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Theorem ostthlem2 20739
Description: Lemma for ostth 20750. Refine ostthlem1 20738 so that it is sufficient to only show equality on the primes. (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
Hypotheses
Ref Expression
qrng.q  |-  Q  =  (flds  QQ )
qabsabv.a  |-  A  =  (AbsVal `  Q )
ostthlem1.1  |-  ( ph  ->  F  e.  A )
ostthlem1.2  |-  ( ph  ->  G  e.  A )
ostthlem2.3  |-  ( (
ph  /\  p  e.  Prime )  ->  ( F `  p )  =  ( G `  p ) )
Assertion
Ref Expression
ostthlem2  |-  ( ph  ->  F  =  G )
Distinct variable groups:    G, p    ph, p    A, p    F, p
Allowed substitution hint:    Q( p)

Proof of Theorem ostthlem2
StepHypRef Expression
1 qrng.q . 2  |-  Q  =  (flds  QQ )
2 qabsabv.a . 2  |-  A  =  (AbsVal `  Q )
3 ostthlem1.1 . 2  |-  ( ph  ->  F  e.  A )
4 ostthlem1.2 . 2  |-  ( ph  ->  G  e.  A )
5 eluz2b2 10257 . . . 4  |-  ( n  e.  ( ZZ>= `  2
)  <->  ( n  e.  NN  /\  1  < 
n ) )
65simplbi 448 . . 3  |-  ( n  e.  ( ZZ>= `  2
)  ->  n  e.  NN )
7 fveq2 5458 . . . . . . 7  |-  ( p  =  1  ->  ( F `  p )  =  ( F ` 
1 ) )
8 fveq2 5458 . . . . . . 7  |-  ( p  =  1  ->  ( G `  p )  =  ( G ` 
1 ) )
97, 8eqeq12d 2272 . . . . . 6  |-  ( p  =  1  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  1 )  =  ( G `  1
) ) )
109imbi2d 309 . . . . 5  |-  ( p  =  1  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) ) ) )
11 fveq2 5458 . . . . . . 7  |-  ( p  =  y  ->  ( F `  p )  =  ( F `  y ) )
12 fveq2 5458 . . . . . . 7  |-  ( p  =  y  ->  ( G `  p )  =  ( G `  y ) )
1311, 12eqeq12d 2272 . . . . . 6  |-  ( p  =  y  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  y )  =  ( G `  y ) ) )
1413imbi2d 309 . . . . 5  |-  ( p  =  y  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  y
)  =  ( G `
 y ) ) ) )
15 fveq2 5458 . . . . . . 7  |-  ( p  =  z  ->  ( F `  p )  =  ( F `  z ) )
16 fveq2 5458 . . . . . . 7  |-  ( p  =  z  ->  ( G `  p )  =  ( G `  z ) )
1715, 16eqeq12d 2272 . . . . . 6  |-  ( p  =  z  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  z )  =  ( G `  z ) ) )
1817imbi2d 309 . . . . 5  |-  ( p  =  z  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  z
)  =  ( G `
 z ) ) ) )
19 fveq2 5458 . . . . . . 7  |-  ( p  =  ( y  x.  z )  ->  ( F `  p )  =  ( F `  ( y  x.  z
) ) )
20 fveq2 5458 . . . . . . 7  |-  ( p  =  ( y  x.  z )  ->  ( G `  p )  =  ( G `  ( y  x.  z
) ) )
2119, 20eqeq12d 2272 . . . . . 6  |-  ( p  =  ( y  x.  z )  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  ( y  x.  z
) )  =  ( G `  ( y  x.  z ) ) ) )
2221imbi2d 309 . . . . 5  |-  ( p  =  ( y  x.  z )  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  (
y  x.  z ) )  =  ( G `
 ( y  x.  z ) ) ) ) )
23 fveq2 5458 . . . . . . 7  |-  ( p  =  n  ->  ( F `  p )  =  ( F `  n ) )
24 fveq2 5458 . . . . . . 7  |-  ( p  =  n  ->  ( G `  p )  =  ( G `  n ) )
