . . . . . . 12  |-  ( y  =  m  ->  ( P pGrp  ( Gs  y )  <->  P pGrp  ( Gs  m ) ) )
41 sseq2 3370 . . . . . . . . . . . 12  |-  ( y  =  m  ->  ( H  C_  y  <->  H  C_  m
) )
4240, 41anbi12d 692 . . . . . . . . . . 11  |-  ( y  =  m  ->  (
( P pGrp  ( Gs  y
)  /\  H  C_  y
)  <->  ( P pGrp  ( Gs  m )  /\  H  C_  m ) ) )
4342elrab 3092 . . . . . . . . . 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 452 . . . . . . . . . . . 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 14945 . . . . . . . . . . . . 13  |-  ( m  e.  (SubGrp `  G
)  ->  m  C_  X
)
4645ad2antrl 709 . . . . . . . . . . . 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 7153 . . . . . . . . . . . 12  |-  ( X  e.  Fin  ->  (
m  C_  X  ->  m  ~<_  X ) )
4844, 46, 47sylc 58 . . . . . . . . . . 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 7329 . . . . . . . . . . . . 13  |-  ( ( X  e.  Fin  /\  m  C_  X )  ->  m  e.  Fin )
5044, 46, 49syl2anc 643 . . . . . . . . . . . 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 11653 . . . . . . . . . . . 12  |-  ( ( m  e.  Fin  /\  X  e.  Fin )  ->  ( ( # `  m
)  <_  ( # `  X
)  <->  m  ~<_  X )
)
5250, 44, 51syl2anc 643 . . . . . . . . . . 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 224 . . . . . . . . . 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 462 . . . . . . . . 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 4232 . . . . . . . 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 2789 . . . . . . 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 4215 . . . . . . . . 9  |-  ( w  =  ( F `  m )  ->  (
w  <_  ( # `  X
)  <->  ( F `  m )  <_  ( # `
 X ) ) )
5857ralrn 5873 . . . . . . . 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 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 ) )
6056, 59sylibr 204 . . . . . 6  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  A. w  e.  ran  F  w  <_ 
( # `  X ) )
61 breq2 4216 . . . . . . . 8  |-  ( z  =  ( # `  X
)  ->  ( w  <_  z  <->  w  <_  ( # `  X ) ) )
6261ralbidv 2725 . . . . . . 7  |-  ( z  =  ( # `  X
)  ->  ( A. w  e.  ran  F  w  <_  z  <->  A. w  e.  ran  F  w  <_ 
( # `  X ) ) )
6362rspcev 3052 . . . . . 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 643 . . . . 5  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  E. z  e.  RR  A. w  e. 
ran  F  w  <_  z )
65 suprzcl 10349 . . . . 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 1184 . . . 4  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F
)
67 fvelrnb 5774 . . . . 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 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 ,  <  )
)
6966, 68sylib 189 . . 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 6089 . . . . . 6  |-  ( y  =  k  ->  ( Gs  y )  =  ( Gs  k ) )
7170breq2d 4224 . . . . 5  |-  ( y  =  k  ->  ( P pGrp  ( Gs  y )  <->  P pGrp  ( Gs  k ) ) )
72 sseq2 3370 . . . . 5  |-  ( y  =  k  ->  ( H  C_  y  <->  H  C_  k
) )
7371, 72anbi12d 692 . . . 4  |-  ( y  =  k  ->  (
( P pGrp  ( Gs  y
)  /\  H  C_  y
)  <->  ( P pGrp  ( Gs  k )  /\  H  C_  k ) ) )
7473rexrab 3098 . . 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 189 . 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 962 . . . . . . 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 15227 . . . . . . 7  |-  ( P pGrp 
S  ->  P  e.  Prime )
7876, 77syl 16 . . . . . 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 733 . . . . . 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 10289 . . . . . . . . . . . . . . . 16  |-  ZZ  C_  RR
8114, 80syl6ss 3360 . . . . . . . . . . . . . . 15  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ran  F  C_  RR )
8281ad2antrr 707 . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . . 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 742 . . . . . . . . . . . . . . . . . 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 741 . . . . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . . 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 741 . . . . . . . . . . . . . . . . . . 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 3358 . . . . . . . . . . . . . . . . . 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 519 . . . . . . . . . . . . . . . . 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 646 . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . 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 5867 . . . . . . . . . . . . . . . 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 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 ) ) ) )  ->  ( F `  m )  e.  ran  F )
9794, 96eqeltrrd 2511 . . . . . . . . . . . . . 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 9969 . . . . . . . . . . . . . 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 1187 . . . . . . . . . . . . 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 742 . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . 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 3092 . . . . . . . . . . . . . . . 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 646 . . . . . . . . . . . . . . 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 5728 . . . . . . . . . . . . . . . 16  |-  ( x  =  k  ->  ( # `
 x )  =  ( # `  k
) )
106 fvex 5742 . . . . . . . . . . . . . . . 16  |-  ( # `  k )  e.  _V
107105, 11, 106fvmpt 5806 . . . . . . . . . . . . . . 15  |-  ( k  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  ->  ( F `  k )  =  ( # `  k
) )
108104, 107syl 16 . . . . . . . . . . . . . 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 2470 . . . . . . . . . . . . 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 4236 . . . . . . . . . . . 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 997 . . . . . . . . . . . . . 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 709 . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . 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 7329 . . . . . . . . . . . . . 14  |-  ( ( m  e.  Fin  /\  k  C_  m )  -> 
k  e.  Fin )
115113, 90, 114syl2anc 643 . . . . . . . . . . . . 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 11639 . . . . . . . . . . . . . 14  |-  ( m  e.  Fin  ->  ( # `
 m )  e. 
