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Theorem ablfac1a 15556
Description: The factors of ablfac1b 15557 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 6029 . . . . . . . 8  |-  ( p  =  P  ->  (
p  pCnt  ( # `  B
) )  =  ( P  pCnt  ( # `  B
) ) )
31, 2oveq12d 6040 . . . . . . 7  |-  ( p  =  P  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
43breq2d 4167 . . . . . 6  |-  ( p  =  P  ->  (
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) )  <->  ( O `  x )  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
54rabbidv 2893 . . . . 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 5684 . . . . . . 7  |-  ( Base `  G )  e.  _V
97, 8eqeltri 2459 . . . . . 6  |-  B  e. 
_V
109rabex 4297 . . . . 5  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  _V
115, 6, 10fvmpt3i 5750 . . . 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 5674 . 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 2389 . . . 4  |-  { x  e.  B  |  ( O `  x )  ||  ( P ^ ( P  pCnt  ( # `  B
) ) ) }  =  { x  e.  B  |  ( O `
 x )  ||  ( P ^ ( P 
pCnt  ( # `  B
) ) ) }
16 eqid 2389 . . . 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 2389 . . . . . . 7  |-  ( P ^ ( P  pCnt  (
# `  B )
) )  =  ( P ^ ( P 
pCnt  ( # `  B
) ) )
22 eqid 2389 . . . . . . 7  |-  ( (
# `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) )  =  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )
237, 14, 6, 17, 19, 20, 21, 22ablfac1lem 15555 . . . . . 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 15554 . . 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 2421 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 1717   {crab 2655   _Vcvv 2901    C_ wss 3265   class class class wbr 4155    e. cmpt 4209   ` cfv 5396  (class class class)co 6022   Fincfn 7047   1c1 8926    x. cmul 8930    / cdiv 9611   NNcn 9934   ^cexp 11311   #chash 11547    || cdivides 12781    gcd cgcd 12935   Primecprime 13008    pCnt cpc 13139   Basecbs 13398   odcod 15092   Abelcabel 15342
This theorem is referenced by:  ablfac1c  15558  ablfac1eu  15560  ablfaclem3  15574
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-disj 4126  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-omul 6667  df-er 6843  df-ec 6845  df-qs 6849  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-oi 7414  df-card 7761  df-acn 7764  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-q 10509  df-rp 10547  df-fz 10978  df-fzo 11068  df-fl 11131  df-mod 11180  df-seq 11253  df-exp 11312  df-fac 11496  df-bc 11523  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-clim 12211  df-sum 12409  df-dvds 12782  df-gcd 12936  df-prm 13009  df-pc 13140  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-0g 13656  df-mnd 14619  df-submnd 14668  df-grp 14741  df-minusg 14742  df-sbg 14743  df-mulg 14744  df-subg 14870  df-eqg 14872  df-ga 14996  df-cntz 15045  df-od 15096  df-lsm 15199  df-pj1 15200  df-cmn 15343  df-abl 15344
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