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Theorem ablfac1a 17318
Description: The factors of ablfac1b 17319 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 6277 . . . . . . . 8  |-  ( p  =  P  ->  (
p  pCnt  ( # `  B
) )  =  ( P  pCnt  ( # `  B
) ) )
31, 2oveq12d 6288 . . . . . . 7  |-  ( p  =  P  ->  (
p ^ ( p 
pCnt  ( # `  B
) ) )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
43breq2d 4451 . . . . . 6  |-  ( p  =  P  ->  (
( O `  x
)  ||  ( p ^ ( p  pCnt  (
# `  B )
) )  <->  ( O `  x )  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )
54rabbidv 3098 . . . . 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 5858 . . . . . . 7  |-  ( Base `  G )  e.  _V
97, 8eqeltri 2538 . . . . . 6  |-  B  e. 
_V
109rabex 4588 . . . . 5  |-  { x  e.  B  |  ( O `  x )  ||  ( p ^ (
p  pCnt  ( # `  B
) ) ) }  e.  _V
115, 6, 10fvmpt3i 5935 . . . 4  |-  ( P  e.  A  ->  ( S `  P )  =  { x  e.  B  |  ( O `  x )  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) } )
1211adantl 464 . . 3  |-  ( (
ph  /\  P  e.  A )  ->  ( S `  P )  =  { x  e.  B  |  ( O `  x )  ||  ( P ^ ( P  pCnt  (
# `  B )
) ) } )
1312fveq2d 5852 . 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 2454 . . . 4  |-  { x  e.  B  |  ( O `  x )  ||  ( P ^ ( P  pCnt  ( # `  B
) ) ) }  =  { x  e.  B  |  ( O `
 x )  ||  ( P ^ ( P 
pCnt  ( # `  B
) ) ) }
16 eqid 2454 . . . 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 463 . . . 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 2454 . . . . . . 7  |-  ( P ^ ( P  pCnt  (
# `  B )
) )  =  ( P ^ ( P 
pCnt  ( # `  B
) ) )
22 eqid 2454 . . . . . . 7  |-  ( (
# `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) )  =  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) )
237, 14, 6, 17, 19, 20, 21, 22ablfac1lem 17317 . . . . . 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 1006 . . . . 5  |-  ( (
ph  /\  P  e.  A )  ->  (
( P ^ ( P  pCnt  ( # `  B
) ) )  e.  NN  /\  ( (
# `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) )  e.  NN ) )
2524simpld 457 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  ( P ^ ( P  pCnt  (
# `  B )
) )  e.  NN )
2624simprd 461 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  (
( # `  B )  /  ( P ^
( P  pCnt  ( # `
 B ) ) ) )  e.  NN )
2723simp2d 1007 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  (
( P ^ ( P  pCnt  ( # `  B
) ) )  gcd  ( ( # `  B
)  /  ( P ^ ( P  pCnt  (
# `  B )
) ) ) )  =  1 )
2823simp3d 1008 . . . 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 17316 . . 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 457 . 2  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 { x  e.  B  |  ( O `
 x )  ||  ( P ^ ( P 
pCnt  ( # `  B
) ) ) } )  =  ( P ^ ( P  pCnt  (
# `  B )
) ) )
3113, 30eqtrd 2495 1  |-  ( (
ph  /\  P  e.  A )  ->  ( # `
 ( S `  P ) )  =  ( P ^ ( P  pCnt  ( # `  B
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {crab 2808   _Vcvv 3106    C_ wss 3461   class class class wbr 4439    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   Fincfn 7509   1c1 9482    x. cmul 9486    / cdiv 10202   NNcn 10531   ^cexp 12151   #chash 12390    || cdvds 14073    gcd cgcd 14231   Primecprime 14304    pCnt cpc 14447   Basecbs 14719   odcod 16751   Abelcabl 17001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-er 7303  df-ec 7305  df-qs 7309  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-acn 8314  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12093  df-exp 12152  df-fac 12339  df-bc 12366  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-clim 13396  df-sum 13594  df-dvds 14074  df-gcd 14232  df-prm 14305  df-pc 14448  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-grp 16259  df-minusg 16260  df-sbg 16261  df-mulg 16262  df-subg 16400  df-eqg 16402  df-ga 16530  df-cntz 16557  df-od 16755  df-lsm 16858  df-pj1 16859  df-cmn 17002  df-abl 17003
This theorem is referenced by:  ablfac1c  17320  ablfac1eu  17322  ablfaclem3  17336
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