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Theorem qaa 19719
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 10346 . 2  |-  ( A  e.  QQ  ->  A  e.  CC )
2 qsscn 10343 . . . . . . 7  |-  QQ  C_  CC
3 1z 10069 . . . . . . . 8  |-  1  e.  ZZ
4 zq 10338 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
53, 4ax-mp 8 . . . . . . 7  |-  1  e.  QQ
6 plyid 19607 . . . . . . 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 19604 . . . . . 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 10348 . . . . . 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 10350 . . . . . 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 10349 . . . . . . 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 19617 . . . 4  |-  ( A  e.  QQ  ->  (
X p  o F  -  ( CC  X.  { A } ) )  e.  (Poly `  QQ ) )
19 peano2cn 9000 . . . . . 6  |-  ( A  e.  CC  ->  ( A  +  1 )  e.  CC )
201, 19syl 15 . . . . 5  |-  ( A  e.  QQ  ->  ( A  +  1 )  e.  CC )
21 fnresi 5377 . . . . . . . . . . 11  |-  (  _I  |`  CC )  Fn  CC
22 df-idp 19587 . . . . . . . . . . . 12  |-  X p  =  (  _I  |`  CC )
2322fneq1i 5354 . . . . . . . . . . 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 5445 . . . . . . . . 9  |-  ( A  e.  QQ  ->  ( CC  X.  { A }
)  Fn  CC )
27 cnex 8834 . . . . . . . . . 10  |-  CC  e.  _V
2827a1i 10 . . . . . . . . 9  |-  ( A  e.  QQ  ->  CC  e.  _V )
29 inidm 3391 . . . . . . . . 9  |-  ( CC 
i^i  CC )  =  CC
3022fveq1i 5542 . . . . . . . . . . 11  |-  ( X p `  ( A  +  1 ) )  =  ( (  _I  |`  CC ) `  ( A  +  1 ) )
31 fvresi 5727 . . . . . . . . . . 11  |-  ( ( A  +  1 )  e.  CC  ->  (
(  _I  |`  CC ) `
 ( A  + 
1 ) )  =  ( A  +  1 ) )
3230, 31syl5eq 2340 . . . . . . . . . 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 5743 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  ( A  +  1
)  e.  CC )  ->  ( ( CC 
X.  { A }
) `  ( A  +  1 ) )  =  A )
3525, 26, 28, 28, 29, 33, 34ofval 6103 . . . . . . . 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 8811 . . . . . . . 8  |-  1  e.  CC
38 pncan2 9074 . . . . . . . 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 2328 . . . . . 6  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  ( A  +  1 ) )  =  1 )
41 ax-1ne0 8822 . . . . . . 7  |-  1  =/=  0
4241a1i 10 . . . . . 6  |-  ( A  e.  QQ  ->  1  =/=  0 )
4340, 42eqnetrd 2477 . . . . 5  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  ( A  +  1 ) )  =/=  0 )
44 ne0p 19605 . . . . 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 3762 . . . 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 5542 . . . . . . . 8  |-  ( X p `  A )  =  ( (  _I  |`  CC ) `  A
)
49 fvresi 5727 . . . . . . . 8  |-  ( A  e.  CC  ->  (
(  _I  |`  CC ) `
 A )  =  A )
5048, 49syl5eq 2340 . . . . . . 7  |-  ( A  e.  CC  ->  (
X p `  A
)  =  A )
5150adantl 452 . . . . . 6  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( X p `  A )  =  A )
52 fvconst2g 5743 . . . . . 6  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( ( CC  X.  { A } ) `  A )  =  A )
5325, 26, 28, 28, 29, 51, 52ofval 6103 . . . . 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 9161 . . . 4  |-  ( A  e.  QQ  ->  ( A  -  A )  =  0 )
5654, 55eqtrd 2328 . . 3  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  A )  =  0 )
57 fveq1 5540 . . . . 5  |-  ( f  =  ( X p  o F  -  ( CC  X.  { A }
) )  ->  (
f `  A )  =  ( ( X p  o F  -  ( CC  X.  { A } ) ) `  A ) )
5857eqeq1d 2304 . . . 4  |-  ( f  =  ( X p  o F  -  ( CC  X.  { A }
) )  ->  (
( f `  A
)  =  0  <->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  A )  =  0 ) )
5958rspcev 2897 . . 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 19718 . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   _Vcvv 2801    \ cdif 3162    C_ wss 3165   {csn 3653    _I cid 4320    X. cxp 4703    |` cres 4707    Fn wfn 5266   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054   ZZcz 10040   QQcq 10332   0 pc0p 19040  Polycply 19582   X pcidp 19583   AAcaa 19710
This theorem is referenced by:  qssaa  19720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-idp 19587  df-coe 19588  df-dgr 19589  df-aa 19711
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