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Theorem sylow2 14939
Description: Sylow's second theorem. See also sylow2b 14936 for the "hard" part of the proof. Any two Sylow  P-subgroups are conjugate to one another, and hence the same size, namely 
P ^ ( P 
pCnt  |  X  | 
) (see fislw 14938). (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
sylow2.x  |-  X  =  ( Base `  G
)
sylow2.f  |-  ( ph  ->  X  e.  Fin )
sylow2.h  |-  ( ph  ->  H  e.  ( P pSyl 
G ) )
sylow2.k  |-  ( ph  ->  K  e.  ( P pSyl 
G ) )
sylow2.a  |-  .+  =  ( +g  `  G )
sylow2.d  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
sylow2  |-  ( ph  ->  E. g  e.  X  H  =  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) ) )
Distinct variable groups:    x,  .-    x, g, 
.+    g, G, x    g, H, x    g, K, x    ph, g    g, X, x
Allowed substitution hints:    ph( x)    P( x, g)    .- ( g)

Proof of Theorem sylow2
StepHypRef Expression
1 sylow2.x . . 3  |-  X  =  ( Base `  G
)
2 sylow2.f . . 3  |-  ( ph  ->  X  e.  Fin )
3 sylow2.h . . . 4  |-  ( ph  ->  H  e.  ( P pSyl 
G ) )
4 slwsubg 14923 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
53, 4syl 15 . . 3  |-  ( ph  ->  H  e.  (SubGrp `  G ) )
6 sylow2.k . . . 4  |-  ( ph  ->  K  e.  ( P pSyl 
G ) )
7 slwsubg 14923 . . . 4  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
86, 7syl 15 . . 3  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
9 sylow2.a . . 3  |-  .+  =  ( +g  `  G )
10 eqid 2285 . . . . 5  |-  ( Gs  H )  =  ( Gs  H )
1110slwpgp 14926 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  P pGrp  ( Gs  H ) )
123, 11syl 15 . . 3  |-  ( ph  ->  P pGrp  ( Gs  H ) )
131, 2, 6slwhash 14937 . . 3  |-  ( ph  ->  ( # `  K
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
14 sylow2.d . . 3  |-  .-  =  ( -g `  G )
151, 2, 5, 8, 9, 12, 13, 14sylow2b 14936 . 2  |-  ( ph  ->  E. g  e.  X  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) ) )
162adantr 451 . . . . . 6  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  X  e.  Fin )
178adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  K  e.  (SubGrp `  G ) )
18 simprl 732 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  g  e.  X
)
19 eqid 2285 . . . . . . . . 9  |-  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  =  ( x  e.  K  |->  ( ( g 
.+  x )  .-  g ) )
201, 9, 14, 19conjsubg 14716 . . . . . . . 8  |-  ( ( K  e.  (SubGrp `  G )  /\  g  e.  X )  ->  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  e.  (SubGrp `  G ) )
2117, 18, 20syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  e.  (SubGrp `  G
) )
221subgss 14624 . . . . . . 7  |-  ( ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  e.  (SubGrp `  G )  ->  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  C_  X
)
2321, 22syl 15 . . . . . 6  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) ) 
C_  X )
24 ssfi 7085 . . . . . 6  |-  ( ( X  e.  Fin  /\  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  C_  X
)  ->  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  e.  Fin )
2516, 23, 24syl2anc 642 . . . . 5  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  e.  Fin )
26 simprr 733 . . . . 5  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
271, 2, 3slwhash 14937 . . . . . . . . 9  |-  ( ph  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
2827, 13eqtr4d 2320 . . . . . . . 8  |-  ( ph  ->  ( # `  H
)  =  ( # `  K ) )
291subgss 14624 . . . . . . . . . . 11  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
305, 29syl 15 . . . . . . . . . 10  |-  ( ph  ->  H  C_  X )
31 ssfi 7085 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  H  C_  X )  ->  H  e.  Fin )
322, 30, 31syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  H  e.  Fin )
331subgss 14624 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
348, 33syl 15 . . . . . . . . . 10  |-  ( ph  ->  K  C_  X )
35 ssfi 7085 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
362, 34, 35syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  K  e.  Fin )
37 hashen 11348 . . . . . . . . 9  |-  ( ( H  e.  Fin  /\  K  e.  Fin )  ->  ( ( # `  H
)  =  ( # `  K )  <->  H  ~~  K ) )
3832, 36, 37syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( ( # `  H
)  =  ( # `  K )  <->  H  ~~  K ) )
3928, 38mpbid 201 . . . . . . 7  |-  ( ph  ->  H  ~~  K )
4039adantr 451 . . . . . 6  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  H  ~~  K
)
411, 9, 14, 19conjsubgen 14717 . . . . . . 7  |-  ( ( K  e.  (SubGrp `  G )  /\  g  e.  X )  ->  K  ~~  ran  ( x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) )
4217, 18, 41syl2anc 642 . . . . . 6  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  K  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
43 entr 6915 . . . . . 6  |-  ( ( H  ~~  K  /\  K  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) ) )  ->  H  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
4440, 42, 43syl2anc 642 . . . . 5  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  H  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
45 fisseneq 7076 . . . . 5  |-  ( ( ran  ( x  e.  K  |->  ( ( g 
.+  x )  .-  g ) )  e. 
Fin  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  /\  H  ~~  ran  ( x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) )  ->  H  =  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
4625, 26, 44, 45syl3anc 1182 . . . 4  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  H  =  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
4746expr 598 . . 3  |-  ( (
ph  /\  g  e.  X )  ->  ( H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  ->  H  =  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )
4847reximdva 2657 . 2  |-  ( ph  ->  ( E. g  e.  X  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  ->  E. g  e.  X  H  =  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )
4915, 48mpd 14 1  |-  ( ph  ->  E. g  e.  X  H  =  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686   E.wrex 2546    C_ wss 3154   class class class wbr 4025    e. cmpt 4079   ran crn 4692   ` cfv 5257  (class class class)co 5860    ~~ cen 6862   Fincfn 6865   ^cexp 11106   #chash 11339    pCnt cpc 12891   Basecbs 13150   ↾s cress 13151   +g cplusg 13210   -gcsg 14367  SubGrpcsubg 14617   pGrp cpgp 14844   pSyl cslw 14845
This theorem is referenced by:  sylow3lem3  14942  sylow3lem6  14945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-disj 3996  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-omul 6486  df-er 6662  df-ec 6664  df-qs 6668  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-oi 7227  df-card 7574  df-acn 7577  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-n0 9968  df-z 10027  df-uz 10233  df-q 10319  df-rp 10357  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-clim 11964  df-sum 12161  df-dvds 12534  df-gcd 12688  df-prm 12761  df-pc 12892  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-0g 13406  df-mnd 14369  df-submnd 14418  df-grp 14491  df-minusg 14492  df-sbg 14493  df-mulg 14494  df-subg 14620  df-eqg 14622  df-ghm 14683  df-ga 14746  df-od 14846  df-pgp 14848  df-slw 14849
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