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Theorem padicabvf 20780
Description: The p-adic absolute value is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
Hypotheses
Ref Expression
qrng.q  |-  Q  =  (flds  QQ )
qabsabv.a  |-  A  =  (AbsVal `  Q )
padic.j  |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( q ^ -u (
q  pCnt  x )
) ) ) )
Assertion
Ref Expression
padicabvf  |-  J : Prime --> A
Distinct variable groups:    x, q, A    x, Q
Allowed substitution hints:    Q( q)    J( x, q)

Proof of Theorem padicabvf
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 qex 10328 . . . 4  |-  QQ  e.  _V
21mptex 5746 . . 3  |-  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( q ^ -u ( q 
pCnt  x ) ) ) )  e.  _V
3 padic.j . . 3  |-  J  =  ( q  e.  Prime  |->  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( q ^ -u (
q  pCnt  x )
) ) ) )
42, 3fnmpti 5372 . 2  |-  J  Fn  Prime
53padicfval 20765 . . . . 5  |-  ( p  e.  Prime  ->  ( J `
 p )  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( p ^ -u (
p  pCnt  x )
) ) ) )
6 prmnn 12761 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  p  e.  NN )
76ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  p  e.  NN )
87nncnd 9762 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  p  e.  CC )
97nnne0d 9790 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  p  =/=  0 )
10 df-ne 2448 . . . . . . . . . 10  |-  ( x  =/=  0  <->  -.  x  =  0 )
11 pcqcl 12909 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  (
x  e.  QQ  /\  x  =/=  0 ) )  ->  ( p  pCnt  x )  e.  ZZ )
1211anassrs 629 . . . . . . . . . 10  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  x  =/=  0
)  ->  ( p  pCnt  x )  e.  ZZ )
1310, 12sylan2br 462 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  (
p  pCnt  x )  e.  ZZ )
148, 9, 13expnegd 11252 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  (
p ^ -u (
p  pCnt  x )
)  =  ( 1  /  ( p ^
( p  pCnt  x
) ) ) )
158, 9, 13exprecd 11253 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  (
( 1  /  p
) ^ ( p 
pCnt  x ) )  =  ( 1  /  (
p ^ ( p 
pCnt  x ) ) ) )
1614, 15eqtr4d 2318 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  (
p ^ -u (
p  pCnt  x )
)  =  ( ( 1  /  p ) ^ ( p  pCnt  x ) ) )
1716ifeq2da 3591 . . . . . 6  |-  ( ( p  e.  Prime  /\  x  e.  QQ )  ->  if ( x  =  0 ,  0 ,  ( p ^ -u (
p  pCnt  x )
) )  =  if ( x  =  0 ,  0 ,  ( ( 1  /  p
) ^ ( p 
pCnt  x ) ) ) )
1817mpteq2dva 4106 . . . . 5  |-  ( p  e.  Prime  ->  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( p ^ -u ( p 
pCnt  x ) ) ) )  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( ( 1  /  p ) ^ ( p  pCnt  x ) ) ) ) )
195, 18eqtrd 2315 . . . 4  |-  ( p  e.  Prime  ->  ( J `
 p )  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( ( 1  /  p
) ^ ( p 
pCnt  x ) ) ) ) )
206nnrecred 9791 . . . . . 6  |-  ( p  e.  Prime  ->  ( 1  /  p )  e.  RR )
216nnred 9761 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  RR )
22 prmuz2 12776 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
23 eluz2b2 10290 . . . . . . . . . 10  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
2423simprbi 450 . . . . . . . . 9  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
2522, 24syl 15 . . . . . . . 8  |-  ( p  e.  Prime  ->  1  < 
p )
26 recgt1i 9653 . . . . . . . 8  |-  ( ( p  e.  RR  /\  1  <  p )  -> 
( 0  <  (
1  /  p )  /\  ( 1  /  p )  <  1
) )
2721, 25, 26syl2anc 642 . . . . . . 7  |-  ( p  e.  Prime  ->  ( 0  <  ( 1  /  p )  /\  (
1  /  p )  <  1 ) )
2827simpld 445 . . . . . 6  |-  ( p  e.  Prime  ->  0  < 
( 1  /  p
) )
2927simprd 449 . . . . . 6  |-  ( p  e.  Prime  ->  ( 1  /  p )  <  1 )
30 0xr 8878 . . . . . . 7  |-  0  e.  RR*
31 ressxr 8876 . . . . . . . 8  |-  RR  C_  RR*
32 1re 8837 . . . . . . . 8  |-  1  e.  RR
3331, 32sselii 3177 . . . . . . 7  |-  1  e.  RR*
34 elioo2 10697 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  (
( 1  /  p
)  e.  ( 0 (,) 1 )  <->  ( (
1  /  p )  e.  RR  /\  0  <  ( 1  /  p
)  /\  ( 1  /  p )  <  1 ) ) )
3530, 33, 34mp2an 653 . . . . . 6  |-  ( ( 1  /  p )  e.  ( 0 (,) 1 )  <->  ( (
1  /  p )  e.  RR  /\  0  <  ( 1  /  p
)  /\  ( 1  /  p )  <  1 ) )
3620, 28, 29, 35syl3anbrc 1136 . . . . 5  |-  ( p  e.  Prime  ->  ( 1  /  p )  e.  ( 0 (,) 1
) )
37 qrng.q . . . . . 6  |-  Q  =  (flds  QQ )
38 qabsabv.a . . . . . 6  |-  A  =  (AbsVal `  Q )
39 eqid 2283 . . . . . 6  |-  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( ( 1  /  p ) ^ ( p  pCnt  x ) ) ) )  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( ( 1  /  p ) ^
( p  pCnt  x
) ) ) )
4037, 38, 39padicabv 20779 . . . . 5  |-  ( ( p  e.  Prime  /\  (
1  /  p )  e.  ( 0 (,) 1 ) )  -> 
( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( ( 1  /  p
) ^ ( p 
pCnt  x ) ) ) )  e.  A )
4136, 40mpdan 649 . . . 4  |-  ( p  e.  Prime  ->  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( ( 1  /  p ) ^ ( p  pCnt  x ) ) ) )  e.  A )
4219, 41eqeltrd 2357 . . 3  |-  ( p  e.  Prime  ->  ( J `
 p )  e.  A )
4342rgen 2608 . 2  |-  A. p  e.  Prime  ( J `  p )  e.  A
44 ffnfv 5685 . 2  |-  ( J : Prime --> A  <->  ( J  Fn  Prime  /\  A. p  e.  Prime  ( J `  p )  e.  A
) )
454, 43, 44mpbir2an 886 1  |-  J : Prime --> A
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   ifcif 3565   class class class wbr 4023    e. cmpt 4077    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738   RR*cxr 8866    < clt 8867   -ucneg 9038    / cdiv 9423   NNcn 9746   2c2 9795   ZZcz 10024   ZZ>=cuz 10230   QQcq 10316   (,)cioo 10656   ^cexp 11104   Primecprime 12758    pCnt cpc 12889   ↾s cress 13149  AbsValcabv 15581  ℂfldccnfld 16377
This theorem is referenced by:  ostth  20788
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  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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-int 3863  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-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  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-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-ioo 10660  df-ico 10662  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-cmn 15091  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-subrg 15543  df-abv 15582  df-cnfld 16378
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