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Theorem padicval 20782
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 20781 . . 3  |-  ( P  e.  Prime  ->  ( J `
 P )  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  x ) ) ) ) )
32fveq1d 5543 . 2  |-  ( P  e.  Prime  ->  ( ( J `  P ) `
 X )  =  ( ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  x
) ) ) ) `
 X ) )
4 eqeq1 2302 . . . 4  |-  ( x  =  X  ->  (
x  =  0  <->  X  =  0 ) )
5 oveq2 5882 . . . . . 6  |-  ( x  =  X  ->  ( P  pCnt  x )  =  ( P  pCnt  X
) )
65negeqd 9062 . . . . 5  |-  ( x  =  X  ->  -u ( P  pCnt  x )  = 
-u ( P  pCnt  X ) )
76oveq2d 5890 . . . 4  |-  ( x  =  X  ->  ( P ^ -u ( P 
pCnt  x ) )  =  ( P ^ -u ( P  pCnt  X ) ) )
84, 7ifbieq2d 3598 . . 3  |-  ( x  =  X  ->  if ( x  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  x ) ) )  =  if ( X  =  0 ,  0 ,  ( P ^ -u ( P 
pCnt  X ) ) ) )
9 eqid 2296 . . 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 8848 . . . 4  |-  0  e.  _V
11 ovex 5899 . . . 4  |-  ( P ^ -u ( P 
pCnt  X ) )  e. 
_V
1210, 11ifex 3636 . . 3  |-  if ( X  =  0 ,  0 ,  ( P ^ -u ( P 
pCnt  X ) ) )  e.  _V
138, 9, 12fvmpt 5618 . 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 2348 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 358    = wceq 1632    e. wcel 1696   ifcif 3578    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   0cc0 8753   -ucneg 9054   QQcq 10332   ^cexp 11120   Primecprime 12774    pCnt cpc 12905
This theorem is referenced by:  padicabvcxp  20797  ostth3  20803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-z 10041  df-q 10333
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