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Theorem sylow2 15188
Description: Sylow's second theorem. See also sylow2b 15185 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 15187). (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 15172 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
53, 4syl 16 . . 3  |-  ( ph  ->  H  e.  (SubGrp `  G ) )
6 sylow2.k . . . 4  |-  ( ph  ->  K  e.  ( P pSyl 
G ) )
7 slwsubg 15172 . . . 4  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
86, 7syl 16 . . 3  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
9 sylow2.a . . 3  |-  .+  =  ( +g  `  G )
10 eqid 2388 . . . . 5  |-  ( Gs  H )  =  ( Gs  H )
1110slwpgp 15175 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  P pGrp  ( Gs  H ) )
123, 11syl 16 . . 3  |-  ( ph  ->  P pGrp  ( Gs  H ) )
131, 2, 6slwhash 15186 . . 3  |-  ( ph  ->  ( # `  K
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
14 sylow2.d . . 3  |-  .-  =  ( -g `  G )
151, 2, 5, 8, 9, 12, 13, 14sylow2b 15185 . 2  |-  ( ph  ->  E. g  e.  X  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) ) )
162adantr 452 . . . . . 6  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  X  e.  Fin )
178adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  K  e.  (SubGrp `  G ) )
18 simprl 733 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  g  e.  X
)
19 eqid 2388 . . . . . . . . 9  |-  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  =  ( x  e.  K  |->  ( ( g 
.+  x )  .-  g ) )
201, 9, 14, 19conjsubg 14965 . . . . . . . 8  |-  ( ( K  e.  (SubGrp `  G )  /\  g  e.  X )  ->  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  e.  (SubGrp `  G ) )
2117, 18, 20syl2anc 643 . . . . . . 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 14873 . . . . . . 7  |-  ( ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  e.  (SubGrp `  G )  ->  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  C_  X
)
2321, 22syl 16 . . . . . 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 7266 . . . . . 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 643 . . . . 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 734 . . . . 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 15186 . . . . . . . . 9  |-  ( ph  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
2827, 13eqtr4d 2423 . . . . . . . 8  |-  ( ph  ->  ( # `  H
)  =  ( # `  K ) )
291subgss 14873 . . . . . . . . . . 11  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
305, 29syl 16 . . . . . . . . . 10  |-  ( ph  ->  H  C_  X )
31 ssfi 7266 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  H  C_  X )  ->  H  e.  Fin )
322, 30, 31syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  H  e.  Fin )
331subgss 14873 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
348, 33syl 16 . . . . . . . . . 10  |-  ( ph  ->  K  C_  X )
35 ssfi 7266 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
362, 34, 35syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  K  e.  Fin )
37 hashen 11559 . . . . . . . . 9  |-  ( ( H  e.  Fin  /\  K  e.  Fin )  ->  ( ( # `  H
)  =  ( # `  K )  <->  H  ~~  K ) )
3832, 36, 37syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( ( # `  H
)  =  ( # `  K )  <->  H  ~~  K ) )
3928, 38mpbid 202 . . . . . . 7  |-  ( ph  ->  H  ~~  K )
4039adantr 452 . . . . . 6  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  H  ~~  K
)
411, 9, 14, 19conjsubgen 14966 . . . . . . 7  |-  ( ( K  e.  (SubGrp `  G )  /\  g  e.  X )  ->  K  ~~  ran  ( x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) )
4217, 18, 41syl2anc 643 . . . . . 6  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  K  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
43 entr 7096 . . . . . 6  |-  ( ( H  ~~  K  /\  K  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) ) )  ->  H  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
4440, 42, 43syl2anc 643 . . . . 5  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  H  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
45 fisseneq 7257 . . . . 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 1184 . . . 4  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  H  =  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
4746expr 599 . . 3  |-  ( (
ph  /\  g  e.  X )  ->  ( H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  ->  H  =  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )
4847reximdva 2762 . 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 15 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2651    C_ wss 3264   class class class wbr 4154    e. cmpt 4208   ran crn 4820   ` cfv 5395  (class class class)co 6021    ~~ cen 7043   Fincfn 7046   ^cexp 11310   #chash 11546    pCnt cpc 13138   Basecbs 13397   ↾s cress 13398   +g cplusg 13457   -gcsg 14616  SubGrpcsubg 14866   pGrp cpgp 15093   pSyl cslw 15094
This theorem is referenced by:  sylow3lem3  15191  sylow3lem6  15194
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-disj 4125  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-omul 6666  df-er 6842  df-ec 6844  df-qs 6848  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-acn 7763  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-q 10508  df-rp 10546  df-fz 10977  df-fzo 11067  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-fac 11495  df-bc 11522  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-sum 12408  df-dvds 12781  df-gcd 12935  df-prm 13008  df-pc 13139  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-0g 13655  df-mnd 14618  df-submnd 14667  df-grp 14740  df-minusg 14741  df-sbg 14742  df-mulg 14743  df-subg 14869  df-eqg 14871  df-ghm 14932  df-ga 14995  df-od 15095  df-pgp 15097  df-slw 15098
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