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Theorem padicabvf 20776
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
Dummy variable  p is distinct from all other variables.
Allowed substitution groups:    Q( q)    J( x, q)

Proof of Theorem padicabvf
StepHypRef Expression
1 qex 10325 . . . 4  |-  QQ  e.  _V
21mptex 5709 . . 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 5339 . 2  |-  J  Fn  Prime
53padicfval 20761 . . . . 5  |-  ( p  e.  Prime  ->  ( J `
 p )  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( p ^ -u (
p  pCnt  x )
) ) ) )
6 prmnn 12757 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  p  e.  NN )
76ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  p  e.  NN )
87nncnd 9759 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  p  e.  CC )
97nnne0d 9787 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  p  =/=  0 )
10 df-ne 2451 . . . . . . . . . 10  |-  ( x  =/=  0  <->  -.  x  =  0 )
11 pcqcl 12905 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  (
x  e.  QQ  /\  x  =/=  0 ) )  ->  ( p  pCnt  x )  e.  ZZ )
1211anassrs 631 . . . . . . . . . 10  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  x  =/=  0
)  ->  ( p  pCnt  x )  e.  ZZ )
1310, 12sylan2br 464 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  (
p  pCnt  x )  e.  ZZ )
148, 9, 13expnegd 11248 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  (
p ^ -u (
p  pCnt  x )
)  =  ( 1  /  ( p ^
( p  pCnt  x
) ) ) )
158, 9, 13exprecd 11249 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  (
( 1  /  p
) ^ ( p 
pCnt  x ) )  =  ( 1  /  (
p ^ ( p 
pCnt  x ) ) ) )
1614, 15eqtr4d 2321 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  x  e.  QQ )  /\  -.  x  =  0 )  ->  (
p ^ -u (
p  pCnt  x )
)  =  ( ( 1  /  p ) ^ ( p  pCnt  x ) ) )
1716ifeq2da 3594 . . . . . 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 4109 . . . . 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 2318 . . . 4  |-  ( p  e.  Prime  ->  ( J `
 p )  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( ( 1  /  p
) ^ ( p 
pCnt  x ) ) ) ) )
206nnrecred 9788 . . . . . 6  |-  ( p  e.  Prime  ->  ( 1  /  p )  e.  RR )
216nnred 9758 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  RR )
22 prmuz2 12772 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
23 eluz2b2 10287 . . . . . . . . . 10  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
2423simprbi 452 . . . . . . . . 9  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
2522, 24syl 17 . . . . . . . 8  |-  ( p  e.  Prime  ->  1  < 
p )
26 recgt1i 9650 . . . . . . . 8  |-  ( ( p  e.  RR  /\  1  <  p )  -> 
( 0  <  (
1  /  p )  /\  ( 1  /  p )  <  1
) )
2721, 25, 26syl2anc 644 . . . . . . 7  |-  ( p  e.  Prime  ->  ( 0  <  ( 1  /  p )  /\  (
1  /  p )  <  1 ) )
2827simpld 447 . . . . . 6  |-  ( p  e.  Prime  ->  0  < 
( 1  /  p
) )
2927simprd 451 . . . . . 6  |-  ( p  e.  Prime  ->  ( 1  /  p )  <  1 )
30 0xr 8875 . . . . . . 7  |-  0  e.  RR*
31 ressxr 8873 . . . . . . . 8  |-  RR  C_  RR*
32 1re 8834 . . . . . . . 8  |-  1  e.  RR
3331, 32sselii 3180 . . . . . . 7  |-  1  e.  RR*
34 elioo2 10693 . . . . . . 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 655 . . . . . 6  |-  ( ( 1  /  p )  e.  ( 0 (,) 1 )  <->  ( (
1  /  p )  e.  RR  /\  0  <  ( 1  /  p
)  /\  ( 1  /  p )  <  1 ) )
3620, 28, 29, 35syl3anbrc 1138 . . . . 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 2286 . . . . . 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 20775 . . . . 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 651 . . . 4  |-  ( p  e.  Prime  ->  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  ( ( 1  /  p ) ^ ( p  pCnt  x ) ) ) )  e.  A )
4219, 41eqeltrd 2360 . . 3  |-  ( p  e.  Prime  ->  ( J `
 p )  e.  A )
4342rgen 2611 . 2  |-  A. p  e.  Prime  ( J `  p )  e.  A
44 ffnfv 5648 . 2  |-  ( J : Prime --> A  <->  ( J  Fn  Prime  /\  A. p  e.  Prime  ( J `  p )  e.  A
) )
454, 43, 44mpbir2an 888 1  |-  J : Prime --> A
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178    /\ wa 360    /\ w3a 936    = wceq 1625    e. wcel 1687    =/= wne 2449   A.wral 2546   ifcif 3568   class class class wbr 4026    e. cmpt 4080    Fn wfn 5218   -->wf 5219   ` cfv 5223  (class class class)co 5821   RRcr 8733   0cc0 8734   1c1 8735   RR*cxr 8863    < clt 8864   -ucneg 9035    / cdiv 9420   NNcn 9743   2c2 9792   ZZcz 10021   ZZ>=cuz 10227   QQcq 10313   (,)cioo 10652   ^cexp 11100   Primecprime 12754    pCnt cpc 12885   ↾s cress 13145  AbsValcabv 15577  ℂfldccnfld 16373
This theorem is referenced by:  ostth  20784
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-cnex 8790  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811  ax-pre-sup 8812  ax-addf 8813  ax-mulf 8814
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-int 3866  df-iun 3910  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-1st 6085  df-2nd 6086  df-tpos 6197  df-iota 6254  df-riota 6301  df-recs 6385  df-rdg 6420  df-1o 6476  df-2o 6477  df-oadd 6480  df-er 6657  df-map 6771  df-en 6861  df-dom 6862  df-sdom 6863  df-fin 6864  df-sup 7191  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037  df-div 9421  df-nn 9744  df-2 9801  df-3 9802  df-4 9803  df-5 9804  df-6 9805  df-7 9806  df-8 9807  df-9 9808  df-10 9809  df-n0 9963  df-z 10022  df-dec 10122  df-uz 10228  df-q 10314  df-rp 10352  df-ioo 10656  df-ico 10658  df-fz 10779  df-fl 10921  df-mod 10970  df-seq 11043  df-exp 11101  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-dvds 12528  df-gcd 12682  df-prm 12755  df-pc 12886  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-starv 13219  df-tset 13223  df-ple 13224  df-ds 13226  df-0g 13400  df-mnd 14363  df-grp 14485  df-minusg 14486  df-subg 14614  df-cmn 15087  df-mgp 15322  df-rng 15336  df-cring 15337  df-ur 15338  df-oppr 15401  df-dvdsr 15419  df-unit 15420  df-invr 15450  df-dvr 15461  df-drng 15510  df-subrg 15539  df-abv 15578  df-cnfld 16374
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