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Theorem sylow2 16124
Description: Sylow's second theorem. See also sylow2b 16121 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 16123). This is part of Metamath 100 proof #72. (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 16108 . . . 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 16108 . . . 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 2442 . . . . 5  |-  ( Gs  H )  =  ( Gs  H )
1110slwpgp 16111 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  P pGrp  ( Gs  H ) )
123, 11syl 16 . . 3  |-  ( ph  ->  P pGrp  ( Gs  H ) )
131, 2, 6slwhash 16122 . . 3  |-  ( ph  ->  ( # `  K
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
14 sylow2.d . . 3  |-  .-  =  ( -g `  G )
151, 2, 5, 8, 9, 12, 13, 14sylow2b 16121 . 2  |-  ( ph  ->  E. g  e.  X  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) ) )
162adantr 465 . . . . . 6  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  X  e.  Fin )
178adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  K  e.  (SubGrp `  G ) )
18 simprl 755 . . . . . . . 8  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  g  e.  X
)
19 eqid 2442 . . . . . . . . 9  |-  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  =  ( x  e.  K  |->  ( ( g 
.+  x )  .-  g ) )
201, 9, 14, 19conjsubg 15777 . . . . . . . 8  |-  ( ( K  e.  (SubGrp `  G )  /\  g  e.  X )  ->  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  e.  (SubGrp `  G ) )
2117, 18, 20syl2anc 661 . . . . . . 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 15681 . . . . . . 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 7532 . . . . . 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 661 . . . . 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 756 . . . . 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 16122 . . . . . . . . 9  |-  ( ph  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
2827, 13eqtr4d 2477 . . . . . . . 8  |-  ( ph  ->  ( # `  H
)  =  ( # `  K ) )
291subgss 15681 . . . . . . . . . . 11  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
305, 29syl 16 . . . . . . . . . 10  |-  ( ph  ->  H  C_  X )
31 ssfi 7532 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  H  C_  X )  ->  H  e.  Fin )
322, 30, 31syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  H  e.  Fin )
331subgss 15681 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
348, 33syl 16 . . . . . . . . . 10  |-  ( ph  ->  K  C_  X )
35 ssfi 7532 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
362, 34, 35syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  K  e.  Fin )
37 hashen 12117 . . . . . . . . 9  |-  ( ( H  e.  Fin  /\  K  e.  Fin )  ->  ( ( # `  H
)  =  ( # `  K )  <->  H  ~~  K ) )
3832, 36, 37syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( ( # `  H
)  =  ( # `  K )  <->  H  ~~  K ) )
3928, 38mpbid 210 . . . . . . 7  |-  ( ph  ->  H  ~~  K )
4039adantr 465 . . . . . 6  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  H  ~~  K
)
411, 9, 14, 19conjsubgen 15778 . . . . . . 7  |-  ( ( K  e.  (SubGrp `  G )  /\  g  e.  X )  ->  K  ~~  ran  ( x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) )
4217, 18, 41syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  K  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
43 entr 7360 . . . . . 6  |-  ( ( H  ~~  K  /\  K  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) ) )  ->  H  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
4440, 42, 43syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  H  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
45 fisseneq 7523 . . . . 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 1218 . . . 4  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  H  =  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
4746expr 615 . . 3  |-  ( (
ph  /\  g  e.  X )  ->  ( H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  ->  H  =  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )
4847reximdva 2827 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2715    C_ wss 3327   class class class wbr 4291    e. cmpt 4349   ran crn 4840   ` cfv 5417  (class class class)co 6090    ~~ cen 7306   Fincfn 7309   ^cexp 11864   #chash 12102    pCnt cpc 13902   Basecbs 14173   ↾s cress 14174   +g cplusg 14237   -gcsg 15412  SubGrpcsubg 15674   pGrp cpgp 16029   pSyl cslw 16030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-disj 4262  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-omul 6924  df-er 7100  df-ec 7102  df-qs 7106  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-oi 7723  df-card 8108  df-acn 8111  df-cda 8336  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-q 10953  df-rp 10991  df-fz 11437  df-fzo 11548  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-fac 12051  df-bc 12078  df-hash 12103  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-clim 12965  df-sum 13163  df-dvds 13535  df-gcd 13690  df-prm 13763  df-pc 13903  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-plusg 14250  df-0g 14379  df-mnd 15414  df-submnd 15464  df-grp 15544  df-minusg 15545  df-sbg 15546  df-mulg 15547  df-subg 15677  df-eqg 15679  df-ghm 15744  df-ga 15807  df-od 16031  df-pgp 16033  df-slw 16034
This theorem is referenced by:  sylow3lem3  16127  sylow3lem6  16130
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