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Theorem qaa 19703
Description: Every rational number is algebraic. (Contributed by Mario Carneiro, 23-Jul-2014.)
Assertion
Ref Expression
qaa  |-  ( A  e.  QQ  ->  A  e.  AA )

Proof of Theorem qaa
Dummy variables  f  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qcn 10330 . 2  |-  ( A  e.  QQ  ->  A  e.  CC )
2 qsscn 10327 . . . . . . 7  |-  QQ  C_  CC
3 1z 10053 . . . . . . . 8  |-  1  e.  ZZ
4 zq 10322 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
53, 4ax-mp 8 . . . . . . 7  |-  1  e.  QQ
6 plyid 19591 . . . . . . 7  |-  ( ( QQ  C_  CC  /\  1  e.  QQ )  ->  X p  e.  (Poly `  QQ ) )
72, 5, 6mp2an 653 . . . . . 6  |-  X p  e.  (Poly `  QQ )
87a1i 10 . . . . 5  |-  ( A  e.  QQ  ->  X p  e.  (Poly `  QQ ) )
9 plyconst 19588 . . . . . 6  |-  ( ( QQ  C_  CC  /\  A  e.  QQ )  ->  ( CC  X.  { A }
)  e.  (Poly `  QQ ) )
102, 9mpan 651 . . . . 5  |-  ( A  e.  QQ  ->  ( CC  X.  { A }
)  e.  (Poly `  QQ ) )
11 qaddcl 10332 . . . . . 6  |-  ( ( x  e.  QQ  /\  y  e.  QQ )  ->  ( x  +  y )  e.  QQ )
1211adantl 452 . . . . 5  |-  ( ( A  e.  QQ  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  ( x  +  y )  e.  QQ )
13 qmulcl 10334 . . . . . 6  |-  ( ( x  e.  QQ  /\  y  e.  QQ )  ->  ( x  x.  y
)  e.  QQ )
1413adantl 452 . . . . 5  |-  ( ( A  e.  QQ  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  ( x  x.  y )  e.  QQ )
15 qnegcl 10333 . . . . . . 7  |-  ( 1  e.  QQ  ->  -u 1  e.  QQ )
165, 15ax-mp 8 . . . . . 6  |-  -u 1  e.  QQ
1716a1i 10 . . . . 5  |-  ( A  e.  QQ  ->  -u 1  e.  QQ )
188, 10, 12, 14, 17plysub 19601 . . . 4  |-  ( A  e.  QQ  ->  (
X p  o F  -  ( CC  X.  { A } ) )  e.  (Poly `  QQ ) )
19 peano2cn 8984 . . . . . 6  |-  ( A  e.  CC  ->  ( A  +  1 )  e.  CC )
201, 19syl 15 . . . . 5  |-  ( A  e.  QQ  ->  ( A  +  1 )  e.  CC )
21 fnresi 5361 . . . . . . . . . . 11  |-  (  _I  |`  CC )  Fn  CC
22 df-idp 19571 . . . . . . . . . . . 12  |-  X p  =  (  _I  |`  CC )
2322fneq1i 5338 . . . . . . . . . . 11  |-  ( X p  Fn  CC  <->  (  _I  |`  CC )  Fn  CC )
2421, 23mpbir 200 . . . . . . . . . 10  |-  X p  Fn  CC
2524a1i 10 . . . . . . . . 9  |-  ( A  e.  QQ  ->  X p  Fn  CC )
26 fnconstg 5429 . . . . . . . . 9  |-  ( A  e.  QQ  ->  ( CC  X.  { A }
)  Fn  CC )
27 cnex 8818 . . . . . . . . . 10  |-  CC  e.  _V
2827a1i 10 . . . . . . . . 9  |-  ( A  e.  QQ  ->  CC  e.  _V )
29 inidm 3378 . . . . . . . . 9  |-  ( CC 
i^i  CC )  =  CC
3022fveq1i 5526 . . . . . . . . . . 11  |-  ( X p `  ( A  +  1 ) )  =  ( (  _I  |`  CC ) `  ( A  +  1 ) )
31 fvresi 5711 . . . . . . . . . . 11  |-  ( ( A  +  1 )  e.  CC  ->  (
(  _I  |`  CC ) `
 ( A  + 
1 ) )  =  ( A  +  1 ) )
3230, 31syl5eq 2327 . . . . . . . . . 10  |-  ( ( A  +  1 )  e.  CC  ->  (
X p `  ( A  +  1 ) )  =  ( A  +  1 ) )
3332adantl 452 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  ( A  +  1
)  e.  CC )  ->  ( X p `
 ( A  + 
1 ) )  =  ( A  +  1 ) )
34 fvconst2g 5727 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  ( A  +  1
)  e.  CC )  ->  ( ( CC 
X.  { A }
) `  ( A  +  1 ) )  =  A )
3525, 26, 28, 28, 29, 33, 34ofval 6087 . . . . . . . 8  |-  ( ( A  e.  QQ  /\  ( A  +  1
)  e.  CC )  ->  ( ( X p  o F  -  ( CC  X.  { A } ) ) `  ( A  +  1
) )  =  ( ( A  +  1 )  -  A ) )
3620, 35mpdan 649 . . . . . . 7  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  ( A  +  1 ) )  =  ( ( A  +  1 )  -  A ) )
