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Theorem ostthlem2 24084
Description: Lemma for ostth 24095. Refine ostthlem1 24083 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 11081 . . 3  |-  ( n  e.  ( ZZ>= `  2
)  ->  n  e.  NN )
6 fveq2 5803 . . . . . . 7  |-  ( p  =  1  ->  ( F `  p )  =  ( F ` 
1 ) )
7 fveq2 5803 . . . . . . 7  |-  ( p  =  1  ->  ( G `  p )  =  ( G ` 
1 ) )
86, 7eqeq12d 2422 . . . . . 6  |-  ( p  =  1  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  1 )  =  ( G `  1
) ) )
98imbi2d 314 . . . . 5  |-  ( p  =  1  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) ) ) )
10 fveq2 5803 . . . . . . 7  |-  ( p  =  y  ->  ( F `  p )  =  ( F `  y ) )
11 fveq2 5803 . . . . . . 7  |-  ( p  =  y  ->  ( G `  p )  =  ( G `  y ) )
1210, 11eqeq12d 2422 . . . . . 6  |-  ( p  =  y  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  y )  =  ( G `  y ) ) )
1312imbi2d 314 . . . . 5  |-  ( p  =  y  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  y
)  =  ( G `
 y ) ) ) )
14 fveq2 5803 . . . . . . 7  |-  ( p  =  z  ->  ( F `  p )  =  ( F `  z ) )
15 fveq2 5803 . . . . . . 7  |-  ( p  =  z  ->  ( G `  p )  =  ( G `  z ) )
1614, 15eqeq12d 2422 . . . . . 6  |-  ( p  =  z  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  z )  =  ( G `  z ) ) )
1716imbi2d 314 . . . . 5  |-  ( p  =  z  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  z
)  =  ( G `
 z ) ) ) )
18 fveq2 5803 . . . . . . 7  |-  ( p  =  ( y  x.  z )  ->  ( F `  p )  =  ( F `  ( y  x.  z
) ) )
19 fveq2 5803 . . . . . . 7  |-  ( p  =  ( y  x.  z )  ->  ( G `  p )  =  ( G `  ( y  x.  z
) ) )
2018, 19eqeq12d 2422 . . . . . 6  |-  ( p  =  ( y  x.  z )  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  ( y  x.  z
) )  =  ( G `  ( y  x.  z ) ) ) )
2120imbi2d 314 . . . . 5  |-  ( p  =  ( y  x.  z )  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  (
y  x.  z ) )  =  ( G `
 ( y  x.  z ) ) ) ) )
22 fveq2 5803 . . . . . . 7  |-  ( p  =  n  ->  ( F `  p )  =  ( F `  n ) )
23 fveq2 5803 . . . . . . 7  |-  ( p  =  n  ->  ( G `  p )  =  ( G `  n ) )
2422, 23eqeq12d 2422 . . . . . 6  |-  ( p  =  n  ->  (
( F `  p
)  =  ( G `
 p )  <->  ( F `  n )  =  ( G `  n ) ) )
2524imbi2d 314 . . . . 5  |-  ( p  =  n  ->  (
( ph  ->  ( F `
 p )  =  ( G `  p
) )  <->  ( ph  ->  ( F `  n
)  =  ( G `
 n ) ) ) )
26 ax-1ne0 9509 . . . . . . 7  |-  1  =/=  0
271qrng1 24078 . . . . . . . 8  |-  1  =  ( 1r `  Q )
281qrng0 24077 . . . . . . . 8  |-  0  =  ( 0g `  Q )
292, 27, 28abv1z 17691 . . . . . . 7  |-  ( ( F  e.  A  /\  1  =/=  0 )  -> 
( F `  1
)  =  1 )
303, 26, 29sylancl 660 . . . . . 6  |-  ( ph  ->  ( F `  1
)  =  1 )
312, 27, 28abv1z 17691 . . . . . . 7  |-  ( ( G  e.  A  /\  1  =/=  0 )  -> 
( G `  1
)  =  1 )
324, 26, 31sylancl 660 . . . . . 6  |-  ( ph  ->  ( G `  1
)  =  1 )
3330, 32eqtr4d 2444 . . . . 5  |-  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) )
34 ostthlem2.3 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( F `  p )  =  ( G `  p ) )
3534expcom 433 . . . . 5  |-  ( p  e.  Prime  ->  ( ph  ->  ( F `  p
)  =  ( G `
 p ) ) )
36 jcab 862 . . . . . 6  |-  ( (
ph  ->  ( ( F `
 y )  =  ( G `  y
)  /\  ( F `  z )  =  ( G `  z ) ) )  <->  ( ( ph  ->  ( F `  y )  =  ( G `  y ) )  /\  ( ph  ->  ( F `  z
)  =  ( G `
 z ) ) ) )
37 oveq12 6241 . . . . . . . . 9  |-  ( ( ( F `  y
)  =  ( G `
 y )  /\  ( F `  z )  =  ( G `  z ) )  -> 
( ( F `  y )  x.  ( F `  z )
)  =  ( ( G `  y )  x.  ( G `  z ) ) )
383adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  F  e.  A )
39 eluzelz 11052 . . . . . . . . . . . . 13  |-  ( y  e.  ( ZZ>= `  2
)  ->  y  e.  ZZ )
4039ad2antrl 726 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  y  e.  ZZ )
41 zq 11149 . . . . . . . . . . . 12  |-  ( y  e.  ZZ  ->  y  e.  QQ )
4240, 41syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  y  e.  QQ )
43 eluzelz 11052 . . . . . . . . . . . . 13  |-  ( z  e.  ( ZZ>= `  2
)  ->  z  e.  ZZ )
4443ad2antll 727 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  z  e.  ZZ )
45 zq 11149 . . . . . . . . . . . 12  |-  ( z  e.  ZZ  ->  z  e.  QQ )
4644, 45syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  z  e.  QQ )
471qrngbas 24075 . . . . . . . . . . . 12  |-  QQ  =  ( Base `  Q )
48 qex 11155 . . . . . . . . . . . . 13  |-  QQ  e.  _V
49 cnfldmul 18636 . . . . . . . . . . . . . 14  |-  x.  =  ( .r ` fld )
501, 49ressmulr 14856 . . . . . . . . . . . . 13  |-  ( QQ  e.  _V  ->  x.  =  ( .r `  Q ) )
5148, 50ax-mp 5 . . . . . . . . . . . 12  |-  x.  =  ( .r `  Q )
522, 47, 51abvmul 17688 . . . . . . . . . . 11  |-  ( ( F  e.  A  /\  y  e.  QQ  /\  z  e.  QQ )  ->  ( F `  ( y  x.  z ) )  =  ( ( F `  y )  x.  ( F `  z )
) )
5338, 42, 46, 52syl3anc 1228 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  ( F `  ( y  x.  z
) )  =  ( ( F `  y
)  x.  ( F `
 z ) ) )
544adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  G  e.  A )
552, 47, 51abvmul 17688 . . . . . . . . . . 11  |-  ( ( G  e.  A  /\  y  e.  QQ  /\  z  e.  QQ )  ->  ( G `  ( y  x.  z ) )  =  ( ( G `  y )  x.  ( G `  z )
) )
5654, 42, 46, 55syl3anc 1228 . . . . . . . . . 10  |-  ( (
ph  /\  ( y  e.  ( ZZ>= `  2 )  /\  z  e.  ( ZZ>=
`  2 ) ) )  ->  ( G `  ( y  x.  z
) )  =  ( ( G `  y
)  x.  ( G `
 z ) ) )
5753, 56eqeq12d 2422 . . . . . . . . 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 433 . . . . . . 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 14328 . . . 4  |-  ( n  e.  NN  ->  ( ph  ->  ( F `  n )  =  ( G `  n ) ) )
6362impcom 428 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  =  ( G `  n
) )
645, 63sylan2 472 . 2  |-  ( (
ph  /\  n  e.  ( ZZ>= `  2 )
)  ->  ( F `  n )  =  ( G `  n ) )
651, 2, 3, 4, 64ostthlem1 24083 1  |-  ( ph  ->  F  =  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840    =/= wne 2596   _Vcvv 3056   ` cfv 5523  (class class class)co 6232   0cc0 9440   1c1 9441    x. cmul 9445   NNcn 10494   2c2 10544   ZZcz 10823   ZZ>=cuz 11043   QQcq 11143   Primecprime 14316   ↾s cress 14732   .rcmulr 14800  AbsValcabv 17675  ℂfldccnfld 18630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-addf 9519  ax-mulf 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-tpos 6910  df-recs 6997  df-rdg 7031  df-1o 7085  df-2o 7086  df-oadd 7089  df-er 7266  df-map 7377  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-q 11144  df-rp 11182  df-ico 11504  df-fz 11642  df-seq 12060  df-exp 12119  df-dvds 14086  df-prm 14317  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-starv 14814  df-tset 14818  df-ple 14819  df-ds 14821  df-unif 14822  df-0g 14946  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-grp 16271  df-minusg 16272  df-subg 16412  df-cmn 17014  df-mgp 17352  df-ur 17364  df-ring 17410  df-cring 17411  df-oppr 17482  df-dvdsr 17500  df-unit 17501  df-invr 17531  df-dvr 17542  df-drng 17608  df-subrg 17637  df-abv 17676  df-cnfld 18631
This theorem is referenced by:  ostth1  24089  ostth3  24094
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