MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pgpssslw Structured version   Unicode version

Theorem pgpssslw 16440
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 997 . . . . . . . . . 10  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  X  e.  Fin )
2 elrabi 3258 . . . . . . . . . . 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 16007 . . . . . . . . . . 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 7740 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  x  C_  X )  ->  x  e.  Fin )
71, 5, 6syl2an 477 . . . . . . . . 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 12396 . . . . . . . . 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 10964 . . . . . . 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 6045 . . . . . 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 5737 . . . . . 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 5876 . . . . . . . 8  |-  ( # `  x )  e.  _V
1615, 11fnmpti 5709 . . . . . . 7  |-  F  Fn  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y
) }
17 simp1 996 . . . . . . . 8  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  H  e.  (SubGrp `  G ) )
18 simp3 998 . . . . . . . 8  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  P pGrp  S )
19 eqimss2 3557 . . . . . . . . . . 11  |-  ( y  =  H  ->  H  C_  y )
2019biantrud 507 . . . . . . . . . 10  |-  ( y  =  H  ->  ( P pGrp  ( Gs  y )  <->  ( P pGrp  ( Gs  y )  /\  H  C_  y ) ) )
21 oveq2 6292 . . . . . . . . . . . 12  |-  ( y  =  H  ->  ( Gs  y )  =  ( Gs  H ) )
22 pgpssslw.2 . . . . . . . . . . . 12  |-  S  =  ( Gs  H )
2321, 22syl6eqr 2526 . . . . . . . . . . 11  |-  ( y  =  H  ->  ( Gs  y )  =  S )
2423breq2d 4459 . . . . . . . . . 10  |-  ( y  =  H  ->  ( P pGrp  ( Gs  y )  <->  P pGrp  S ) )
2520, 24bitr3d 255 . . . . . . . . 9  |-  ( y  =  H  ->  (
( P pGrp  ( Gs  y
)  /\  H  C_  y
)  <->  P pGrp  S )
)
2625elrab 3261 . . . . . . . 8  |-  ( H  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  <->  ( H  e.  (SubGrp `  G )  /\  P pGrp  S )
)
2717, 18, 26sylanbrc 664 . . . . . . 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 6018 . . . . . . 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 663 . . . . . 6  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ( F `  H )  e.  ran  F )
30 ne0i 3791 . . . . . 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 12396 . . . . . . . 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 10853 . . . . . 6  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ( # `  X
)  e.  RR )
35 fveq2 5866 . . . . . . . . . . 11  |-  ( x  =  m  ->  ( # `
 x )  =  ( # `  m
) )
36 fvex 5876 . . . . . . . . . . 11  |-  ( # `  m )  e.  _V
3735, 11, 36fvmpt 5950 . . . . . . . . . 10  |-  ( m  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  ->  ( F `  m )  =  ( # `  m
) )
3837adantl 466 . . . . . . . . 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 6292 . . . . . . . . . . . . 13  |-  ( y  =  m  ->  ( Gs  y )  =  ( Gs  m ) )
4039breq2d 4459 . . . . . . . . . . . 12  |-  ( y  =  m  ->  ( P pGrp  ( Gs  y )  <->  P pGrp  ( Gs  m ) ) )
41 sseq2 3526 . . . . . . . . . . . 12  |-  ( y  =  m  ->  ( H  C_  y  <->  H  C_  m
) )
4240, 41anbi12d 710 . . . . . . . . . . 11  |-  ( y  =  m  ->  (
( P pGrp  ( Gs  y
)  /\  H  C_  y
)  <->  ( P pGrp  ( Gs  m )  /\  H  C_  m ) ) )
4342elrab 3261 . . . . . . . . . 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 465 . . . . . . . . . . . 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 16007 . . . . . . . . . . . . 13  |-  ( m  e.  (SubGrp `  G
)  ->  m  C_  X
)
4645ad2antrl 727 . . . . . . . . . . . 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 7561 . . . . . . . . . . . 12  |-  ( X  e.  Fin  ->  (
m  C_  X  ->  m  ~<_  X ) )
4844, 46, 47sylc 60 . . . . . . . . . . 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 7740 . . . . . . . . . . . . 13  |-  ( ( X  e.  Fin  /\  m  C_  X )  ->  m  e.  Fin )
5044, 46, 49syl2anc 661 . . . . . . . . . . . 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 12415 . . . . . . . . . . . 12  |-  ( ( m  e.  Fin  /\  X  e.  Fin )  ->  ( ( # `  m
)  <_  ( # `  X
)  <->  m  ~<_  X )
)
