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Theorem qaa 20232
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 10580 . 2  |-  ( A  e.  QQ  ->  A  e.  CC )
2 qsscn 10577 . . . . . . 7  |-  QQ  C_  CC
3 1z 10303 . . . . . . . 8  |-  1  e.  ZZ
4 zq 10572 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
53, 4ax-mp 8 . . . . . . 7  |-  1  e.  QQ
6 plyid 20120 . . . . . . 7  |-  ( ( QQ  C_  CC  /\  1  e.  QQ )  ->  X p  e.  (Poly `  QQ ) )
72, 5, 6mp2an 654 . . . . . 6  |-  X p  e.  (Poly `  QQ )
87a1i 11 . . . . 5  |-  ( A  e.  QQ  ->  X p  e.  (Poly `  QQ ) )
9 plyconst 20117 . . . . . 6  |-  ( ( QQ  C_  CC  /\  A  e.  QQ )  ->  ( CC  X.  { A }
)  e.  (Poly `  QQ ) )
102, 9mpan 652 . . . . 5  |-  ( A  e.  QQ  ->  ( CC  X.  { A }
)  e.  (Poly `  QQ ) )
11 qaddcl 10582 . . . . . 6  |-  ( ( x  e.  QQ  /\  y  e.  QQ )  ->  ( x  +  y )  e.  QQ )
1211adantl 453 . . . . 5  |-  ( ( A  e.  QQ  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  ( x  +  y )  e.  QQ )
13 qmulcl 10584 . . . . . 6  |-  ( ( x  e.  QQ  /\  y  e.  QQ )  ->  ( x  x.  y
)  e.  QQ )
1413adantl 453 . . . . 5  |-  ( ( A  e.  QQ  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  ( x  x.  y )  e.  QQ )
15 qnegcl 10583 . . . . . . 7  |-  ( 1  e.  QQ  ->  -u 1  e.  QQ )
165, 15ax-mp 8 . . . . . 6  |-  -u 1  e.  QQ
1716a1i 11 . . . . 5  |-  ( A  e.  QQ  ->  -u 1  e.  QQ )
188, 10, 12, 14, 17plysub 20130 . . . 4  |-  ( A  e.  QQ  ->  (
X p  o F  -  ( CC  X.  { A } ) )  e.  (Poly `  QQ ) )
19 peano2cn 9230 . . . . . 6  |-  ( A  e.  CC  ->  ( A  +  1 )  e.  CC )
201, 19syl 16 . . . . 5  |-  ( A  e.  QQ  ->  ( A  +  1 )  e.  CC )
21 fnresi 5554 . . . . . . . . . . 11  |-  (  _I  |`  CC )  Fn  CC
22 df-idp 20100 . . . . . . . . . . . 12  |-  X p  =  (  _I  |`  CC )
2322fneq1i 5531 . . . . . . . . . . 11  |-  ( X p  Fn  CC  <->  (  _I  |`  CC )  Fn  CC )
2421, 23mpbir 201 . . . . . . . . . 10  |-  X p  Fn  CC
2524a1i 11 . . . . . . . . 9  |-  ( A  e.  QQ  ->  X p  Fn  CC )
26 fnconstg 5623 . . . . . . . . 9  |-  ( A  e.  QQ  ->  ( CC  X.  { A }
)  Fn  CC )
27 cnex 9063 . . . . . . . . . 10  |-  CC  e.  _V
2827a1i 11 . . . . . . . . 9  |-  ( A  e.  QQ  ->  CC  e.  _V )
29 inidm 3542 . . . . . . . . 9  |-  ( CC 
i^i  CC )  =  CC
3022fveq1i 5721 . . . . . . . . . . 11  |-  ( X p `  ( A  +  1 ) )  =  ( (  _I  |`  CC ) `  ( A  +  1 ) )
31 fvresi 5916 . . . . . . . . . . 11  |-  ( ( A  +  1 )  e.  CC  ->  (
(  _I  |`  CC ) `
 ( A  + 
1 ) )  =  ( A  +  1 ) )
3230, 31syl5eq 2479 . . . . . . . . . 10  |-  ( ( A  +  1 )  e.  CC  ->  (
X p `  ( A  +  1 ) )  =  ( A  +  1 ) )
3332adantl 453 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  ( A  +  1
)  e.  CC )  ->  ( X p `
 ( A  + 
1 ) )  =  ( A  +  1 ) )
34 fvconst2g 5937 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  ( A  +  1
)  e.  CC )  ->  ( ( CC 
X.  { A }
) `  ( A  +  1 ) )  =  A )
3525, 26, 28, 28, 29, 33, 34ofval 6306 . . . . . . . 8  |-  ( ( A  e.  QQ  /\  ( A  +  1
)  e.  CC )  ->  ( ( X p  o F  -  ( CC  X.  { A } ) ) `  ( A  +  1
) )  =  ( ( A  +  1 )  -  A ) )
3620, 35mpdan 650 . . . . . . 7  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  ( A  +  1 ) )  =  ( ( A  +  1 )  -  A ) )
