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Theorem ostth1 20798
Description: - Lemma for ostth 20804: trivial case. (Not that the proof is trivial, but that we are proving that the function is trivial.) If  F is equal to  1 on the primes, then by complete induction and the multiplicative property abvmul 15610 of the absolute value,  F is equal to  1 on all the integers, and ostthlem1 20792 extends this to the other rational numbers. (Contributed by Mario Carneiro, 10-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 )
) ) ) )
ostth.k  |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )
ostth.1  |-  ( ph  ->  F  e.  A )
ostth1.2  |-  ( ph  ->  A. n  e.  NN  -.  1  <  ( F `
 n ) )
ostth1.3  |-  ( ph  ->  A. n  e.  Prime  -.  ( F `  n
)  <  1 )
Assertion
Ref Expression
ostth1  |-  ( ph  ->  F  =  K )
Distinct variable groups:    n, K    x, n, q, ph    A, n, q, x    Q, n, x    n, F, q, x
Allowed substitution hints:    Q( q)    J( x, n, q)    K( x, q)

Proof of Theorem ostth1
StepHypRef Expression
1 qrng.q . 2  |-  Q  =  (flds  QQ )
2 qabsabv.a . 2  |-  A  =  (AbsVal `  Q )
3 ostth.1 . 2  |-  ( ph  ->  F  e.  A )
41qdrng 20785 . . 3  |-  Q  e.  DivRing
51qrngbas 20784 . . . 4  |-  QQ  =  ( Base `  Q )
61qrng0 20786 . . . 4  |-  0  =  ( 0g `  Q )
7 ostth.k . . . 4  |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )
82, 5, 6, 7abvtriv 15622 . . 3  |-  ( Q  e.  DivRing  ->  K  e.  A
)
94, 8mp1i 11 . 2  |-  ( ph  ->  K  e.  A )
10 ostth1.3 . . . . 5  |-  ( ph  ->  A. n  e.  Prime  -.  ( F `  n
)  <  1 )
1110r19.21bi 2654 . . . 4  |-  ( (
ph  /\  n  e.  Prime )  ->  -.  ( F `  n )  <  1 )
12 prmnn 12777 . . . . 5  |-  ( n  e.  Prime  ->  n  e.  NN )
13 ostth1.2 . . . . . 6  |-  ( ph  ->  A. n  e.  NN  -.  1  <  ( F `
 n ) )
1413r19.21bi 2654 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  -.  1  <  ( F `  n
) )
1512, 14sylan2 460 . . . 4  |-  ( (
ph  /\  n  e.  Prime )  ->  -.  1  <  ( F `  n
) )
16 nnq 10345 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  QQ )
1712, 16syl 15 . . . . . 6  |-  ( n  e.  Prime  ->  n  e.  QQ )
182, 5abvcl 15605 . . . . . 6  |-  ( ( F  e.  A  /\  n  e.  QQ )  ->  ( F `  n
)  e.  RR )
193, 17, 18syl2an 463 . . . . 5  |-  ( (
ph  /\  n  e.  Prime )  ->  ( F `  n )  e.  RR )
20 1re 8853 . . . . 5  |-  1  e.  RR
21 lttri3 8921 . . . . 5  |-  ( ( ( F `  n
)  e.  RR  /\  1  e.  RR )  ->  ( ( F `  n )  =  1  <-> 
( -.  ( F `
 n )  <  1  /\  -.  1  <  ( F `  n
) ) ) )
2219, 20, 21sylancl 643 . . . 4  |-  ( (
ph  /\  n  e.  Prime )  ->  ( ( F `  n )  =  1  <->  ( -.  ( F `  n )  <  1  /\  -.  1  <  ( F `  n ) ) ) )
2311, 15, 22mpbir2and 888 . . 3  |-  ( (
ph  /\  n  e.  Prime )  ->  ( F `  n )  =  1 )
2412adantl 452 . . . 4  |-  ( (
ph  /\  n  e.  Prime )  ->  n  e.  NN )
25 eqeq1 2302 . . . . . . . 8  |-  ( x  =  n  ->  (
x  =  0  <->  n  =  0 ) )
2625ifbid 3596 . . . . . . 7  |-  ( x  =  n  ->  if ( x  =  0 ,  0 ,  1 )  =  if ( n  =  0 ,  0 ,  1 ) )
27 c0ex 8848 . . . . . . . 8  |-  0  e.  _V
28 1ex 8849 . . . . . . . 8  |-  1  e.  _V
2927, 28ifex 3636 . . . . . . 7  |-  if ( n  =  0 ,  0 ,  1 )  e.  _V
3026, 7, 29fvmpt 5618 . . . . . 6  |-  ( n  e.  QQ  ->  ( K `  n )  =  if ( n  =  0 ,  0 ,  1 ) )
3116, 30syl 15 . . . . 5  |-  ( n  e.  NN  ->  ( K `  n )  =  if ( n  =  0 ,  0 ,  1 ) )
32 nnne0 9794 . . . . . . 7  |-  ( n  e.  NN  ->  n  =/=  0 )
3332neneqd 2475 . . . . . 6  |-  ( n  e.  NN  ->  -.  n  =  0 )
34 iffalse 3585 . . . . . 6  |-  ( -.  n  =  0  ->  if ( n  =  0 ,  0 ,  1 )  =  1 )
3533, 34syl 15 . . . . 5  |-  ( n  e.  NN  ->  if ( n  =  0 ,  0 ,  1 )  =  1 )
3631, 35eqtrd 2328 . . . 4  |-  ( n  e.  NN  ->  ( K `  n )  =  1 )
3724, 36syl 15 . . 3  |-  ( (
ph  /\  n  e.  Prime )  ->  ( K `  n )  =  1 )
3823, 37eqtr4d 2331 . 2  |-  ( (
ph  /\  n  e.  Prime )  ->  ( F `  n )  =  ( K `  n ) )
391, 2, 3, 9, 38ostthlem2 20793 1  |-  ( ph  ->  F  =  K )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   ifcif 3578   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    < clt 8883   -ucneg 9054   NNcn 9762   QQcq 10332   ^cexp 11120   Primecprime 12774    pCnt cpc 12905   ↾s cress 13165   DivRingcdr 15528  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-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-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-ico 10678  df-fz 10799  df-seq 11063  df-exp 11121  df-dvds 12548  df-prm 12775  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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