5250, 44, 51syl2anc 661 . . . . . . . . . . 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 232 . . . . . . . . . 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 475 . . . . . . . . 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 4467 . . . . . . . 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 2878 . . . . . . 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 4450 . . . . . . . . 9  |-  ( w  =  ( F `  m )  ->  (
w  <_  ( # `  X
)  <->  ( F `  m )  <_  ( # `
 X ) ) )
5857ralrn 6024 . . . . . . . 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 212 . . . . . 6  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  A. w  e.  ran  F  w  <_ 
( # `  X ) )
61 breq2 4451 . . . . . . . 8  |-  ( z  =  ( # `  X
)  ->  ( w  <_  z  <->  w  <_  ( # `  X ) ) )
6261ralbidv 2903 . . . . . . 7  |-  ( z  =  ( # `  X
)  ->  ( A. w  e.  ran  F  w  <_  z  <->  A. w  e.  ran  F  w  <_ 
( # `  X ) ) )
6362rspcev 3214 . . . . . 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 661 . . . . 5  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  E. z  e.  RR  A. w  e. 
ran  F  w  <_  z )
65 suprzcl 10940 . . . . 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 1228 . . . 4  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  sup ( ran  F ,  RR ,  <  )  e.  ran  F
)
67 fvelrnb 5915 . . . . 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 196 . . 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 6292 . . . . . 6  |-  ( y  =  k  ->  ( Gs  y )  =  ( Gs  k ) )
7170breq2d 4459 . . . . 5  |-  ( y  =  k  ->  ( P pGrp  ( Gs  y )  <->  P pGrp  ( Gs  k ) ) )
72 sseq2 3526 . . . . 5  |-  ( y  =  k  ->  ( H  C_  y  <->  H  C_  k
) )
7371, 72anbi12d 710 . . . 4  |-  ( y  =  k  ->  (
( P pGrp  ( Gs  y
)  /\  H  C_  y
)  <->  ( P pGrp  ( Gs  k )  /\  H  C_  k ) ) )
7473rexrab 3267 . . 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 196 . 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 1001 . . . . . . 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 16419 . . . . . . 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 755 . . . . . 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 10871 . . . . . . . . . . . . . . . 16  |-  ZZ  C_  RR
8114, 80syl6ss 3516 . . . . . . . . . . . . . . 15  |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S
)  ->  ran  F  C_  RR )
8281ad2antrr 725 . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . 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 763 . . . . . . . . . . . . . . . . . . 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 3514 . . . . . . . . . . . . . . . . . 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 532 . . . . . . . . . . . . . . . . 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 664 . . . . . . . . . . . . . . . 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 6018 . . . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . . . 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 2556 . . . . . . . . . . . . . 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 10504 . . . . . . . . . . . . . 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 1231 . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . 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 3261 . . . . . . . . . . . . . . . 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 664 . . . . . . . . . . . . . . 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 5866 . . . . . . . . . . . . . . . 16  |-  ( x  =  k  ->  ( # `
 x )  =  ( # `  k
) )
106 fvex 5876 . . . . . . . . . . . . . . . 16  |-  ( # `  k )  e.  _V
107105, 11, 106fvmpt 5950 . . . . . . . . . . . . . . 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 2510 . . . . . . . . . . . . 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 4471 . . . . . . . . . . . 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 1036 . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . 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 7740 . . . . . . . . . . . . . 14  |-  ( ( m  e.  Fin  /\  k  C_  m )  -> 
k  e.  Fin )
115113, 90, 114syl2anc 661 . . . . . . . . . . . . 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 12396 . . . . . . . . . . . . . 14  |-  ( m  e.  Fin  ->  ( # `
 m )  e. 