2523, 24eqeq12d 2272 . . . . . 6  |-  ( p  =  n  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  n )  =  ( G `  n ) ) )
2625imbi2d 309 . . . . 5  |-  ( p  =  n  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  n
)  =  ( G `
 n ) ) ) )
27 ax-1ne0 8774 . . . . . . 7  |-  1  =/=  0
281qrng1 20733 . . . . . . . 8  |-  1  =  ( 1r `  Q )
291qrng0 20732 . . . . . . . 8  |-  0  =  ( 0g `  Q )
302, 28, 29abv1z 15559 . . . . . . 7  |-  ( ( F  e.  A  /\  1  =/=  0 )  -> 
( F `  1
)  =  1 )
313, 27, 30sylancl 646 . . . . . 6  |-  ( ph  ->  ( F `  1
)  =  1 )
322, 28, 29abv1z 15559 . . . . . . 7  |-  ( ( G  e.  A  /\  1  =/=  0 )  -> 
( G `  1
)  =  1 )
334, 27, 32sylancl 646 . . . . . 6  |-  ( ph  ->  ( G `  1
)  =  1 )
3431, 33eqtr4d 2293 . . . . 5  |-  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) )
35 ostthlem2.3 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( F `  p )  =  ( G `  p ) )
3635expcom 426 . . . . 5  |-  ( p  e.  Prime  ->  ( ph  ->  ( F `  p
)  =  ( G `
 p ) ) )
37 jcab 836 . . . . . 6  |-  ( (
ph  ->  ( ( F `
 y )  =  ( G `  y
)  /\  ( F `  z )  =  ( G `  z ) ) )  <->  ( ( ph  ->  ( F `  y )  =  ( G `  y ) )  /\  ( ph  ->  ( F `  z
)  =  ( G `
 z ) ) ) )
38 oveq12 5801 . . . . . . . . 9  |-  ( ( ( F `  y
)  =  ( G `
 y )  /\  ( F `  z )  =  ( G `  z ) )  -> 
( ( F `  y )  x.  ( F `  z )
)  =  ( ( G `  y )  x.  ( G `  z ) ) )
393adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  F  e.  A )
40 eluzelz 10205 . . . . . . . . . . . . 13  |-  ( y  e.  ( ZZ>= `  2
)  ->  y  e.  ZZ )
4140ad2antrl 711 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  y  e.  ZZ )
42 zq 10289 . . . . . . . . . . . 12  |-  ( y  e.  ZZ  ->  y  e.  QQ )
4341, 42syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  y  e.  QQ )
44 eluzelz 10205 . . . . . . . . . . . . 13  |-  ( z  e.  ( ZZ>= `  2
)  ->  z  e.  ZZ )
4544ad2antll 712 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  z  e.  ZZ )
46 zq 10289 . . . . . . . . . . . 12  |-  ( z  e.  ZZ  ->  z  e.  QQ )
4745, 46syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  z  e.  QQ )
481qrngbas 20730 . . . . . . . . . . . 12  |-  QQ  =  ( Base `  Q )
49 qex 10295 . . . . . . . . . . . . 13  |-  QQ  e.  _V
50 cnfldmul 16347 . . . . . . . . . . . . . 14  |-  x.  =  ( .r ` fld )
511, 50ressmulr 13223 . . . . . . . . . . . . 13  |-  ( QQ  e.  _V  ->  x.  =  ( .r `  Q ) )
5249, 51ax-mp 10 . . . . . . . . . . . 12  |-  x.  =  ( .r `  Q )
532, 48, 52abvmul 15556 . . . . . . . . . . 11  |-  ( ( F  e.  A  /\  y  e.  QQ  /\  z  e.  QQ )  ->  ( F `  ( y  x.  z ) )  =  ( ( F `  y )  x.  ( F `  z )
) )
5439, 43, 47, 53syl3anc 1187 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  ( F `  ( y  x.  z
) )  =  ( ( F `  y
)  x.  ( F `
 z ) ) )
554adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  G  e.  A )
562, 48, 52abvmul 15556 . . . . . . . . . . 11  |-  ( ( G  e.  A  /\  y  e.  QQ  /\  z  e.  QQ )  ->  ( G `  ( y  x.  z ) )  =  ( ( G `  y )  x.  ( G `  z )
