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Theorem ostthlem2 22836
Description: Lemma for ostth 22847. Refine ostthlem1 22835 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 10923 . . . 4  |-  ( n  e.  ( ZZ>= `  2
)  <->  ( n  e.  NN  /\  1  < 
n ) )
65simplbi 457 . . 3  |-  ( n  e.  ( ZZ>= `  2
)  ->  n  e.  NN )
7 fveq2 5688 . . . . . . 7  |-  ( p  =  1  ->  ( F `  p )  =  ( F ` 
1 ) )
8 fveq2 5688 . . . . . . 7  |-  ( p  =  1  ->  ( G `  p )  =  ( G ` 
1 ) )
97, 8eqeq12d 2455 . . . . . 6  |-  ( p  =  1  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  1 )  =  ( G `  1
) ) )
109imbi2d 316 . . . . 5  |-  ( p  =  1  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) ) ) )
11 fveq2 5688 . . . . . . 7  |-  ( p  =  y  ->  ( F `  p )  =  ( F `  y ) )
12 fveq2 5688 . . . . . . 7  |-  ( p  =  y  ->  ( G `  p )  =  ( G `  y ) )
1311, 12eqeq12d 2455 . . . . . 6  |-  ( p  =  y  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  y )  =  ( G `  y ) ) )
1413imbi2d 316 . . . . 5  |-  ( p  =  y  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  y
)  =  ( G `
 y ) ) ) )
15 fveq2 5688 . . . . . . 7  |-  ( p  =  z  ->  ( F `  p )  =  ( F `  z ) )
16 fveq2 5688 . . . . . . 7  |-  ( p  =  z  ->  ( G `  p )  =  ( G `  z ) )
1715, 16eqeq12d 2455 . . . . . 6  |-  ( p  =  z  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  z )  =  ( G `  z ) ) )
1817imbi2d 316 . . . . 5  |-  ( p  =  z  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  z
)  =  ( G `
 z ) ) ) )
19 fveq2 5688 . . . . . . 7  |-  ( p  =  ( y  x.  z )  ->  ( F `  p )  =  ( F `  ( y  x.  z
) ) )
20 fveq2 5688 . . . . . . 7  |-  ( p  =  ( y  x.  z )  ->  ( G `  p )  =  ( G `  ( y  x.  z
) ) )
2119, 20eqeq12d 2455 . . . . . 6  |-  ( p  =  ( y  x.  z )  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  ( y  x.  z
) )  =  ( G `  ( y  x.  z ) ) ) )
2221imbi2d 316 . . . . 5  |-  ( p  =  ( y  x.  z )  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  (
y  x.  z ) )  =  ( G `
 ( y  x.  z ) ) ) ) )
23 fveq2 5688 . . . . . . 7  |-  ( p  =  n  ->  ( F `  p )  =  ( F `  n ) )
24 fveq2 5688 . . . . . . 7  |-  ( p  =  n  ->  ( G `  p )  =  ( G `  n ) )
2523, 24eqeq12d 2455 . . . . . 6  |-  ( p  =  n  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  n )  =  ( G `  n ) ) )
2625imbi2d 316 . . . . 5  |-  ( p  =  n  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  n
)  =  ( G `
 n ) ) ) )
27 ax-1ne0 9347 . . . . . . 7  |-  1  =/=  0
281qrng1 22830 . . . . . . . 8  |-  1  =  ( 1r `  Q )
291qrng0 22829 . . . . . . . 8  |-  0  =  ( 0g `  Q )
302, 28, 29abv1z 16897 . . . . . . 7  |-  ( ( F  e.  A  /\  1  =/=  0 )  -> 
( F `  1
)  =  1 )
313, 27, 30sylancl 657 . . . . . 6  |-  ( ph  ->  ( F `  1
)  =  1 )
322, 28, 29abv1z 16897 . . . . . . 7  |-  ( ( G  e.  A  /\  1  =/=  0 )  -> 
( G `  1
)  =  1 )
334, 27, 32sylancl 657 . . . . . 6  |-  ( ph  ->  ( G `  1
)  =  1 )
3431, 33eqtr4d 2476 . . . . 5  |-  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) )
35 ostthlem2.3 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( F `  p )  =  ( G `  p ) )
3635expcom 435 . . . . 5  |-  ( p  e.  Prime  ->  ( ph  ->  ( F `  p
)  =  ( G `
 p ) ) )
37 jcab 853 . . . . . 6  |-  ( (
ph  ->  ( ( F `
 y )  =  ( G `  y
)  /\  ( F `  z )  =  ( G `  z ) ) )  <->  ( ( ph  ->  ( F `  y )  =  ( G `  y ) )  /\  ( ph  ->  ( F `  z
