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Theorem sylow2 17263
Description: Sylow's second theorem. See also sylow2b 17260 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 17262). 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.f . . . . 5  |-  ( ph  ->  X  e.  Fin )
21adantr 466 . . . 4  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  X  e.  Fin )
3 sylow2.k . . . . . . . 8  |-  ( ph  ->  K  e.  ( P pSyl 
G ) )
4 slwsubg 17247 . . . . . . . 8  |-  ( K  e.  ( P pSyl  G
)  ->  K  e.  (SubGrp `  G ) )
53, 4syl 17 . . . . . . 7  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
65adantr 466 . . . . . 6  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  K  e.  (SubGrp `  G ) )
7 simprl 762 . . . . . 6  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  g  e.  X
)
8 sylow2.x . . . . . . 7  |-  X  =  ( Base `  G
)
9 sylow2.a . . . . . . 7  |-  .+  =  ( +g  `  G )
10 sylow2.d . . . . . . 7  |-  .-  =  ( -g `  G )
11 eqid 2422 . . . . . . 7  |-  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  =  ( x  e.  K  |->  ( ( g 
.+  x )  .-  g ) )
128, 9, 10, 11conjsubg 16899 . . . . . 6  |-  ( ( K  e.  (SubGrp `  G )  /\  g  e.  X )  ->  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  e.  (SubGrp `  G ) )
136, 7, 12syl2anc 665 . . . . 5  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  e.  (SubGrp `  G
) )
148subgss 16803 . . . . 5  |-  ( ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  e.  (SubGrp `  G )  ->  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  C_  X
)
1513, 14syl 17 . . . 4  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) ) 
C_  X )
16 ssfi 7794 . . . 4  |-  ( ( X  e.  Fin  /\  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) )  C_  X
)  ->  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  e.  Fin )
172, 15, 16syl2anc 665 . . 3  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  e.  Fin )
18 simprr 764 . . 3  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
19 sylow2.h . . . . . . . 8  |-  ( ph  ->  H  e.  ( P pSyl 
G ) )
208, 1, 19slwhash 17261 . . . . . . 7  |-  ( ph  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
218, 1, 3slwhash 17261 . . . . . . 7  |-  ( ph  ->  ( # `  K
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
2220, 21eqtr4d 2466 . . . . . 6  |-  ( ph  ->  ( # `  H
)  =  ( # `  K ) )
23 slwsubg 17247 . . . . . . . . . 10  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
2419, 23syl 17 . . . . . . . . 9  |-  ( ph  ->  H  e.  (SubGrp `  G ) )
258subgss 16803 . . . . . . . . 9  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
2624, 25syl 17 . . . . . . . 8  |-  ( ph  ->  H  C_  X )
27 ssfi 7794 . . . . . . . 8  |-  ( ( X  e.  Fin  /\  H  C_  X )  ->  H  e.  Fin )
281, 26, 27syl2anc 665 . . . . . . 7  |-  ( ph  ->  H  e.  Fin )
298subgss 16803 . . . . . . . . 9  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
305, 29syl 17 . . . . . . . 8  |-  ( ph  ->  K  C_  X )
31 ssfi 7794 . . . . . . . 8  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
321, 30, 31syl2anc 665 . . . . . . 7  |-  ( ph  ->  K  e.  Fin )
33 hashen 12529 . . . . . . 7  |-  ( ( H  e.  Fin  /\  K  e.  Fin )  ->  ( ( # `  H
)  =  ( # `  K )  <->  H  ~~  K ) )
3428, 32, 33syl2anc 665 . . . . . 6  |-  ( ph  ->  ( ( # `  H
)  =  ( # `  K )  <->  H  ~~  K ) )
3522, 34mpbid 213 . . . . 5  |-  ( ph  ->  H  ~~  K )
3635adantr 466 . . . 4  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  H  ~~  K
)
378, 9, 10, 11conjsubgen 16900 . . . . 5  |-  ( ( K  e.  (SubGrp `  G )  /\  g  e.  X )  ->  K  ~~  ran  ( x  e.  K  |->  ( ( g 
.+  x )  .-  g ) ) )
386, 7, 37syl2anc 665 . . . 4  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  K  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
39 entr 7624 . . . 4  |-  ( ( H  ~~  K  /\  K  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) ) )  ->  H  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
4036, 38, 39syl2anc 665 . . 3  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  H  ~~  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
41 fisseneq 7785 . . 3  |-  ( ( 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
) ) )
4217, 18, 40, 41syl3anc 1264 . 2  |-  ( (
ph  /\  ( g  e.  X  /\  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) ) )  ->  H  =  ran  ( x  e.  K  |->  ( ( g  .+  x )  .-  g
) ) )
43 eqid 2422 . . . . 5  |-  ( Gs  H )  =  ( Gs  H )
4443slwpgp 17250 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  P pGrp  ( Gs  H ) )
4519, 44syl 17 . . 3  |-  ( ph  ->  P pGrp  ( Gs  H ) )
468, 1, 24, 5, 9, 45, 21, 10sylow2b 17260 . 2  |-  ( ph  ->  E. g  e.  X  H  C_  ran  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) ) )
4742, 46reximddv 2901 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 187    /\ wa 370    = wceq 1437    e. wcel 1868   E.wrex 2776    C_ wss 3436   class class class wbr 4420    |-> cmpt 4479   ran crn 4850   ` cfv 5597  (class class class)co 6301    ~~ cen 7570   Fincfn 7573   ^cexp 12271   #chash 12514    pCnt cpc 14771   Basecbs 15106   ↾s cress 15107   +g cplusg 15175   -gcsg 16656  SubGrpcsubg 16796   pGrp cpgp 17154   pSyl cslw 17156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-disj 4392  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-omul 7191  df-er 7367  df-ec 7369  df-qs 7373  df-map 7478  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-acn 8377  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272  df-fac 12459  df-bc 12487  df-hash 12515  df-cj 13148  df-re 13149  df-im 13150  df-sqrt 13284  df-abs 13285  df-clim 13537  df-sum 13738  df-dvds 14291  df-gcd 14454  df-prm 14608  df-pc 14772  df-ndx 15109  df-slot 15110  df-base 15111  df-sets 15112  df-ress 15113  df-plusg 15188  df-0g 15325  df-mgm 16473  df-sgrp 16512  df-mnd 16522  df-submnd 16568  df-grp 16658  df-minusg 16659  df-sbg 16660  df-mulg 16661  df-subg 16799  df-eqg 16801  df-ghm 16866  df-ga 16929  df-od 17157  df-pgp 17161  df-slw 17163
This theorem is referenced by:  sylow3lem3  17266  sylow3lem6  17269
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