37 ax-1cn 8795 . . . . . . . 8  |-  1  e.  CC
38 pncan2 9058 . . . . . . . 8  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  A
)  =  1 )
391, 37, 38sylancl 643 . . . . . . 7  |-  ( A  e.  QQ  ->  (
( A  +  1 )  -  A )  =  1 )
4036, 39eqtrd 2315 . . . . . 6  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  ( A  +  1 ) )  =  1 )
41 ax-1ne0 8806 . . . . . . 7  |-  1  =/=  0
4241a1i 10 . . . . . 6  |-  ( A  e.  QQ  ->  1  =/=  0 )
4340, 42eqnetrd 2464 . . . . 5  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  ( A  +  1 ) )  =/=  0 )
44 ne0p 19589 . . . . 5  |-  ( ( ( A  +  1 )  e.  CC  /\  ( ( X p  o F  -  ( CC  X.  { A }
) ) `  ( A  +  1 ) )  =/=  0 )  ->  ( X p  o F  -  ( CC  X.  { A }
) )  =/=  0 p )
4520, 43, 44syl2anc 642 . . . 4  |-  ( A  e.  QQ  ->  (
X p  o F  -  ( CC  X.  { A } ) )  =/=  0 p )
46 eldifsn 3749 . . . 4  |-  ( ( X p  o F  -  ( CC  X.  { A } ) )  e.  ( (Poly `  QQ )  \  { 0 p } )  <->  ( (
X p  o F  -  ( CC  X.  { A } ) )  e.  (Poly `  QQ )  /\  ( X p  o F  -  ( CC  X.  { A }
) )  =/=  0 p ) )
4718, 45, 46sylanbrc 645 . . 3  |-  ( A  e.  QQ  ->  (
X p  o F  -  ( CC  X.  { A } ) )  e.  ( (Poly `  QQ )  \  { 0 p } ) )
4822fveq1i 5526 . . . . . . . 8  |-  ( X p `  A )  =  ( (  _I  |`  CC ) `  A
)
49 fvresi 5711 . . . . . . . 8  |-  ( A  e.  CC  ->  (
(  _I  |`  CC ) `
 A )  =  A )
5048, 49syl5eq 2327 . . . . . . 7  |-  ( A  e.  CC  ->  (
X p `  A
)  =  A )
5150adantl 452 . . . . . 6  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( X p `  A )  =  A )
52 fvconst2g 5727 . . . . . 6  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( ( CC  X.  { A } ) `  A )  =  A )
5325, 26, 28, 28, 29, 51, 52ofval 6087 . . . . 5  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( ( X p  o F  -  ( CC  X.  { A }
) ) `  A
)  =  ( A  -  A ) )
541, 53mpdan 649 . . . 4  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  A )  =  ( A  -  A ) )
551subidd 9145 . . . 4  |-  ( A  e.  QQ  ->  ( A  -  A )  =  0 )
5654, 55eqtrd 2315 . . 3  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  A )  =  0 )
57 fveq1 5524 . . . . 5  |-  ( f  =  ( X p  o F  -  ( CC  X.  { A }
) )  ->  (
f `  A )  =  ( ( X p  o F  -  ( CC  X.  { A } ) ) `  A ) )
5857eqeq1d 2291 . . . 4  |-  ( f  =  ( X p  o F  -  ( CC  X.  { A }
) )  ->  (
( f `  A
)  =  0  <->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  A )  =  0 ) )
5958rspcev 2884 . . 3  |-  ( ( ( X p  o F  -  ( CC  X.  { A } ) )  e.  ( (Poly `  QQ )  \  {
0 p } )  /\  ( ( X p  o F  -  ( CC  X.  { A } ) ) `  A )  =  0 )  ->  E. f  e.  ( (Poly `  QQ )  \  { 0 p } ) ( f `
 A )  =  0 )
6047, 56, 59syl2anc 642 . 2  |-  ( A  e.  QQ  ->  E. f  e.  ( (Poly `  QQ )  \  { 0 p } ) ( f `
 A )  =  0 )
61 elqaa 19702 . 2  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  { 0 p } ) ( f `
 A )  =  0 ) )
621, 60, 61sylanbrc 645 1  |-  ( A  e.  QQ  ->  A  e.  AA )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640    _I cid 4304    X. cxp 4687    |` cres 4691    Fn wfn 5250   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038   ZZcz 10024   QQcq 10316   0 pc0p 19024  Polycply 19566   X pcidp 19567   AAcaa 19694
This theorem is referenced by:  qssaa  19704
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-idp 19571  df-coe 19572  df-dgr 19573  df-aa 19695
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