)  =  ( G `
 z ) ) ) )
38 oveq12 6099 . . . . . . . . 9  |-  ( ( ( F `  y
)  =  ( G `
 y )  /\  ( F `  z )  =  ( G `  z ) )  -> 
( ( F `  y )  x.  ( F `  z )
)  =  ( ( G `  y )  x.  ( G `  z ) ) )
393adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  F  e.  A )
40 eluzelz 10866 . . . . . . . . . . . . 13  |-  ( y  e.  ( ZZ>= `  2
)  ->  y  e.  ZZ )
4140ad2antrl 722 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  y  e.  ZZ )
42 zq 10955 . . . . . . . . . . . 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 10866 . . . . . . . . . . . . 13  |-  ( z  e.  ( ZZ>= `  2
)  ->  z  e.  ZZ )
4544ad2antll 723 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  z  e.  ZZ )
46 zq 10955 . . . . . . . . . . . 12  |-  ( z  e.  ZZ  ->  z  e.  QQ )
4745, 46syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  z  e.  QQ )
481qrngbas 22827 . . . . . . . . . . . 12  |-  QQ  =  ( Base `  Q )
49 qex 10961 . . . . . . . . . . . . 13  |-  QQ  e.  _V
50 cnfldmul 17783 . . . . . . . . . . . . . 14  |-  x.  =  ( .r ` fld )
511, 50ressmulr 14287 . . . . . . . . . . . . 13  |-  ( QQ  e.  _V  ->  x.  =  ( .r `  Q ) )
5249, 51ax-mp 5 . . . . . . . . . . . 12  |-  x.  =  ( .r `  Q )
532, 48, 52abvmul 16894 . . . . . . . . . . 11  |-  ( ( F  e.  A  /\  y  e.  QQ  /\  z  e.  QQ )  ->  ( F `  ( y  x.  z ) )  =  ( ( F `  y )  x.  ( F `  z )
) )
5439, 43, 47, 53syl3anc 1213 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  ( F `  ( y  x.  z
) )  =  ( ( F `  y
)  x.  ( F `
 z ) ) )
554adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  G  e.  A )
562, 48, 52abvmul 16894 . . . . . . . . . . 11  |-  ( ( G  e.  A  /\  y  e.  QQ  /\  z  e.  QQ )  ->  ( G `  ( y  x.  z ) )  =  ( ( G `  y )  x.  ( G `  z )
) )
5755, 43, 47, 56syl3anc 1213 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  ( G `  ( y  x.  z
) )  =  ( ( G `  y
)  x.  ( G `
 z ) ) )
5854, 57eqeq12d 2455 . . . . . . . . 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 221 . . . . . . . 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 435 . . . . . . 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 26 . . . . . 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 218 . . . . 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 13771 . . . 4  |-  ( n  e.  NN  ->  ( ph  ->  ( F `  n )  =  ( G `  n ) ) )
6463impcom 430 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  =  ( G `  n
) )
656, 64sylan2 471 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  2 )
)  ->  ( F `  n )  =  ( G `  n ) )
661, 2, 3, 4, 65ostthlem1 22835 1  |-  ( ph  ->  F  =  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   _Vcvv 2970   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   0cc0 9278   1c1 9279    x. cmul 9283    < clt 9414   NNcn 10318   2c2 10367   ZZcz 10642   ZZ>=cuz 10857   QQcq 10949   Primecprime 13759   ↾s cress 14171   .rcmulr 14235  AbsValcabv 16881  ℂfldccnfld 17777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-ico 11302  df-fz 11434  df-seq 11803  df-exp 11862  df-dvds 13532  df-prm 13760  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-subg 15671  df-cmn 16272  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-drng 16814  df-subrg 16843  df-abv 16882  df-cnfld 17778
This theorem is referenced by:  ostth1  22841  ostth3  22846
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