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Theorem sylow2 14772
Description: Sylow's second theorem. See also sylow2b 14769 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 14771). (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 14756 . . . 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 14756 . . . 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 2253 . . . . 5  |-  ( Gs  H )  =  ( Gs  H )
1110slwpgp 14759 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  P pGrp  ( Gs  H ) )
123, 11syl 17 . . 3  |-  ( ph  ->  P pGrp  ( Gs  H ) )
131, 2, 6slwhash 14770 . . 3  |-  ( ph  ->  ( # `  K
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
14 sylow2.d . . 3  |-  .-  =  ( -g `  G )
151, 2, 5, 8, 9, 12, 13, 14sylow2b 14769 . 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 2253 . . . . . . . . 9  |-  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  =  ( x  e.  K  |->  ( ( g 
.+  x )  .-  g ) )
201, 9, 14, 19conjsubg 14549 . . . . . . . 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 14457 . . . . . . 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 6968 . . . . . 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 14770 . . . . . . . . 9  |-  ( ph  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
2827, 13eqtr4d 2288 . . . . . . . 8  |-  ( ph  ->  ( # `  H
)  =  ( # `  K ) )
291subgss 14457 . . . . . . . . . . 11  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
305, 29syl 17 . . . . . . . . . 10  |-  ( ph  ->  H  C_  X )
31 ssfi 6968 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  H  C_  X )  ->  H  e.  Fin )
322, 30, 31syl2anc 645 . . . . . . . . 9  |-  ( ph  ->  H  e.  Fin )
331subgss 14457 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
348, 33syl 17 . . . . . . . . . 10  |-  ( ph  ->  K  C_  X )
35 ssfi 6968 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
362, 34, 35syl2anc 645 . . . . . . . . 9  |-  ( ph  ->  K  e.  Fin )
37 hashen 11224 . . . . . . . . 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 14550 . . . . . . 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 6798 . . . . . 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 6959 . . . . 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 2617 . 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 2510    C_ wss 3078   class class class wbr 3920    e. cmpt 3974   ran crn 4581   ` cfv 4592  (class class class)co 5710    ~~ cen 6746   Fincfn 6749   ^cexp 10982   #chash 11215    pCnt cpc 12763   Basecbs 13022   ↾s cress 13023   +g cplusg 13082   -gcsg 14200  SubGrpcsubg 14450   pGrp cpgp 14677   pSyl cslw 14678
This theorem is referenced by:  sylow3lem3  14775  sylow3lem6  14778
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-disj 3892  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-omul 6370  df-er 6546  df-ec 6548  df-qs 6552  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-acn 7459  df-cda 7678  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-q 10196  df-rp 10234  df-fz 10661  df-fzo 10749  df-fl 10803  df-mod 10852  df-seq 10925  df-exp 10983  df-fac 11167  df-bc 11194  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-sum 12036  df-divides 12406  df-gcd 12560  df-prime 12633  df-pc 12764  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-0g 13278  df-mnd 14202  df-submnd 14251  df-grp 14324  df-minusg 14325  df-sbg 14326  df-mulg 14327  df-subg 14453  df-eqg 14455  df-ghm 14516  df-ga 14579  df-od 14679  df-pgp 14681  df-slw 14682
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