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Theorem padicabvf 20796
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 10344 . . . 4  |-  QQ  e.  _V
21mptex 5762 . . 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 5388 . 2  |-  J  Fn  Prime
53padicfval 20781 . . . . 5  |-  ( p  e.  Prime  ->  ( J `
 p )  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( p ^ -u (
p  pCnt  x )
) ) ) )
6 prmnn 12777 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  p  e.  NN )
76ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  p  e.  NN )
87nncnd 9778 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  p  e.  CC )
97nnne0d 9806 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  p  =/=  0 )
10 df-ne 2461 . . . . . . . . . 10  |-  ( x  =/=  0  <->  -.  x  =  0 )
11 pcqcl 12925 . . . . . . . . . . 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 11268 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  (
p ^ -u (
p  pCnt  x )
)  =  ( 1  /  ( p ^
( p  pCnt  x
) ) ) )
158, 9, 13exprecd 11269 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  (
( 1  /  p
) ^ ( p 
pCnt  x ) )  =  ( 1  /  (
p ^ ( p 
pCnt  x ) ) ) )
1614, 15eqtr4d 2331 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  (
p ^ -u (
p  pCnt  x )
)  =  ( ( 1  /  p ) ^ ( p  pCnt  x ) ) )
1716ifeq2da 3604 . . . . . 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 4122 . . . . 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 2328 . . . 4  |-  ( p  e.  Prime  ->  ( J `
 p )  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( ( 1  /  p
) ^ ( p 
pCnt  x ) ) ) ) )
206nnrecred 9807 . . . . . 6  |-  ( p  e.  Prime  ->  ( 1  /  p )  e.  RR )
216nnred 9777 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  RR )
22 prmuz2 12792 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
23 eluz2b2 10306 . . . . . . . . . 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 9669 . . . . . . . 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 8894 . . . . . . 7  |-  0  e.  RR*
31 ressxr 8892 . . . . . . . 8  |-  RR  C_  RR*
32 1re 8853 . . . . . . . 8  |-  1  e.  RR
3331, 32sselii 3190 . . . . . . 7  |-  1  e.  RR*
34 elioo2 10713 . . . . . . 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 2296 . . . . . 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 20795 . . . . 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 2370 . . 3  |-  ( p  e.  Prime  ->  ( J `
 p )  e.  A )
4342rgen 2621 . 2  |-  A. p  e.  Prime  ( J `  p )  e.  A
44 ffnfv 5701 . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   ifcif 3578   class class class wbr 4039    e. cmpt 4093    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754   RR*cxr 8882    < clt 8883   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   ZZcz 10040   ZZ>=cuz 10246   QQcq 10332   (,)cioo 10672   ^cexp 11120   Primecprime 12774    pCnt cpc 12905   ↾s cress 13165  AbsValcabv 15597  ℂfldccnfld 16393
This theorem is referenced by:  ostth  20804
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  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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-int 3879  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-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  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-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-ioo 10676  df-ico 10678  df-fz 10799  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-subg 14634  df-cmn 15107  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-subrg 15559  df-abv 15598  df-cnfld 16394
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