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Theorem ablfac1a 15590
Description: The factors of ablfac1b 15591 are of prime power order. (Contributed by Mario Carneiro, 26-Apr-2016.)
Hypotheses
Ref Expression
ablfac1.b  |-  B  =  ( Base `  G
)
ablfac1.o  |-  O  =  ( od `  G
)
ablfac1.s  |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) } )
ablfac1.g  |-  ( ph  ->  G  e.  Abel )
ablfac1.f  |-  ( ph  ->  B  e.  Fin )
ablfac1.1  |-  ( ph  ->  A  C_  Prime )
Assertion
Ref Expression
ablfac1a  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 ( S `  P ) )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
Distinct variable groups:    x, p, B    ph, p, x    A, p, x    O, p, x    P, p, x    G, p, x
Allowed substitution hints:    S( x, p)

Proof of Theorem ablfac1a
StepHypRef Expression
1 id 20 . . . . . . . 8  |-  ( p  =  P  ->  p  =  P )
2 oveq1 6055 . . . . . . . 8  |-  ( p  =  P  ->  (
p  pCnt  ( # `  B
) )  =  ( P  pCnt  ( # `  B
) ) )
31, 2oveq12d 6066 . . . . . . 7  |-  ( p  =  P  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
43breq2d 4192 . . . . . 6  |-  ( p  =  P  ->  (
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) )  <->  ( O `  x )  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
54rabbidv 2916 . . . . 5  |-  ( p  =  P  ->  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  =  { x  e.  B  |  ( O `
 x )  ||  ( P ^ ( P 
pCnt  ( # `  B
) ) ) } )
6 ablfac1.s . . . . 5  |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  (
p ^ ( p 
pCnt  ( # `  B
) ) ) } )
7 ablfac1.b . . . . . . 7  |-  B  =  ( Base `  G
)
8 fvex 5709 . . . . . . 7  |-  ( Base `  G )  e.  _V
97, 8eqeltri 2482 . . . . . 6  |-  B  e. 
_V
109rabex 4322 . . . . 5  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  _V
115, 6, 10fvmpt3i 5776 . . . 4  |-  ( P  e.  A  ->  ( S `  P )  =  { x  e.  B  |  ( O `  x )  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) } )
1211adantl 453 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  ( S `  P )  =  { x  e.  B  |  ( O `  x )  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) } )
1312fveq2d 5699 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 ( S `  P ) )  =  ( # `  {
x  e.  B  | 
( O `  x
)  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) } ) )
14 ablfac1.o . . . 4  |-  O  =  ( od `  G
)
15 eqid 2412 . . . 4  |-  { x  e.  B  |  ( O `  x )  ||  ( P ^ ( P  pCnt  ( # `  B
) ) ) }  =  { x  e.  B  |  ( O `
 x )  ||  ( P ^ ( P 
pCnt  ( # `  B
) ) ) }
16 eqid 2412 . . . 4  |-  { x  e.  B  |  ( O `  x )  ||  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) ) }  =  { x  e.  B  |  ( O `
 x )  ||  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) ) }
17 ablfac1.g . . . . 5  |-  ( ph  ->  G  e.  Abel )
1817adantr 452 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  G  e.  Abel )
19 ablfac1.f . . . . . . 7  |-  ( ph  ->  B  e.  Fin )
20 ablfac1.1 . . . . . . 7  |-  ( ph  ->  A  C_  Prime )
21 eqid 2412 . . . . . . 7  |-  ( P ^ ( P  pCnt  (
# `  B )
) )  =  ( P ^ ( P 
pCnt  ( # `  B
) ) )
22 eqid 2412 . . . . . . 7  |-  ( (
# `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) )  =  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )
237, 14, 6, 17, 19, 20, 21, 22ablfac1lem 15589 . . . . . 6  |-  ( (
ph  /\  P  e.  A )  ->  (
( ( P ^
( P  pCnt  ( # `
 B ) ) )  e.  NN  /\  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )  e.  NN )  /\  (
( P ^ ( P  pCnt  ( # `  B
) ) )  gcd  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )  =  1  /\  ( # `
 B )  =  ( ( P ^
( P  pCnt  ( # `
 B ) ) )  x.  ( (
# `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) ) ) ) )
2423simp1d 969 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  (
( P ^ ( P  pCnt  ( # `  B
) ) )  e.  NN  /\  ( (
# `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) )  e.  NN ) )
2524simpld 446 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( P ^ ( P  pCnt  (
# `  B )
) )  e.  NN )
2624simprd 450 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  (
( # `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) )  e.  NN )
2723simp2d 970 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  (
( P ^ ( P  pCnt  ( # `  B
) ) )  gcd  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )  =  1 )
2823simp3d 971 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 B )  =  ( ( P ^
( P  pCnt  ( # `
 B ) ) )  x.  ( (
# `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) ) ) )
297, 14, 15, 16, 18, 25, 26, 27, 28ablfacrp2 15588 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  (
( # `  { x  e.  B  |  ( O `  x )  ||  ( P ^ ( P  pCnt  ( # `  B
) ) ) } )  =  ( P ^ ( P  pCnt  (
# `  B )
) )  /\  ( # `
 { x  e.  B  |  ( O `
 x )  ||  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) ) } )  =  ( (
# `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) ) ) )
3029simpld 446 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 { x  e.  B  |  ( O `
 x )  ||  ( P ^ ( P 
pCnt  ( # `  B
) ) ) } )  =  ( P ^ ( P  pCnt  (
# `  B )
) ) )
3113, 30eqtrd 2444 1  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 ( S `  P ) )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2678   _Vcvv 2924    C_ wss 3288   class class class wbr 4180    e. cmpt 4234   ` cfv 5421  (class class class)co 6048   Fincfn 7076   1c1 8955    x. cmul 8959    / cdiv 9641   NNcn 9964   ^cexp 11345   #chash 11581    || cdivides 12815    gcd cgcd 12969   Primecprime 13042    pCnt cpc 13173   Basecbs 13432   odcod 15126   Abelcabel 15376
This theorem is referenced by:  ablfac1c  15592  ablfac1eu  15594  ablfaclem3  15608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-disj 4151  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6872  df-ec 6874  df-qs 6878  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-acn 7793  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-q 10539  df-rp 10577  df-fz 11008  df-fzo 11099  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-fac 11530  df-bc 11557  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-sum 12443  df-dvds 12816  df-gcd 12970  df-prm 13043  df-pc 13174  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-0g 13690  df-mnd 14653  df-submnd 14702  df-grp 14775  df-minusg 14776  df-sbg 14777  df-mulg 14778  df-subg 14904  df-eqg 14906  df-ga 15030  df-cntz 15079  df-od 15130  df-lsm 15233  df-pj1 15234  df-cmn 15377  df-abl 15378
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