NN0 )
117 hashcl 11639 . . . . . . . . . . . . . 14  |-  ( k  e.  Fin  ->  ( # `
 k )  e. 
NN0 )
118 nn0re 10230 . . . . . . . . . . . . . . 15  |-  ( (
# `  m )  e.  NN0  ->  ( # `  m
)  e.  RR )
119 nn0re 10230 . . . . . . . . . . . . . . 15  |-  ( (
# `  k )  e.  NN0  ->  ( # `  k
)  e.  RR )
120 lenlt 9154 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  m
)  e.  RR  /\  ( # `  k )  e.  RR )  -> 
( ( # `  m
)  <_  ( # `  k
)  <->  -.  ( # `  k
)  <  ( # `  m
) ) )
121118, 119, 120syl2an 464 . . . . . . . . . . . . . 14  |-  ( ( ( # `  m
)  e.  NN0  /\  ( # `  k )  e.  NN0 )  -> 
( ( # `  m
)  <_  ( # `  k
)  <->  -.  ( # `  k
)  <  ( # `  m
) ) )
122116, 117, 121syl2an 464 . . . . . . . . . . . . 13  |-  ( ( m  e.  Fin  /\  k  e.  Fin )  ->  ( ( # `  m
)  <_  ( # `  k
)  <->  -.  ( # `  k
)  <  ( # `  m
) ) )
123113, 115, 122syl2anc 643 . . . . . . . . . . . 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 202 . . . . . . . . . . 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 7293 . . . . . . . . . . . . . 14  |-  ( ( m  e.  Fin  /\  k  C.  m )  -> 
k  ~<  m )
126125ex 424 . . . . . . . . . . . . 13  |-  ( m  e.  Fin  ->  (
k  C.  m  ->  k 
~<  m ) )
127113, 126syl 16 . . . . . . . . . . . 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 11655 . . . . . . . . . . . . 13  |-  ( ( k  e.  Fin  /\  m  e.  Fin )  ->  ( ( # `  k
)  <  ( # `  m
)  <->  k  ~<  m
) )
129115, 113, 128syl2anc 643 . . . . . . . . . . . 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 226 . . . . . . . . . . 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 170 . . . . . . . . . 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 3446 . . . . . . . . . . . 12  |-  ( k 
C_  m  <->  ( k  C.  m  \/  k  =  m ) )
13390, 132sylib 189 . . . . . . . . . . 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 367 . . . . . . . . . 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 599 . . . . . . . 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 446 . . . . . . . . . 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 452 . . . . . . . . 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 6089 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( Gs  k )  =  ( Gs  m ) )
140139breq2d 4224 . . . . . . . . . 10  |-  ( k  =  m  ->  ( P pGrp  ( Gs  k )  <->  P pGrp  ( Gs  m ) ) )
141 eqimss 3400 . . . . . . . . . . 11  |-  ( k  =  m  ->  k  C_  m )
142141biantrurd 495 . . . . . . . . . 10  |-  ( k  =  m  ->  ( P pGrp  ( Gs  m )  <->  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )
143140, 142bitrd 245 . . . . . . . . 9  |-  ( k  =  m  ->  ( P pGrp  ( Gs  k )  <->  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )
144138, 143syl5ibcom 212 . . . . . . . 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 184 . . . . . . 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 2789 . . . . . 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 15242 . . . . . 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 1138 . . . . 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 450 . . . . 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 519 . . . 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 424 . . 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 2815 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   {crab 2709    C_ wss 3320    C. wpss 3321   (/)c0 3628   class class class wbr 4212    e. cmpt 4266   ran crn 4879    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    ~<_ cdom 7107    ~< csdm 7108   Fincfn 7109   supcsup 7445   RRcr 8989    < clt 9120    <_ cle 9121   NN0cn0 10221   ZZcz 10282   #chash 11618   Primecprime 13079   Basecbs 13469   ↾s cress 13470  SubGrpcsubg 14938   pGrp cpgp 15165   pSyl cslw 15166
This theorem is referenced by:  slwn0  15249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-hash 11619  df-subg 14941  df-pgp 15169  df-slw 15170
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