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Theorem sylow2 14864
Description: Sylow's second theorem. See also sylow2b 14861 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 14863). (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 14848 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
53, 4syl 17 . . 3  |-  ( ph  ->  H  e.  (SubGrp `  G ) )
6 sylow2.k . . . 4  |-  ( ph  ->  K  e.  ( P pSyl 
G ) )
7 slwsubg 14848 . . . 4  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
86, 7syl 17 . . 3  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
9 sylow2.a . . 3  |-  .+  =  ( +g  `  G )
10 eqid 2256 . . . . 5  |-  ( Gs  H )  =  ( Gs  H )
1110slwpgp 14851 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  P pGrp  ( Gs  H ) )
123, 11syl 17 . . 3  |-  ( ph  ->  P pGrp  ( Gs  H ) )
131, 2, 6slwhash 14862 . . 3  |-  ( ph  ->  ( # `  K
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
14 sylow2.d . . 3  |-  .-  =  ( -g `  G )
151, 2, 5, 8, 9, 12, 13, 14sylow2b 14861 . 2  |-  ( ph  ->  E. g  e.  X  H  C_  ran  (  x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) ) )
162adantr 453 . . . . . 6  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  (  x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) ) )  ->  X  e.  Fin )
178adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  (  x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) ) )  ->  K  e.  (SubGrp `  G ) )
18 simprl 735 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  (  x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) ) )  ->  g  e.  X )
19 eqid 2256 . . . . . . . . 9  |-  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  =  ( x  e.  K  |->  ( ( g 
.+  x )  .-  g ) )
201, 9, 14, 19conjsubg 14641 . . . . . . . 8  |-  ( ( K  e.  (SubGrp `  G )  /\  g  e.  X )  ->  ran  (  x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  e.  (SubGrp `  G ) )
2117, 18, 20syl2anc 645 . . . . . . 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 14549 . . . . . . 7  |-  ( ran  (  x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  e.  (SubGrp `  G )  ->  ran  (  x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  C_  X
)
2321, 22syl 17 . . . . . 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 7016 . . . . . 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 645 . . . . 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 736 . . . . 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 14862 . . . . . . . . 9  |-  ( ph  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
2827, 13eqtr4d 2291 . . . . . . . 8  |-  ( ph  ->  ( # `  H
)  =  ( # `  K ) )
291subgss 14549 . . . . . . . . . . 11  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
305, 29syl 17 . . . . . . . . . 10  |-  ( ph  ->  H  C_  X )
31 ssfi 7016 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  H  C_  X )  ->  H  e.  Fin )
322, 30, 31syl2anc 645 . . . . . . . . 9  |-  ( ph  ->  H  e.  Fin )
331subgss 14549 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
348, 33syl 17 . . . . . . . . . 10  |-  ( ph  ->  K  C_  X )
35 ssfi 7016 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
362, 34, 35syl2anc 645 . . . . . . . . 9  |-  ( ph  ->  K  e.  Fin )
37 hashen 11277 . . . . . . . . 9  |-  ( ( H  e.  Fin  /\  K  e.  Fin )  ->  ( ( # `  H
)  =  ( # `  K )  <->  H  ~~  K ) )
3832, 36, 37syl2anc 645 . . . . . . . 8  |-  ( ph  ->  ( ( # `  H
)  =  ( # `  K )  <->  H  ~~  K ) )
3928, 38mpbid 203 . . . . . . 7  |-  ( ph  ->  H  ~~  K )
4039adantr 453 . . . . . 6  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  (  x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) ) )  ->  H  ~~  K )
411, 9, 14, 19conjsubgen 14642 . . . . . . 7  |-  ( ( K  e.  (SubGrp `  G )  /\  g  e.  X )  ->  K  ~~  ran  (  x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) )
4217, 18, 41syl2anc 645 . . . . . 6  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  (  x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) ) )  ->  K  ~~  ran  (  x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) )
43 entr 6846 . . . . . 6  |-  ( ( H  ~~  K  /\  K  ~~  ran  (  x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) ) )  ->  H  ~~  ran  (  x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) )
4440, 42, 43syl2anc 645 . . . . 5  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  (  x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) ) )  ->  H  ~~  ran  (  x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) )
45 fisseneq 7007 . . . . 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 1187 . . . 4  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  (  x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) ) )  ->  H  =  ran  (  x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) )
4746expr 601 . . 3  |-  ( (
ph  /\  g  e.  X )  ->  ( H  C_  ran  (  x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  ->  H  =  ran  (  x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )
4847reximdva 2626 . 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 16 1  |-  ( ph  ->  E. g  e.  X  H  =  ran  (  x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2517    C_ wss 3094   class class class wbr 3963    e. cmpt 4017   ran crn 4627   ` cfv 4638  (class class class)co 5757    ~~ cen 6793   Fincfn 6796   ^cexp 11035   #chash 11268    pCnt cpc 12816   Basecbs 13075   ↾s cress 13076   +g cplusg 13135   -gcsg 14292  SubGrpcsubg 14542   pGrp cpgp 14769   pSyl cslw 14770
This theorem is referenced by:  sylow3lem3  14867  sylow3lem6  14870
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-disj 3935  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-omul 6417  df-er 6593  df-ec 6595  df-qs 6599  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-oi 7158  df-card 7505  df-acn 7508  df-cda 7727  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-q 10249  df-rp 10287  df-fz 10714  df-fzo 10802  df-fl 10856  df-mod 10905  df-seq 10978  df-exp 11036  df-fac 11220  df-bc 11247  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-clim 11892  df-sum 12089  df-divides 12459  df-gcd 12613  df-prime 12686  df-pc 12817  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-0g 13331  df-mnd 14294  df-submnd 14343  df-grp 14416  df-minusg 14417  df-sbg 14418  df-mulg 14419  df-subg 14545  df-eqg 14547  df-ghm 14608  df-ga 14671  df-od 14771  df-pgp 14773  df-slw 14774
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