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Theorem ablfac1a 16687
Description: The factors of ablfac1b 16688 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 22 . . . . . . . 8  |-  ( p  =  P  ->  p  =  P )
2 oveq1 6202 . . . . . . . 8  |-  ( p  =  P  ->  (
p  pCnt  ( # `  B
) )  =  ( P  pCnt  ( # `  B
) ) )
31, 2oveq12d 6213 . . . . . . 7  |-  ( p  =  P  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
43breq2d 4407 . . . . . 6  |-  ( p  =  P  ->  (
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) )  <->  ( O `  x )  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
54rabbidv 3064 . . . . 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 5804 . . . . . . 7  |-  ( Base `  G )  e.  _V
97, 8eqeltri 2536 . . . . . 6  |-  B  e. 
_V
109rabex 4546 . . . . 5  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  _V
115, 6, 10fvmpt3i 5882 . . . 4  |-  ( P  e.  A  ->  ( S `  P )  =  { x  e.  B  |  ( O `  x )  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) } )
1211adantl 466 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  ( S `  P )  =  { x  e.  B  |  ( O `  x )  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) } )
1312fveq2d 5798 . 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 2452 . . . 4  |-  { x  e.  B  |  ( O `  x )  ||  ( P ^ ( P  pCnt  ( # `  B
) ) ) }  =  { x  e.  B  |  ( O `
 x )  ||  ( P ^ ( P 
pCnt  ( # `  B
) ) ) }
16 eqid 2452 . . . 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 465 . . . 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 2452 . . . . . . 7  |-  ( P ^ ( P  pCnt  (
# `  B )
) )  =  ( P ^ ( P 
pCnt  ( # `  B
) ) )
22 eqid 2452 . . . . . . 7  |-  ( (
# `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) )  =  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )
237, 14, 6, 17, 19, 20, 21, 22ablfac1lem 16686 . . . . . 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 1000 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  (
( P ^ ( P  pCnt  ( # `  B
) ) )  e.  NN  /\  ( (
# `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) )  e.  NN ) )
2524simpld 459 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( P ^ ( P  pCnt  (
# `  B )
) )  e.  NN )
2624simprd 463 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  (
( # `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) )  e.  NN )
2723simp2d 1001 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  (
( P ^ ( P  pCnt  ( # `  B
) ) )  gcd  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )  =  1 )
2823simp3d 1002 . . . 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 16685 . . 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 459 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 { x  e.  B  |  ( O `
 x )  ||  ( P ^ ( P 
pCnt  ( # `  B
) ) ) } )  =  ( P ^ ( P  pCnt  (
# `  B )
) ) )
3113, 30eqtrd 2493 1  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 ( S `  P ) )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2800   _Vcvv 3072    C_ wss 3431   class class class wbr 4395    |-> cmpt 4453   ` cfv 5521  (class class class)co 6195   Fincfn 7415   1c1 9389    x. cmul 9393    / cdiv 10099   NNcn 10428   ^cexp 11977   #chash 12215    || cdivides 13648    gcd cgcd 13803   Primecprime 13876    pCnt cpc 14016   Basecbs 14287   odcod 16144   Abelcabel 16394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-disj 4366  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-omul 7030  df-er 7206  df-ec 7208  df-qs 7212  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-oi 7830  df-card 8215  df-acn 8218  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-n0 10686  df-z 10753  df-uz 10968  df-q 11060  df-rp 11098  df-fz 11550  df-fzo 11661  df-fl 11754  df-mod 11821  df-seq 11919  df-exp 11978  df-fac 12164  df-bc 12191  df-hash 12216  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-clim 13079  df-sum 13277  df-dvds 13649  df-gcd 13804  df-prm 13877  df-pc 14017  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-0g 14494  df-mnd 15529  df-submnd 15579  df-grp 15659  df-minusg 15660  df-sbg 15661  df-mulg 15662  df-subg 15792  df-eqg 15794  df-ga 15922  df-cntz 15949  df-od 16148  df-lsm 16251  df-pj1 16252  df-cmn 16395  df-abl 16396
This theorem is referenced by:  ablfac1c  16689  ablfac1eu  16691  ablfaclem3  16705
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