37 ax-1cn 9040 . . . . . . . 8  |-  1  e.  CC
38 pncan2 9304 . . . . . . . 8  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  A
)  =  1 )
391, 37, 38sylancl 644 . . . . . . 7  |-  ( A  e.  QQ  ->  (
( A  +  1 )  -  A )  =  1 )
4036, 39eqtrd 2467 . . . . . 6  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  ( A  +  1 ) )  =  1 )
41 ax-1ne0 9051 . . . . . . 7  |-  1  =/=  0
4241a1i 11 . . . . . 6  |-  ( A  e.  QQ  ->  1  =/=  0 )
4340, 42eqnetrd 2616 . . . . 5  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  ( A  +  1 ) )  =/=  0 )
44 ne0p 20118 . . . . 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 643 . . . 4  |-  ( A  e.  QQ  ->  (
X p  o F  -  ( CC  X.  { A } ) )  =/=  0 p )
46 eldifsn 3919 . . . 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 646 . . 3  |-  ( A  e.  QQ  ->  (
X p  o F  -  ( CC  X.  { A } ) )  e.  ( (Poly `  QQ )  \  { 0 p } ) )
4822fveq1i 5721 . . . . . . . 8  |-  ( X p `  A )  =  ( (  _I  |`  CC ) `  A
)
49 fvresi 5916 . . . . . . . 8  |-  ( A  e.  CC  ->  (
(  _I  |`  CC ) `
 A )  =  A )
5048, 49syl5eq 2479 . . . . . . 7  |-  ( A  e.  CC  ->  (
X p `  A
)  =  A )
5150adantl 453 . . . . . 6  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( X p `  A )  =  A )
52 fvconst2g 5937 . . . . . 6  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( ( CC  X.  { A } ) `  A )  =  A )
5325, 26, 28, 28, 29, 51, 52ofval 6306 . . . . 5  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( ( X p  o F  -  ( CC  X.  { A }
) ) `  A
)  =  ( A  -  A ) )
541, 53mpdan 650 . . . 4  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  A )  =  ( A  -  A ) )
551subidd 9391 . . . 4  |-  ( A  e.  QQ  ->  ( A  -  A )  =  0 )
5654, 55eqtrd 2467 . . 3  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  A )  =  0 )
57 fveq1 5719 . . . . 5  |-  ( f  =  ( X p  o F  -  ( CC  X.  { A }
) )  ->  (
f `  A )  =  ( ( X p  o F  -  ( CC  X.  { A } ) ) `  A ) )
5857eqeq1d 2443 . . . 4  |-  ( f  =  ( X p  o F  -  ( CC  X.  { A }
) )  ->  (
( f `  A
)  =  0  <->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  A )  =  0 ) )
5958rspcev 3044 . . 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 643 . 2  |-  ( A  e.  QQ  ->  E. f  e.  ( (Poly `  QQ )  \  { 0 p } ) ( f `
 A )  =  0 )
61 elqaa 20231 . 2  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  { 0 p } ) ( f `
 A )  =  0 ) )
621, 60, 61sylanbrc 646 1  |-  ( A  e.  QQ  ->  A  e.  AA )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   _Vcvv 2948    \ cdif 3309    C_ wss 3312   {csn 3806    _I cid 4485    X. cxp 4868    |` cres 4872    Fn wfn 5441   ` cfv 5446  (class class class)co 6073    o Fcof 6295   CCcc 8980   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    - cmin 9283   -ucneg 9284   ZZcz 10274   QQcq 10566   0 pc0p 19553  Polycply 20095   X pcidp 20096   AAcaa 20223
This theorem is referenced by:  qssaa  20233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-0p 19554  df-ply 20099  df-idp 20100  df-coe 20101  df-dgr 20102  df-aa 20224
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