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Theorem sylow2 14900
Description: Sylow's second theorem. See also sylow2b 14897 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 14899). (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 14884 . . . 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 14884 . . . 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 2258 . . . . 5  |-  ( Gs  H )  =  ( Gs  H )
1110slwpgp 14887 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  P pGrp  ( Gs  H ) )
123, 11syl 17 . . 3  |-  ( ph  ->  P pGrp  ( Gs  H ) )
131, 2, 6slwhash 14898 . . 3  |-  ( ph  ->  ( # `  K
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
14 sylow2.d . . 3  |-  .-  =  ( -g `  G )
151, 2, 5, 8, 9, 12, 13, 14sylow2b 14897 . 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 2258 . . . . . . . . 9  |-  ( x  e.  K  |->  ( ( g  .+  x ) 
.-  g ) )  =  ( x  e.  K  |->  ( ( g 
.+  x )  .-  g ) )
201, 9, 14, 19conjsubg 14677 . . . . . . . 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 14585 . . . . . . 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 7051 . . . . . 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 14898 . . . . . . . . 9  |-  ( ph  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
2827, 13eqtr4d 2293 . . . . . . . 8  |-  ( ph  ->  ( # `  H
)  =  ( # `  K ) )
291subgss 14585 . . . . . . . . . . 11  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
305, 29syl 17 . . . . . . . . . 10  |-  ( ph  ->  H  C_  X )
31 ssfi 7051 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  H  C_  X )  ->  H  e.  Fin )
322, 30, 31syl2anc 645 . . . . . . . . 9  |-  ( ph  ->  H  e.  Fin )
331subgss 14585 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp `  G
)  ->  K  C_  X
)
348, 33syl 17 . . . . . . . . . 10  |-  ( ph  ->  K  C_  X )
35 ssfi 7051 . . . . . . . . . 10  |-  ( ( X  e.  Fin  /\  K  C_  X )  ->  K  e.  Fin )
362, 34, 35syl2anc 645 . . . . . . . . 9  |-  ( ph  ->  K  e.  Fin )
37 hashen 11313 . . . . . . . . 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 14678 . . . . . . 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 6881 . . . . . 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 7042 . . . . 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 2630 . 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 2519    C_ wss 3127   class class class wbr 3997    e. cmpt 4051   ran crn 4662   ` cfv 4673  (class class class)co 5792    ~~ cen 6828   Fincfn 6831   ^cexp 11071   #chash 11304    pCnt cpc 12852   Basecbs 13111   ↾s cress 13112   +g cplusg 13171   -gcsg 14328  SubGrpcsubg 14578   pGrp cpgp 14805   pSyl cslw 14806
This theorem is referenced by:  sylow3lem3  14903  sylow3lem6  14906
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-disj 3968  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-omul 6452  df-er 6628  df-ec 6630  df-qs 6634  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-acn 7543  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9934  df-z 9993  df-uz 10199  df-q 10285  df-rp 10323  df-fz 10750  df-fzo 10838  df-fl 10892  df-mod 10941  df-seq 11014  df-exp 11072  df-fac 11256  df-bc 11283  df-hash 11305  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-clim 11928  df-sum 12125  df-divides 12495  df-gcd 12649  df-prime 12722  df-pc 12853  df-ndx 13114  df-slot 13115  df-base 13116  df-sets 13117  df-ress 13118  df-plusg 13184  df-0g 13367  df-mnd 14330  df-submnd 14379  df-grp 14452  df-minusg 14453  df-sbg 14454  df-mulg 14455  df-subg 14581  df-eqg 14583  df-ghm 14644  df-ga 14707  df-od 14807  df-pgp 14809  df-slw 14810
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