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Theorem ostthlem2 23791
Description: Lemma for ostth 23802. Refine ostthlem1 23790 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 eluz2nn 11130 . . 3  |-  ( n  e.  ( ZZ>= `  2
)  ->  n  e.  NN )
6 fveq2 5856 . . . . . . 7  |-  ( p  =  1  ->  ( F `  p )  =  ( F ` 
1 ) )
7 fveq2 5856 . . . . . . 7  |-  ( p  =  1  ->  ( G `  p )  =  ( G ` 
1 ) )
86, 7eqeq12d 2465 . . . . . 6  |-  ( p  =  1  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  1 )  =  ( G `  1
) ) )
98imbi2d 316 . . . . 5  |-  ( p  =  1  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) ) ) )
10 fveq2 5856 . . . . . . 7  |-  ( p  =  y  ->  ( F `  p )  =  ( F `  y ) )
11 fveq2 5856 . . . . . . 7  |-  ( p  =  y  ->  ( G `  p )  =  ( G `  y ) )
1210, 11eqeq12d 2465 . . . . . 6  |-  ( p  =  y  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  y )  =  ( G `  y ) ) )
1312imbi2d 316 . . . . 5  |-  ( p  =  y  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  y
)  =  ( G `
 y ) ) ) )
14 fveq2 5856 . . . . . . 7  |-  ( p  =  z  ->  ( F `  p )  =  ( F `  z ) )
15 fveq2 5856 . . . . . . 7  |-  ( p  =  z  ->  ( G `  p )  =  ( G `  z ) )
1614, 15eqeq12d 2465 . . . . . 6  |-  ( p  =  z  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  z )  =  ( G `  z ) ) )
1716imbi2d 316 . . . . 5  |-  ( p  =  z  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  z
)  =  ( G `
 z ) ) ) )
18 fveq2 5856 . . . . . . 7  |-  ( p  =  ( y  x.  z )  ->  ( F `  p )  =  ( F `  ( y  x.  z
) ) )
19 fveq2 5856 . . . . . . 7  |-  ( p  =  ( y  x.  z )  ->  ( G `  p )  =  ( G `  ( y  x.  z
) ) )
2018, 19eqeq12d 2465 . . . . . 6  |-  ( p  =  ( y  x.  z )  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  ( y  x.  z
) )  =  ( G `  ( y  x.  z ) ) ) )
2120imbi2d 316 . . . . 5  |-  ( p  =  ( y  x.  z )  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  (
y  x.  z ) )  =  ( G `
 ( y  x.  z ) ) ) ) )
22 fveq2 5856 . . . . . . 7  |-  ( p  =  n  ->  ( F `  p )  =  ( F `  n ) )
23 fveq2 5856 . . . . . . 7  |-  ( p  =  n  ->  ( G `  p )  =  ( G `  n ) )
2422, 23eqeq12d 2465 . . . . . 6  |-  ( p  =  n  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  n )  =  ( G `  n ) ) )
2524imbi2d 316 . . . . 5  |-  ( p  =  n  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  n
)  =  ( G `
 n ) ) ) )
26 ax-1ne0 9564 . . . . . . 7  |-  1  =/=  0
271qrng1 23785 . . . . . . . 8  |-  1  =  ( 1r `  Q )
281qrng0 23784 . . . . . . . 8  |-  0  =  ( 0g `  Q )
292, 27, 28abv1z 17460 . . . . . . 7  |-  ( ( F  e.  A  /\  1  =/=  0 )  -> 
( F `  1
)  =  1 )
303, 26, 29sylancl 662 . . . . . 6  |-  ( ph  ->  ( F `  1
)  =  1 )
312, 27, 28abv1z 17460 . . . . . . 7  |-  ( ( G  e.  A  /\  1  =/=  0 )  -> 
( G `  1
)  =  1 )
324, 26, 31sylancl 662 . . . . . 6  |-  ( ph  ->  ( G `  1
)  =  1 )
3330, 32eqtr4d 2487 . . . . 5  |-  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) )
34 ostthlem2.3 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( F `  p )  =  ( G `  p ) )
3534expcom 435 . . . . 5  |-  ( p  e.  Prime  ->  ( ph  ->  ( F `  p
)  =  ( G `
 p ) ) )
36 jcab 863 . . . . . 6  |-  ( (
ph  ->  ( ( F `
 y )  =  ( G `  y
)  /\  ( F `  z )  =  ( G `  z ) ) )  <->  ( ( ph  ->  ( F `  y )  =  ( G `  y ) )  /\  ( ph  ->  ( F `  z
)  =  ( G `
 z ) ) ) )
37 oveq12 6290 . . . . . . . . 9  |-  ( ( ( F `  y
)  =  ( G `
 y )  /\  ( F `  z )  =  ( G `  z ) )  -> 
( ( F `  y )  x.  ( F `  z )
)  =  ( ( G `  y )  x.  ( G `  z ) ) )
383adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  F  e.  A )
39 eluzelz 11101 . . . . . . . . . . . . 13  |-  ( y  e.  ( ZZ>= `  2
)  ->  y  e.  ZZ )
4039ad2antrl 727 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  y  e.  ZZ )
41 zq 11199 . . . . . . . . . . . 12  |-  ( y  e.  ZZ  ->  y  e.  QQ )
4240, 41syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  y  e.  QQ )
43 eluzelz 11101 . . . . . . . . . . . . 13  |-  ( z  e.  ( ZZ>= `  2
)  ->  z  e.  ZZ )
4443ad2antll 728 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  z  e.  ZZ )
45 zq 11199 . . . . . . . . . . . 12  |-  ( z  e.  ZZ  ->  z  e.  QQ )
4644, 45syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  z  e.  QQ )
471qrngbas 23782 . . . . . . . . . . . 12  |-  QQ  =  ( Base `  Q )
48 qex 11205 . . . . . . . . . . . . 13  |-  QQ  e.  _V
49 cnfldmul 18405 . . . . . . . . . . . . . 14  |-  x.  =  ( .r ` fld )
501, 49ressmulr 14732 . . . . . . . . . . . . 13  |-  ( QQ  e.  _V  ->  x.  =  ( .r `  Q ) )
5148, 50ax-mp 5 . . . . . . . . . . . 12  |-  x.  =  ( .r `  Q )
522, 47, 51abvmul 17457 . . . . . . . . . . 11  |-  ( ( F  e.  A  /\  y  e.  QQ  /\  z  e.  QQ )  ->  ( F `  ( y  x.  z ) )  =  ( ( F `  y )  x.  ( F `  z )
) )
5338, 42, 46, 52syl3anc 1229 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  ( F `  ( y  x.  z
) )  =  ( ( F `  y
)  x.  ( F `
 z ) ) )
544adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  G  e.  A )
552, 47, 51abvmul 17457 . . . . . . . . . . 11  |-  ( ( G  e.  A  /\  y  e.  QQ  /\  z  e.  QQ )  ->  ( G `  ( y  x.  z ) )  =  ( ( G `  y )  x.  ( G `  z )
) )
5654, 42, 46, 55syl3anc 1229 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  ( G `  ( y  x.  z
) )  =  ( ( G `  y
)  x.  ( G `
 z ) ) )
5753, 56eqeq12d 2465 . . . . . . . . 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 ) ) ) )
5837, 57syl5ibr 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 ) ) ) )
5958expcom 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 ) ) ) ) )
6059a2d 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 ) ) ) ) )
6136, 60syl5bir 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 ) ) ) ) )
629, 13, 17, 21, 25, 33, 35, 61prmind 14211 . . . 4  |-  ( n  e.  NN  ->  ( ph  ->  ( F `  n )  =  ( G `  n ) ) )
6362impcom 430 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  =  ( G `  n
) )
645, 63sylan2 474 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  2 )
)  ->  ( F `  n )  =  ( G `  n ) )
651, 2, 3, 4, 64ostthlem1 23790 1  |-  ( ph  ->  F  =  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   _Vcvv 3095   ` cfv 5578  (class class class)co 6281   0cc0 9495   1c1 9496    x. cmul 9500   NNcn 10543   2c2 10592   ZZcz 10871   ZZ>=cuz 11092   QQcq 11193   Primecprime 14199   ↾s cress 14615   .rcmulr 14680  AbsValcabv 17444  ℂfldccnfld 18399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-q 11194  df-rp 11232  df-ico 11546  df-fz 11684  df-seq 12090  df-exp 12149  df-dvds 13969  df-prm 14200  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-starv 14694  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-0g 14821  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-grp 16036  df-minusg 16037  df-subg 16177  df-cmn 16779  df-mgp 17121  df-ur 17133  df-ring 17179  df-cring 17180  df-oppr 17251  df-dvdsr 17269  df-unit 17270  df-invr 17300  df-dvr 17311  df-drng 17377  df-subrg 17406  df-abv 17445  df-cnfld 18400
This theorem is referenced by:  ostth1  23796  ostth3  23801
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