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Theorem padicval 21301
Description: Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
Hypothesis
Ref Expression
padicval.j  |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( q ^ -u (
q  pCnt  x )
) ) ) )
Assertion
Ref Expression
padicval  |-  ( ( P  e.  Prime  /\  X  e.  QQ )  ->  (
( J `  P
) `  X )  =  if ( X  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  X ) ) ) )
Distinct variable groups:    x, q, P    x, X
Allowed substitution hints:    J( x, q)    X( q)

Proof of Theorem padicval
StepHypRef Expression
1 padicval.j . . . 4  |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( q ^ -u (
q  pCnt  x )
) ) ) )
21padicfval 21300 . . 3  |-  ( P  e.  Prime  ->  ( J `
 P )  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  x ) ) ) ) )
32fveq1d 5722 . 2  |-  ( P  e.  Prime  ->  ( ( J `  P ) `
 X )  =  ( ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  x
) ) ) ) `
 X ) )
4 eqeq1 2441 . . . 4  |-  ( x  =  X  ->  (
x  =  0  <->  X  =  0 ) )
5 oveq2 6081 . . . . . 6  |-  ( x  =  X  ->  ( P  pCnt  x )  =  ( P  pCnt  X
) )
65negeqd 9290 . . . . 5  |-  ( x  =  X  ->  -u ( P  pCnt  x )  = 
-u ( P  pCnt  X ) )
76oveq2d 6089 . . . 4  |-  ( x  =  X  ->  ( P ^ -u ( P 
pCnt  x ) )  =  ( P ^ -u ( P  pCnt  X ) ) )
84, 7ifbieq2d 3751 . . 3  |-  ( x  =  X  ->  if ( x  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  x ) ) )  =  if ( X  =  0 ,  0 ,  ( P ^ -u ( P 
pCnt  X ) ) ) )
9 eqid 2435 . . 3  |-  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( P ^ -u ( P 
pCnt  x ) ) ) )  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( P ^ -u ( P 
pCnt  x ) ) ) )
10 c0ex 9075 . . . 4  |-  0  e.  _V
11 ovex 6098 . . . 4  |-  ( P ^ -u ( P 
pCnt  X ) )  e. 
_V
1210, 11ifex 3789 . . 3  |-  if ( X  =  0 ,  0 ,  ( P ^ -u ( P 
pCnt  X ) ) )  e.  _V
138, 9, 12fvmpt 5798 . 2  |-  ( X  e.  QQ  ->  (
( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  x ) ) ) ) `  X
)  =  if ( X  =  0 ,  0 ,  ( P ^ -u ( P 
pCnt  X ) ) ) )
143, 13sylan9eq 2487 1  |-  ( ( P  e.  Prime  /\  X  e.  QQ )  ->  (
( J `  P
) `  X )  =  if ( X  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  X ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ifcif 3731    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   0cc0 8980   -ucneg 9282   QQcq 10564   ^cexp 11372   Primecprime 13069    pCnt cpc 13200
This theorem is referenced by:  padicabvcxp  21316  ostth3  21322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-z 10273  df-q 10565
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