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Theorem qaa 21925
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 11081 . 2  |-  ( A  e.  QQ  ->  A  e.  CC )
2 qsscn 11078 . . . . . . 7  |-  QQ  C_  CC
3 1z 10790 . . . . . . . 8  |-  1  e.  ZZ
4 zq 11073 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
53, 4ax-mp 5 . . . . . . 7  |-  1  e.  QQ
6 plyid 21813 . . . . . . 7  |-  ( ( QQ  C_  CC  /\  1  e.  QQ )  ->  Xp  e.  (Poly `  QQ ) )
72, 5, 6mp2an 672 . . . . . 6  |-  Xp  e.  (Poly `  QQ )
87a1i 11 . . . . 5  |-  ( A  e.  QQ  ->  Xp  e.  (Poly `  QQ ) )
9 plyconst 21810 . . . . . 6  |-  ( ( QQ  C_  CC  /\  A  e.  QQ )  ->  ( CC  X.  { A }
)  e.  (Poly `  QQ ) )
102, 9mpan 670 . . . . 5  |-  ( A  e.  QQ  ->  ( CC  X.  { A }
)  e.  (Poly `  QQ ) )
11 qaddcl 11083 . . . . . 6  |-  ( ( x  e.  QQ  /\  y  e.  QQ )  ->  ( x  +  y )  e.  QQ )
1211adantl 466 . . . . 5  |-  ( ( A  e.  QQ  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  ( x  +  y )  e.  QQ )
13 qmulcl 11085 . . . . . 6  |-  ( ( x  e.  QQ  /\  y  e.  QQ )  ->  ( x  x.  y
)  e.  QQ )
1413adantl 466 . . . . 5  |-  ( ( A  e.  QQ  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  ( x  x.  y )  e.  QQ )
15 qnegcl 11084 . . . . . . 7  |-  ( 1  e.  QQ  ->  -u 1  e.  QQ )
165, 15ax-mp 5 . . . . . 6  |-  -u 1  e.  QQ
1716a1i 11 . . . . 5  |-  ( A  e.  QQ  ->  -u 1  e.  QQ )
188, 10, 12, 14, 17plysub 21823 . . . 4  |-  ( A  e.  QQ  ->  (
Xp  oF  -  ( CC  X.  { A } ) )  e.  (Poly `  QQ ) )
19 peano2cn 9655 . . . . . 6  |-  ( A  e.  CC  ->  ( A  +  1 )  e.  CC )
201, 19syl 16 . . . . 5  |-  ( A  e.  QQ  ->  ( A  +  1 )  e.  CC )
21 fnresi 5639 . . . . . . . . . . 11  |-  (  _I  |`  CC )  Fn  CC
22 df-idp 21793 . . . . . . . . . . . 12  |-  Xp  =  (  _I  |`  CC )
2322fneq1i 5616 . . . . . . . . . . 11  |-  ( Xp  Fn  CC  <->  (  _I  |`  CC )  Fn  CC )
2421, 23mpbir 209 . . . . . . . . . 10  |-  Xp  Fn  CC
2524a1i 11 . . . . . . . . 9  |-  ( A  e.  QQ  ->  Xp  Fn  CC )
26 fnconstg 5709 . . . . . . . . 9  |-  ( A  e.  QQ  ->  ( CC  X.  { A }
)  Fn  CC )
27 cnex 9477 . . . . . . . . . 10  |-  CC  e.  _V
2827a1i 11 . . . . . . . . 9  |-  ( A  e.  QQ  ->  CC  e.  _V )
29 inidm 3670 . . . . . . . . 9  |-  ( CC 
i^i  CC )  =  CC
3022fveq1i 5803 . . . . . . . . . . 11  |-  ( Xp `  ( A  +  1 ) )  =  ( (  _I  |`  CC ) `  ( A  +  1 ) )
31 fvresi 6016 . . . . . . . . . . 11  |-  ( ( A  +  1 )  e.  CC  ->  (
(  _I  |`  CC ) `
 ( A  + 
1 ) )  =  ( A  +  1 ) )
3230, 31syl5eq 2507 . . . . . . . . . 10  |-  ( ( A  +  1 )  e.  CC  ->  (
Xp `  ( A  +  1 ) )  =  ( A  +  1 ) )
3332adantl 466 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  ( A  +  1
)  e.  CC )  ->  ( Xp `  ( A  + 
1 ) )  =  ( A  +  1 ) )
34 fvconst2g 6043 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  ( A  +  1
)  e.  CC )  ->  ( ( CC 
X.  { A }
) `  ( A  +  1 ) )  =  A )
3525, 26, 28, 28, 29, 33, 34ofval 6442 . . . . . . . 8  |-  ( ( A  e.  QQ  /\  ( A  +  1
)  e.  CC )  ->  ( ( Xp  oF  -  ( CC  X.  { A } ) ) `  ( A  +  1
) )  =  ( ( A  +  1 )  -  A ) )
3620, 35mpdan 668 . . . . . . 7  |-  ( A  e.  QQ  ->  (
( Xp  oF  -  ( CC 
X.  { A }
) ) `  ( A  +  1 ) )  =  ( ( A  +  1 )  -  A ) )
37 ax-1cn 9454 . . . . . . . 8  |-  1  e.  CC
38 pncan2 9731 . . . . . . . 8  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  A
)  =  1 )
391, 37, 38sylancl 662 . . . . . . 7  |-  ( A  e.  QQ  ->  (
( A  +  1 )  -  A )  =  1 )
4036, 39eqtrd 2495 . . . . . 6  |-  ( A  e.  QQ  ->  (
