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Theorem ostthlem2 21322
Description: Lemma for ostth 21333. Refine ostthlem1 21321 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
Dummy variables  n  y  z are mutually distinct and distinct from all other variables.
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 10548 . . . 4  |-  ( n  e.  ( ZZ>= `  2
)  <->  ( n  e.  NN  /\  1  < 
n ) )
65simplbi 447 . . 3  |-  ( n  e.  ( ZZ>= `  2
)  ->  n  e.  NN )
7 fveq2 5728 . . . . . . 7  |-  ( p  =  1  ->  ( F `  p )  =  ( F ` 
1 ) )
8 fveq2 5728 . . . . . . 7  |-  ( p  =  1  ->  ( G `  p )  =  ( G ` 
1 ) )
97, 8eqeq12d 2450 . . . . . 6  |-  ( p  =  1  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  1 )  =  ( G `  1
) ) )
109imbi2d 308 . . . . 5  |-  ( p  =  1  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) ) ) )
11 fveq2 5728 . . . . . . 7  |-  ( p  =  y  ->  ( F `  p )  =  ( F `  y ) )
12 fveq2 5728 . . . . . . 7  |-  ( p  =  y  ->  ( G `  p )  =  ( G `  y ) )
1311, 12eqeq12d 2450 . . . . . 6  |-  ( p  =  y  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  y )  =  ( G `  y ) ) )
1413imbi2d 308 . . . . 5  |-  ( p  =  y  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  y
)  =  ( G `
 y ) ) ) )
15 fveq2 5728 . . . . . . 7  |-  ( p  =  z  ->  ( F `  p )  =  ( F `  z ) )
16 fveq2 5728 . . . . . . 7  |-  ( p  =  z  ->  ( G `  p )  =  ( G `  z ) )
1715, 16eqeq12d 2450 . . . . . 6  |-  ( p  =  z  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  z )  =  ( G `  z ) ) )
1817imbi2d 308 . . . . 5  |-  ( p  =  z  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  z
)  =  ( G `
 z ) ) ) )
19 fveq2 5728 . . . . . . 7  |-  ( p  =  ( y  x.  z )  ->  ( F `  p )  =  ( F `  ( y  x.  z
) ) )
20 fveq2 5728 . . . . . . 7  |-  ( p  =  ( y  x.  z )  ->  ( G `  p )  =  ( G `  ( y  x.  z
) ) )
2119, 20eqeq12d 2450 . . . . . 6  |-  ( p  =  ( y  x.  z )  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  ( y  x.  z
) )  =  ( G `  ( y  x.  z ) ) ) )
2221imbi2d 308 . . . . 5  |-  ( p  =  ( y  x.  z )  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  (
y  x.  z ) )  =  ( G `
 ( y  x.  z ) ) ) ) )
23 fveq2 5728 . . . . . . 7  |-  ( p  =  n  ->  ( F `  p )  =  ( F `  n ) )
24 fveq2 5728 . . . . . . 7  |-  ( p  =  n  ->  ( G `  p )  =  ( G `  n ) )
2523, 24eqeq12d 2450 . . . . . 6  |-  ( p  =  n  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  n )  =  ( G `  n ) ) )
2625imbi2d 308 . . . . 5  |-  ( p  =  n  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  n
)  =  ( G `
 n ) ) ) )
27 ax-1ne0 9059 . . . . . . 7  |-  1  =/=  0
281qrng1 21316 . . . . . . . 8  |-  1  =  ( 1r `  Q )
291qrng0 21315 . . . . . . . 8  |-  0  =  ( 0g `  Q )
302, 28, 29abv1z 15920 . . . . . . 7  |-  ( ( F  e.  A  /\  1  =/=  0 )  -> 
( F `  1
)  =  1 )
313, 27, 30sylancl 644 . . . . . 6  |-  ( ph  ->  ( F `  1
)  =  1 )
322, 28, 29abv1z 15920 . . . . . . 7  |-  ( ( G  e.  A  /\  1  =/=  0 )  -> 
( G `  1
)  =  1 )
334, 27, 32sylancl 644 . . . . . 6  |-  ( ph  ->  ( G `  1
)  =  1 )
3431, 33eqtr4d 2471 . . . . 5  |-  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) )
35 ostthlem2.3 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( F `  p )  =  ( G `  p ) )
3635expcom 425 . . . . 5  |-  ( p  e.  Prime  ->  ( ph  ->  ( F `  p
)  =  ( G `
 p ) ) )
37 jcab 834 . . . . . 6  |-  ( (
ph  ->  ( ( F `
 y )  =  ( G `  y
