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Theorem sylow2 16519
Description: Sylow's second theorem. See also sylow2b 16516 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 16518). 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 16503 . . . 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 16503 . . . 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 2467 . . . . 5  |-  ( Gs  H )  =  ( Gs  H )
1110slwpgp 16506 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  P pGrp  ( Gs  H ) )
123, 11syl 16 . . 3  |-  ( ph  ->  P pGrp  ( Gs  H ) )
131, 2, 6slwhash 16517 . . 3  |-  ( ph  ->  ( # `  K
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
14 sylow2.d . . 3  |-  .-  =  ( -g `  G )
151, 2, 5, 8, 9, 12, 13, 14sylow2b 16516 . 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 2467 . . . . . . . . 9  |-  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  =  ( x  e.  K  |->  ( ( g 
.+  x )  .-  g ) )
201, 9, 14, 19conjsubg 16170 . . . . . . . 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 16074 . . . . . . 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 7752 . . . . . 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 16517 . . . . . . . . 9  |-  ( ph  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
2827, 13eqtr4d 2511 . . . . . . . 8  |-  ( ph  ->  ( # `  H
)  =  ( # `  K ) )
291subgss 16074 . . . . . . . . . . 11  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
305, 29syl 16 . . . . . . . . . 10  |-  ( ph  ->  H  C_  X )
31 ssfi 7752 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  H  C_  X )  ->  H  e.  Fin )
322, 30, 31syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  H  e.  Fin )
331subgss 16074 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
348, 33syl 16 . . . . . . . . . 10  |-  ( ph  ->  K  C_  X )
35 ssfi 7752 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
362, 34, 35syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  K  e.  Fin )
37 hashen 12400 . . . . . . . . 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 16171 . . . . . . 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 7579 . . . . . 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 7743 . . . . 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 1228 . . . 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 2942 . 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 1379    e. wcel 1767   E.wrex 2818    C_ wss 3481   class class class wbr 4453    |-> cmpt 4511   ran crn 5006   ` cfv 5594  (class class class)co 6295    ~~ cen 7525   Fincfn 7528   ^cexp 12146   #chash 12385    pCnt cpc 14236   Basecbs 14507   ↾s cress 14508   +g cplusg 14572   -gcsg 15927  SubGrpcsubg 16067   pGrp cpgp 16424   pSyl cslw 16425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147  df-er 7323  df-ec 7325  df-qs 7329  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-acn 8335  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-dvds 13865  df-gcd 14021  df-prm 14094  df-pc 14237  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-subg 16070  df-eqg 16072  df-ghm 16137  df-ga 16200  df-od 16426  df-pgp 16428  df-slw 16429
This theorem is referenced by:  sylow3lem3  16522  sylow3lem6  16525
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