( Xp  oF  -  ( CC 
X.  { A }
) ) `  ( A  +  1 ) )  =  1 )
41 ax-1ne0 9465 . . . . . . 7  |-  1  =/=  0
4241a1i 11 . . . . . 6  |-  ( A  e.  QQ  ->  1  =/=  0 )
4340, 42eqnetrd 2745 . . . . 5  |-  ( A  e.  QQ  ->  (
( Xp  oF  -  ( CC 
X.  { A }
) ) `  ( A  +  1 ) )  =/=  0 )
44 ne0p 21811 . . . . 5  |-  ( ( ( A  +  1 )  e.  CC  /\  ( ( Xp  oF  -  ( CC  X.  { A }
) ) `  ( A  +  1 ) )  =/=  0 )  ->  ( Xp  oF  -  ( CC  X.  { A }
) )  =/=  0p )
4520, 43, 44syl2anc 661 . . . 4  |-  ( A  e.  QQ  ->  (
Xp  oF  -  ( CC  X.  { A } ) )  =/=  0p )
46 eldifsn 4111 . . . 4  |-  ( ( Xp  oF  -  ( CC  X.  { A } ) )  e.  ( (Poly `  QQ )  \  { 0p } )  <->  ( (
Xp  oF  -  ( CC  X.  { A } ) )  e.  (Poly `  QQ )  /\  ( Xp  oF  -  ( CC  X.  { A }
) )  =/=  0p ) )
4718, 45, 46sylanbrc 664 . . 3  |-  ( A  e.  QQ  ->  (
Xp  oF  -  ( CC  X.  { A } ) )  e.  ( (Poly `  QQ )  \  { 0p } ) )
4822fveq1i 5803 . . . . . . . 8  |-  ( Xp `  A )  =  ( (  _I  |`  CC ) `  A
)
49 fvresi 6016 . . . . . . . 8  |-  ( A  e.  CC  ->  (
(  _I  |`  CC ) `
 A )  =  A )
5048, 49syl5eq 2507 . . . . . . 7  |-  ( A  e.  CC  ->  (
Xp `  A
)  =  A )
5150adantl 466 . . . . . 6  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( Xp `  A )  =  A )
52 fvconst2g 6043 . . . . . 6  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( ( CC  X.  { A } ) `  A )  =  A )
5325, 26, 28, 28, 29, 51, 52ofval 6442 . . . . 5  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) ) `  A
)  =  ( A  -  A ) )
541, 53mpdan 668 . . . 4  |-  ( A  e.  QQ  ->  (
( Xp  oF  -  ( CC 
X.  { A }
) ) `  A
)  =  ( A  -  A ) )
551subidd 9821 . . . 4  |-  ( A  e.  QQ  ->  ( A  -  A )  =  0 )
5654, 55eqtrd 2495 . . 3  |-  ( A  e.  QQ  ->  (
( Xp  oF  -  ( CC 
X.  { A }
) ) `  A
)  =  0 )
57 fveq1 5801 . . . . 5  |-  ( f  =  ( Xp  oF  -  ( CC  X.  { A }
) )  ->  (
f `  A )  =  ( ( Xp  oF  -  ( CC  X.  { A } ) ) `  A ) )
5857eqeq1d 2456 . . . 4  |-  ( f  =  ( Xp  oF  -  ( CC  X.  { A }
) )  ->  (
( f `  A
)  =  0  <->  (
( Xp  oF  -  ( CC 
X.  { A }
) ) `  A
)  =  0 ) )
5958rspcev 3179 . . 3  |-  ( ( ( Xp  oF  -  ( CC 
X.  { A }
) )  e.  ( (Poly `  QQ )  \  { 0p }
)  /\  ( (
Xp  oF  -  ( CC  X.  { A } ) ) `
 A )  =  0 )  ->  E. f  e.  ( (Poly `  QQ )  \  { 0p } ) ( f `
 A )  =  0 )
6047, 56, 59syl2anc 661 . 2  |-  ( A  e.  QQ  ->  E. f  e.  ( (Poly `  QQ )  \  { 0p } ) ( f `
 A )  =  0 )
61 elqaa 21924 . 2  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  { 0p } ) ( f `
 A )  =  0 ) )
621, 60, 61sylanbrc 664 1  |-  ( A  e.  QQ  ->  A  e.  AA )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   E.wrex 2800   _Vcvv 3078    \ cdif 3436    C_ wss 3439   {csn 3988    _I cid 4742    X. cxp 4949    |` cres 4953    Fn wfn 5524   ` cfv 5529  (class class class)co 6203    oFcof 6431   CCcc 9394   0cc0 9396   1c1 9397    + caddc 9399    x. cmul 9401    - cmin 9709   -ucneg 9710   ZZcz 10760   QQcq 11067   0pc0p 21283  Polycply 21788   Xpcidp 21789   AAcaa 21916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474  ax-addf 9475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-q 11068  df-rp 11106  df-fz 11558  df-fzo 11669  df-fl 11762  df-mod 11829  df-seq 11927  df-exp 11986  df-hash 12224  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087  df-rlim 13088  df-sum 13285  df-0p 21284  df-ply 21792  df-idp 21793  df-coe 21794  df-dgr 21795  df-aa 21917
This theorem is referenced by:  qssaa  21926
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