) )
5755, 43, 47, 56syl3anc 1187 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  ( G `  ( y  x.  z
) )  =  ( ( G `  y
)  x.  ( G `
 z ) ) )
5854, 57eqeq12d 2272 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  ( ( F `  ( y  x.  z ) )  =  ( G `  (
y  x.  z ) )  <->  ( ( F `
 y )  x.  ( F `  z
) )  =  ( ( G `  y
)  x.  ( G `
 z ) ) ) )
5938, 58syl5ibr 214 . . . . . . . 8  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  ( (
( F `  y
)  =  ( G `
 y )  /\  ( F `  z )  =  ( G `  z ) )  -> 
( F `  (
y  x.  z ) )  =  ( G `
 ( y  x.  z ) ) ) )
6059expcom 426 . . . . . . 7  |-  ( ( y  e.  ( ZZ>= ` 
2 )  /\  z  e.  ( ZZ>= `  2 )
)  ->  ( ph  ->  ( ( ( F `
 y )  =  ( G `  y
)  /\  ( F `  z )  =  ( G `  z ) )  ->  ( F `  ( y  x.  z
) )  =  ( G `  ( y  x.  z ) ) ) ) )
6160a2d 25 . . . . . 6  |-  ( ( y  e.  ( ZZ>= ` 
2 )  /\  z  e.  ( ZZ>= `  2 )
)  ->  ( ( ph  ->  ( ( F `
 y )  =  ( G `  y
)  /\  ( F `  z )  =  ( G `  z ) ) )  ->  ( ph  ->  ( F `  ( y  x.  z
) )  =  ( G `  ( y  x.  z ) ) ) ) )
6237, 61syl5bir 211 . . . . 5  |-  ( ( y  e.  ( ZZ>= ` 
2 )  /\  z  e.  ( ZZ>= `  2 )
)  ->  ( (
( ph  ->  ( F `
 y )  =  ( G `  y
) )  /\  ( ph  ->  ( F `  z )  =  ( G `  z ) ) )  ->  ( ph  ->  ( F `  ( y  x.  z
) )  =  ( G `  ( y  x.  z ) ) ) ) )
6310, 14, 18, 22, 26, 34, 36, 62prmind 12732 . . . 4  |-  ( n  e.  NN  ->  ( ph  ->  ( F `  n )  =  ( G `  n ) ) )
6463impcom 421 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  =  ( G `  n
) )
656, 64sylan2 462 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  2 )
)  ->  ( F `  n )  =  ( G `  n ) )
661, 2, 3, 4, 65ostthlem1 20738 1  |-  ( ph  ->  F  =  G )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421   _Vcvv 2763   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   0cc0 8705   1c1 8706    x. cmul 8710    < clt 8835   NNcn 9714   2c2 9763   ZZcz 9991   ZZ>=cuz 10197   QQcq 10283   Primecprime 12720   ↾s cress 13111   .rcmulr 13171  AbsValcabv 15543  ℂfldccnfld 16339
This theorem is referenced by:  ostth1  20744  ostth3  20749
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-addf 8784  ax-mulf 8785
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-tpos 6168  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9933  df-z 9992  df-dec 10092  df-uz 10198  df-q 10284  df-rp 10322  df-ico 10628  df-fz 10749  df-seq 11013  df-exp 11071  df-divides 12494  df-prime 12721  df-struct 13112  df-ndx 13113  df-slot 13114  df-base 13115  df-sets 13116  df-ress 13117  df-plusg 13183  df-mulr 13184  df-starv 13185  df-tset 13189  df-ple 13190  df-ds 13192  df-0g 13366  df-mnd 14329  df-grp 14451  df-minusg 14452  df-subg 14580  df-cmn 15053  df-mgp 15288  df-ring 15302  df-cring 15303  df-ur 15304  df-oppr 15367  df-dvdsr 15385  df-unit 15386  df-invr 15416  df-dvr 15427  df-drng 15476  df-subrg 15505  df-abv 15544  df-cnfld 16340
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