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Theorem ablfac1a 15403
Description: The factors of ablfac1b 15404 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 19 . . . . . . . 8  |-  ( p  =  P  ->  p  =  P )
2 oveq1 5952 . . . . . . . 8  |-  ( p  =  P  ->  (
p  pCnt  ( # `  B
) )  =  ( P  pCnt  ( # `  B
) ) )
31, 2oveq12d 5963 . . . . . . 7  |-  ( p  =  P  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
43breq2d 4116 . . . . . 6  |-  ( p  =  P  ->  (
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) )  <->  ( O `  x )  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
54rabbidv 2856 . . . . 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 5622 . . . . . . 7  |-  ( Base `  G )  e.  _V
97, 8eqeltri 2428 . . . . . 6  |-  B  e. 
_V
109rabex 4246 . . . . 5  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  _V
115, 6, 10fvmpt3i 5688 . . . 4  |-  ( P  e.  A  ->  ( S `  P )  =  { x  e.  B  |  ( O `  x )  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) } )
1211adantl 452 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  ( S `  P )  =  { x  e.  B  |  ( O `  x )  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) } )
1312fveq2d 5612 . 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 2358 . . . 4  |-  { x  e.  B  |  ( O `  x )  ||  ( P ^ ( P  pCnt  ( # `  B
) ) ) }  =  { x  e.  B  |  ( O `
 x )  ||  ( P ^ ( P 
pCnt  ( # `  B
) ) ) }
16 eqid 2358 . . . 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 451 . . . 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 2358 . . . . . . 7  |-  ( P ^ ( P  pCnt  (
# `  B )
) )  =  ( P ^ ( P 
pCnt  ( # `  B
) ) )
22 eqid 2358 . . . . . . 7  |-  ( (
# `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) )  =  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )
237, 14, 6, 17, 19, 20, 21, 22ablfac1lem 15402 . . . . . 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 967 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  (
( P ^ ( P  pCnt  ( # `  B
) ) )  e.  NN  /\  ( (
# `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) )  e.  NN ) )
2524simpld 445 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( P ^ ( P  pCnt  (
# `  B )
) )  e.  NN )
2624simprd 449 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  (
( # `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) )  e.  NN )
2723simp2d 968 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  (
( P ^ ( P  pCnt  ( # `  B
) ) )  gcd  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )  =  1 )
2823simp3d 969 . . . 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 15401 . . 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 445 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 { x  e.  B  |  ( O `
 x )  ||  ( P ^ ( P 
pCnt  ( # `  B
) ) ) } )  =  ( P ^ ( P  pCnt  (
# `  B )
) ) )
3113, 30eqtrd 2390 1  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 ( S `  P ) )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   {crab 2623   _Vcvv 2864    C_ wss 3228   class class class wbr 4104    e. cmpt 4158   ` cfv 5337  (class class class)co 5945   Fincfn 6951   1c1 8828    x. cmul 8832    / cdiv 9513   NNcn 9836   ^cexp 11197   #chash 11430    || cdivides 12628    gcd cgcd 12782   Primecprime 12855    pCnt cpc 12986   Basecbs 13245   odcod 14939   Abelcabel 15189
This theorem is referenced by:  ablfac1c  15405  ablfac1eu  15407  ablfaclem3  15421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-disj 4075  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-omul 6571  df-er 6747  df-ec 6749  df-qs 6753  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-acn 7665  df-cda 7884  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-q 10409  df-rp 10447  df-fz 10875  df-fzo 10963  df-fl 11017  df-mod 11066  df-seq 11139  df-exp 11198  df-fac 11382  df-bc 11409  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-clim 12058  df-sum 12256  df-dvds 12629  df-gcd 12783  df-prm 12856  df-pc 12987  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-0g 13503  df-mnd 14466  df-submnd 14515  df-grp 14588  df-minusg 14589  df-sbg 14590  df-mulg 14591  df-subg 14717  df-eqg 14719  df-ga 14843  df-cntz 14892  df-od 14943  df-lsm 15046  df-pj1 15047  df-cmn 15190  df-abl 15191
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