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Theorem padicval 20766
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 20765 . . 3  |-  ( P  e.  Prime  ->  ( J `
 P )  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  x ) ) ) ) )
32fveq1d 5527 . 2  |-  ( P  e.  Prime  ->  ( ( J `  P ) `
 X )  =  ( ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  x
) ) ) ) `
 X ) )
4 eqeq1 2289 . . . 4  |-  ( x  =  X  ->  (
x  =  0  <->  X  =  0 ) )
5 oveq2 5866 . . . . . 6  |-  ( x  =  X  ->  ( P  pCnt  x )  =  ( P  pCnt  X
) )
65negeqd 9046 . . . . 5  |-  ( x  =  X  ->  -u ( P  pCnt  x )  = 
-u ( P  pCnt  X ) )
76oveq2d 5874 . . . 4  |-  ( x  =  X  ->  ( P ^ -u ( P 
pCnt  x ) )  =  ( P ^ -u ( P  pCnt  X ) ) )
84, 7ifbieq2d 3585 . . 3  |-  ( x  =  X  ->  if ( x  =  0 ,  0 ,  ( P ^ -u ( P  pCnt  x ) ) )  =  if ( X  =  0 ,  0 ,  ( P ^ -u ( P 
pCnt  X ) ) ) )
9 eqid 2283 . . 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 8832 . . . 4  |-  0  e.  _V
11 ovex 5883 . . . 4  |-  ( P ^ -u ( P 
pCnt  X ) )  e. 
_V
1210, 11ifex 3623 . . 3  |-  if ( X  =  0 ,  0 ,  ( P ^ -u ( P 
pCnt  X ) ) )  e.  _V
138, 9, 12fvmpt 5602 . 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 2335 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 1623    e. wcel 1684   ifcif 3565    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   0cc0 8737   -ucneg 9038   QQcq 10316   ^cexp 11104   Primecprime 12758    pCnt cpc 12889
This theorem is referenced by:  padicabvcxp  20781  ostth3  20787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-z 10025  df-q 10317
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