)  /\  ( F `  z )  =  ( G `  z ) ) )  <->  ( ( ph  ->  ( F `  y )  =  ( G `  y ) )  /\  ( ph  ->  ( F `  z
)  =  ( G `
 z ) ) ) )
38 oveq12 6090 . . . . . . . . 9  |-  ( ( ( F `  y
)  =  ( G `
 y )  /\  ( F `  z )  =  ( G `  z ) )  -> 
( ( F `  y )  x.  ( F `  z )
)  =  ( ( G `  y )  x.  ( G `  z ) ) )
393adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  F  e.  A )
40 eluzelz 10496 . . . . . . . . . . . . 13  |-  ( y  e.  ( ZZ>= `  2
)  ->  y  e.  ZZ )
4140ad2antrl 709 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  y  e.  ZZ )
42 zq 10580 . . . . . . . . . . . 12  |-  ( y  e.  ZZ  ->  y  e.  QQ )
4341, 42syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  y  e.  QQ )
44 eluzelz 10496 . . . . . . . . . . . . 13  |-  ( z  e.  ( ZZ>= `  2
)  ->  z  e.  ZZ )
4544ad2antll 710 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  z  e.  ZZ )
46 zq 10580 . . . . . . . . . . . 12  |-  ( z  e.  ZZ  ->  z  e.  QQ )
4745, 46syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  z  e.  QQ )
481qrngbas 21313 . . . . . . . . . . . 12  |-  QQ  =  ( Base `  Q )
49 qex 10586 . . . . . . . . . . . . 13  |-  QQ  e.  _V
50 cnfldmul 16709 . . . . . . . . . . . . . 14  |-  x.  =  ( .r ` fld )
511, 50ressmulr 13582 . . . . . . . . . . . . 13  |-  ( QQ  e.  _V  ->  x.  =  ( .r `  Q ) )
5249, 51ax-mp 8 . . . . . . . . . . . 12  |-  x.  =  ( .r `  Q )
532, 48, 52abvmul 15917 . . . . . . . . . . 11  |-  ( ( F  e.  A  /\  y  e.  QQ  /\  z  e.  QQ )  ->  ( F `  ( y  x.  z ) )  =  ( ( F `  y )  x.  ( F `  z )
) )
5439, 43, 47, 53syl3anc 1184 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  ( F `  ( y  x.  z
) )  =  ( ( F `  y
)  x.  ( F `
 z ) ) )
554adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  G  e.  A )
562, 48, 52abvmul 15917 . . . . . . . . . . 11  |-  ( ( G  e.  A  /\  y  e.  QQ  /\  z  e.  QQ )  ->  ( G `  ( y  x.  z ) )  =  ( ( G `  y )  x.  ( G `  z )
) )
5755, 43, 47, 56syl3anc 1184 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  ( G `  ( y  x.  z
) )  =  ( ( G `  y
)  x.  ( G `
 z ) ) )
5854, 57eqeq12d 2450 . . . . . . . . 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 213 . . . . . . . 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 425 . . . . . . 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 24 . . . . . 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 210 . . . . 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 13091 . . . 4  |-  ( n  e.  NN  ->  ( ph  ->  ( F `  n )  =  ( G `  n ) ) )
6463impcom 420 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  =  ( G `  n
) )
656, 64sylan2 461 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  2 )
)  ->  ( F `  n )  =  ( G `  n ) )
661, 2, 3, 4, 65ostthlem1 21321 1  |-  ( ph  ->  F  =  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   0cc0 8990   1c1 8991    x. cmul 8995    < clt 9120   NNcn 10000   2c2 10049   ZZcz 10282   ZZ>=cuz 10488   QQcq 10574   Primecprime 13079   ↾s cress 13470   .rcmulr 13530  AbsValcabv 15904  ℂfldccnfld 16703
This theorem is referenced by:  ostth1  21327  ostth3  21332
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-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-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-ico 10922  df-fz 11044  df-seq 11324  df-exp 11383  df-dvds 12853  df-prm 13080  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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