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Theorem ostth1 20778
Description: - Lemma for ostth 20784: 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 15590 of the absolute value,  F is equal to  1 on all the integers, and ostthlem1 20772 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 groups:    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 20765 . . 3  |-  Q  e.  DivRing
51qrngbas 20764 . . . 4  |-  QQ  =  ( Base `  Q )
61qrng0 20766 . . . 4  |-  0  =  ( 0g `  Q )
7 ostth.k . . . 4  |-  K  =  ( x  e.  QQ  |->  if ( x  =  0 ,  0 ,  1 ) )
82, 5, 6, 7abvtriv 15602 . . 3  |-  ( Q  e.  DivRing  ->  K  e.  A
)
94, 8mp1i 13 . 2  |-  ( ph  ->  K  e.  A )
10 ostth1.3 . . . . 5  |-  ( ph  ->  A. n  e.  Prime  -.  ( F `  n
)  <  1 )
1110r19.21bi 2644 . . . 4  |-  ( (
ph  /\  n  e.  Prime )  ->  -.  ( F `  n )  <  1 )
12 prmnn 12757 . . . . 5  |-  ( n  e.  Prime  ->  n  e.  NN )
13 ostth1.2 . . . . . 6  |-  ( ph  ->  A. n  e.  NN  -.  1  <  ( F `
 n ) )
1413r19.21bi 2644 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  -.  1  <  ( F `  n
) )
1512, 14sylan2 462 . . . 4  |-  ( (
ph  /\  n  e.  Prime )  ->  -.  1  <  ( F `  n
) )
16 nnq 10326 . . . . . . 7  |-  ( n  e.  NN  ->  n  e.  QQ )
1712, 16syl 17 . . . . . 6  |-  ( n  e.  Prime  ->  n  e.  QQ )
182, 5abvcl 15585 . . . . . 6  |-  ( ( F  e.  A  /\  n  e.  QQ )  ->  ( F `  n
)  e.  RR )
193, 17, 18syl2an 465 . . . . 5  |-  ( (
ph  /\  n  e.  Prime )  ->  ( F `  n )  e.  RR )
20 1re 8834 . . . . 5  |-  1  e.  RR
21 lttri3 8902 . . . . 5  |-  ( ( ( F `  n
)  e.  RR  /\  1  e.  RR )  ->  ( ( F `  n )  =  1  <-> 
( -.  ( F `
 n )  <  1  /\  -.  1  <  ( F `  n
) ) ) )
2219, 20, 21sylancl 645 . . . 4  |-  ( (
ph  /\  n  e.  Prime )  ->  ( ( F `  n )  =  1  <->  ( -.  ( F `  n )  <  1  /\  -.  1  <  ( F `  n ) ) ) )
2311, 15, 22mpbir2and 890 . . 3  |-  ( (
ph  /\  n  e.  Prime )  ->  ( F `  n )  =  1 )
2412adantl 454 . . . 4  |-  ( (
ph  /\  n  e.  Prime )  ->  n  e.  NN )
25 eqeq1 2292 . . . . . . . 8  |-  ( x  =  n  ->  (
x  =  0  <->  n  =  0 ) )
2625ifbid 3586 . . . . . . 7  |-  ( x  =  n  ->  if ( x  =  0 ,  0 ,  1 )  =  if ( n  =  0 ,  0 ,  1 ) )
27 c0ex 8829 . . . . . . . 8  |-  0  e.  _V
28 1ex 8830 . . . . . . . 8  |-  1  e.  _V
2927, 28ifex 3626 . . . . . . 7  |-  if ( n  =  0 ,  0 ,  1 )  e.  _V
3026, 7, 29fvmpt 5565 . . . . . 6  |-  ( n  e.  QQ  ->  ( K `  n )  =  if ( n  =  0 ,  0 ,  1 ) )
3116, 30syl 17 . . . . 5  |-  ( n  e.  NN  ->  ( K `  n )  =  if ( n  =  0 ,  0 ,  1 ) )
32 nnne0 9775 . . . . . . 7  |-  ( n  e.  NN  ->  n  =/=  0 )
3332neneqd 2465 . . . . . 6  |-  ( n  e.  NN  ->  -.  n  =  0 )
34 iffalse 3575 . . . . . 6  |-  ( -.  n  =  0  ->  if ( n  =  0 ,  0 ,  1 )  =  1 )
3533, 34syl 17 . . . . 5  |-  ( n  e.  NN  ->  if ( n  =  0 ,  0 ,  1 )  =  1 )
3631, 35eqtrd 2318 . . . 4  |-  ( n  e.  NN  ->  ( K `  n )  =  1 )
3724, 36syl 17 . . 3  |-  ( (
ph  /\  n  e.  Prime )  ->  ( K `  n )  =  1 )
3823, 37eqtr4d 2321 . 2  |-  ( (
ph  /\  n  e.  Prime )  ->  ( F `  n )  =  ( K `  n ) )
391, 2, 3, 9, 38ostthlem2 20773 1  |-  ( ph  ->  F  =  K )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1625    e. wcel 1687   A.wral 2546   ifcif 3568   class class class wbr 4026    e. cmpt 4080   ` cfv 5223  (class class class)co 5821   RRcr 8733   0cc0 8734   1c1 8735    < clt 8864   -ucneg 9035   NNcn 9743   QQcq 10313   ^cexp 11100   Primecprime 12754    pCnt cpc 12885   ↾s cress 13145   DivRingcdr 15508  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-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-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-ico 10658  df-fz 10779  df-seq 11043  df-exp 11101  df-dvds 12528  df-prm 12755  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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