NN0 )
117 hashcl 12396 . . . . . . . . . . . . . 14  |-  ( k  e.  Fin  ->  ( # `
 k )  e. 
NN0 )
118 nn0re 10804 . . . . . . . . . . . . . . 15  |-  ( (
# `  m )  e.  NN0  ->  ( # `  m
)  e.  RR )
119 nn0re 10804 . . . . . . . . . . . . . . 15  |-  ( (
# `  k )  e.  NN0  ->  ( # `  k
)  e.  RR )
120 lenlt 9663 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  m
)  e.  RR  /\  ( # `  k )  e.  RR )  -> 
( ( # `  m
)  <_  ( # `  k
)  <->  -.  ( # `  k
)  <  ( # `  m
) ) )
121118, 119, 120syl2an 477 . . . . . . . . . . . . . 14  |-  ( ( ( # `  m
)  e.  NN0  /\  ( # `  k )  e.  NN0 )  -> 
( ( # `  m
)  <_  ( # `  k
)  <->  -.  ( # `  k
)  <  ( # `  m
) ) )
122116, 117, 121syl2an 477 . . . . . . . . . . . . 13  |-  ( ( m  e.  Fin  /\  k  e.  Fin )  ->  ( ( # `  m
)  <_  ( # `  k
)  <->  -.  ( # `  k
)  <  ( # `  m
) ) )
123113, 115, 122syl2anc 661 . . . . . . . . . . . 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 210 . . . . . . . . . . 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 7703 . . . . . . . . . . . . . 14  |-  ( ( m  e.  Fin  /\  k  C.  m )  -> 
k  ~<  m )
126125ex 434 . . . . . . . . . . . . 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 12417 . . . . . . . . . . . . 13  |-  ( ( k  e.  Fin  /\  m  e.  Fin )  ->  ( ( # `  k
)  <  ( # `  m
)  <->  k  ~<  m
) )
129115, 113, 128syl2anc 661 . . . . . . . . . . . 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 234 . . . . . . . . . . 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 177 . . . . . . . . . 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 3603 . . . . . . . . . . . 12  |-  ( k 
C_  m  <->  ( k  C.  m  \/  k  =  m ) )
13390, 132sylib 196 . . . . . . . . . . 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 377 . . . . . . . . . 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 615 . . . . . . . 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 459 . . . . . . . . . 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 465 . . . . . . . . 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 6292 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( Gs  k )  =  ( Gs  m ) )
140139breq2d 4459 . . . . . . . . . 10  |-  ( k  =  m  ->  ( P pGrp  ( Gs  k )  <->  P pGrp  ( Gs  m ) ) )
141 eqimss 3556 . . . . . . . . . . 11  |-  ( k  =  m  ->  k  C_  m )
142141biantrurd 508 . . . . . . . . . 10  |-  ( k  =  m  ->  ( P pGrp  ( Gs  m )  <->  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )
143140, 142bitrd 253 . . . . . . . . 9  |-  ( k  =  m  ->  ( P pGrp  ( Gs  k )  <->  ( k  C_  m  /\  P pGrp  ( Gs  m ) ) ) )
144138, 143syl5ibcom 220 . . . . . . . 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 191 . . . . . . 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 2878 . . . . . 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 16434 . . . . . 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 1180 . . . . 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 463 . . . . 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 532 . . . 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 434 . . 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 2934 . 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 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818    C_ wss 3476    C. wpss 3477   (/)c0 3785   class class class wbr 4447    |-> cmpt 4505   ran crn 5000    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284    ~<_ cdom 7514    ~< csdm 7515   Fincfn 7516   supcsup 7900   RRcr 9491    < clt 9628    <_ cle 9629   NN0cn0 10795   ZZcz 10864   #chash 12373   Primecprime 14076   Basecbs 14490   ↾s cress 14491  SubGrpcsubg 16000   pGrp cpgp 16357   pSyl cslw 16358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-hash 12374  df-subg 16003  df-pgp 16361  df-slw 16362
This theorem is referenced by:  slwn0  16441
  Copyright terms: Public domain W3C validator