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Theorem padicabvf 21325
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 10586 . . . 4  |-  QQ  e.  _V
21mptex 5966 . . 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 5573 . 2  |-  J  Fn  Prime
53padicfval 21310 . . . . 5  |-  ( p  e.  Prime  ->  ( J `
 p )  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( p ^ -u (
p  pCnt  x )
) ) ) )
6 prmnn 13082 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  p  e.  NN )
76ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  p  e.  NN )
87nncnd 10016 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  p  e.  CC )
97nnne0d 10044 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  p  =/=  0 )
10 df-ne 2601 . . . . . . . . . 10  |-  ( x  =/=  0  <->  -.  x  =  0 )
11 pcqcl 13230 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  (
x  e.  QQ  /\  x  =/=  0 ) )  ->  ( p  pCnt  x )  e.  ZZ )
1211anassrs 630 . . . . . . . . . 10  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  x  =/=  0
)  ->  ( p  pCnt  x )  e.  ZZ )
1310, 12sylan2br 463 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  (
p  pCnt  x )  e.  ZZ )
148, 9, 13expnegd 11530 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  (
p ^ -u (
p  pCnt  x )
)  =  ( 1  /  ( p ^
( p  pCnt  x
) ) ) )
158, 9, 13exprecd 11531 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  (
( 1  /  p
) ^ ( p 
pCnt  x ) )  =  ( 1  /  (
p ^ ( p 
pCnt  x ) ) ) )
1614, 15eqtr4d 2471 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  (
p ^ -u (
p  pCnt  x )
)  =  ( ( 1  /  p ) ^ ( p  pCnt  x ) ) )
1716ifeq2da 3765 . . . . . 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 4295 . . . . 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 2468 . . . 4  |-  ( p  e.  Prime  ->  ( J `
 p )  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( ( 1  /  p
) ^ ( p 
pCnt  x ) ) ) ) )
206nnrecred 10045 . . . . . 6  |-  ( p  e.  Prime  ->  ( 1  /  p )  e.  RR )
216nnred 10015 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  RR )
22 prmuz2 13097 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
23 eluz2b2 10548 . . . . . . . . . 10  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
2423simprbi 451 . . . . . . . . 9  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
2522, 24syl 16 . . . . . . . 8  |-  ( p  e.  Prime  ->  1  < 
p )
26 recgt1i 9907 . . . . . . . 8  |-  ( ( p  e.  RR  /\  1  <  p )  -> 
( 0  <  (
1  /  p )  /\  ( 1  /  p )  <  1
) )
2721, 25, 26syl2anc 643 . . . . . . 7  |-  ( p  e.  Prime  ->  ( 0  <  ( 1  /  p )  /\  (
1  /  p )  <  1 ) )
2827simpld 446 . . . . . 6  |-  ( p  e.  Prime  ->  0  < 
( 1  /  p
) )
2927simprd 450 . . . . . 6  |-  ( p  e.  Prime  ->  ( 1  /  p )  <  1 )
30 0xr 9131 . . . . . . 7  |-  0  e.  RR*
31 1re 9090 . . . . . . . 8  |-  1  e.  RR
3231rexri 9137 . . . . . . 7  |-  1  e.  RR*
33 elioo2 10957 . . . . . . 7  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  (
( 1  /  p
)  e.  ( 0 (,) 1 )  <->  ( (
1  /  p )  e.  RR  /\  0  <  ( 1  /  p
)  /\  ( 1  /  p )  <  1 ) ) )
3430, 32, 33mp2an 654 . . . . . 6  |-  ( ( 1  /  p )  e.  ( 0 (,) 1 )  <->  ( (
1  /  p )  e.  RR  /\  0  <  ( 1  /  p
)  /\  ( 1  /  p )  <  1 ) )
3520, 28, 29, 34syl3anbrc 1138 . . . . 5  |-  ( p  e.  Prime  ->  ( 1  /  p )  e.  ( 0 (,) 1
) )
36 qrng.q . . . . . 6  |-  Q  =  (flds  QQ )
37 qabsabv.a . . . . . 6  |-  A  =  (AbsVal `  Q )
38 eqid 2436 . . . . . 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
) ) ) )
3936, 37, 38padicabv 21324 . . . . 5  |-  ( ( p  e.  Prime  /\  (
1  /  p )  e.  ( 0 (,) 1 ) )  -> 
( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( ( 1  /  p
) ^ ( p 
pCnt  x ) ) ) )  e.  A )
4035, 39mpdan 650 . . . 4  |-  ( p  e.  Prime  ->  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( ( 1  /  p ) ^ ( p  pCnt  x ) ) ) )  e.  A )
4119, 40eqeltrd 2510 . . 3  |-  ( p  e.  Prime  ->  ( J `
 p )  e.  A )
4241rgen 2771 . 2  |-  A. p  e.  Prime  ( J `  p )  e.  A
43 ffnfv 5894 . 2  |-  ( J : Prime --> A  <->  ( J  Fn  Prime  /\  A. p  e.  Prime  ( J `  p )  e.  A
) )
444, 42, 43mpbir2an 887 1  |-  J : Prime --> A
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   ifcif 3739   class class class wbr 4212    e. cmpt 4266    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   RRcr 8989   0cc0 8990   1c1 8991   RR*cxr 9119    < clt 9120   -ucneg 9292    / cdiv 9677   NNcn 10000   2c2 10049   ZZcz 10282   ZZ>=cuz 10488   QQcq 10574   (,)cioo 10916   ^cexp 11382   Primecprime 13079    pCnt cpc 13210   ↾s cress 13470  AbsValcabv 15904  ℂfldccnfld 16703
This theorem is referenced by:  ostth  21333
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-ioo 10920  df-ico 10922  df-fz 11044  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-prm 13080  df-pc 13211  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-subg 14941  df-cmn 15414  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-dvr 15788  df-drng 15837  df-subrg 15866  df-abv 15905  df-cnfld 16704
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