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Theorem qaa 20107
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 10520 . 2  |-  ( A  e.  QQ  ->  A  e.  CC )
2 qsscn 10517 . . . . . . 7  |-  QQ  C_  CC
3 1z 10243 . . . . . . . 8  |-  1  e.  ZZ
4 zq 10512 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
53, 4ax-mp 8 . . . . . . 7  |-  1  e.  QQ
6 plyid 19995 . . . . . . 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 19992 . . . . . 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 10522 . . . . . 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 10524 . . . . . 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 10523 . . . . . . 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 20005 . . . 4  |-  ( A  e.  QQ  ->  (
X p  o F  -  ( CC  X.  { A } ) )  e.  (Poly `  QQ ) )
19 peano2cn 9170 . . . . . 6  |-  ( A  e.  CC  ->  ( A  +  1 )  e.  CC )
201, 19syl 16 . . . . 5  |-  ( A  e.  QQ  ->  ( A  +  1 )  e.  CC )
21 fnresi 5502 . . . . . . . . . . 11  |-  (  _I  |`  CC )  Fn  CC
22 df-idp 19975 . . . . . . . . . . . 12  |-  X p  =  (  _I  |`  CC )
2322fneq1i 5479 . . . . . . . . . . 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 5571 . . . . . . . . 9  |-  ( A  e.  QQ  ->  ( CC  X.  { A }
)  Fn  CC )
27 cnex 9004 . . . . . . . . . 10  |-  CC  e.  _V
2827a1i 11 . . . . . . . . 9  |-  ( A  e.  QQ  ->  CC  e.  _V )
29 inidm 3493 . . . . . . . . 9  |-  ( CC 
i^i  CC )  =  CC
3022fveq1i 5669 . . . . . . . . . . 11  |-  ( X p `  ( A  +  1 ) )  =  ( (  _I  |`  CC ) `  ( A  +  1 ) )
31 fvresi 5863 . . . . . . . . . . 11  |-  ( ( A  +  1 )  e.  CC  ->  (
(  _I  |`  CC ) `
 ( A  + 
1 ) )  =  ( A  +  1 ) )
3230, 31syl5eq 2431 . . . . . . . . . 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 5884 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  ( A  +  1
)  e.  CC )  ->  ( ( CC 
X.  { A }
) `  ( A  +  1 ) )  =  A )
3525, 26, 28, 28, 29, 33, 34ofval 6253 . . . . . . . 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 8981 . . . . . . . 8  |-  1  e.  CC
38 pncan2 9244 . . . . . . . 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 2419 . . . . . 6  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  ( A  +  1 ) )  =  1 )
41 ax-1ne0 8992 . . . . . . 7  |-  1  =/=  0
4241a1i 11 . . . . . 6  |-  ( A  e.  QQ  ->  1  =/=  0 )
4340, 42eqnetrd 2568 . . . . 5  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  ( A  +  1 ) )  =/=  0 )
44 ne0p 19993 . . . . 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 3870 . . . 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 5669 . . . . . . . 8  |-  ( X p `  A )  =  ( (  _I  |`  CC ) `  A
)
49 fvresi 5863 . . . . . . . 8  |-  ( A  e.  CC  ->  (
(  _I  |`  CC ) `
 A )  =  A )
5048, 49syl5eq 2431 . . . . . . 7  |-  ( A  e.  CC  ->  (
X p `  A
)  =  A )
5150adantl 453 . . . . . 6  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( X p `  A )  =  A )
52 fvconst2g 5884 . . . . . 6  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( ( CC  X.  { A } ) `  A )  =  A )
5325, 26, 28, 28, 29, 51, 52ofval 6253 . . . . 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 9331 . . . 4  |-  ( A  e.  QQ  ->  ( A  -  A )  =  0 )
5654, 55eqtrd 2419 . . 3  |-  ( A  e.  QQ  ->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  A )  =  0 )
57 fveq1 5667 . . . . 5  |-  ( f  =  ( X p  o F  -  ( CC  X.  { A }
) )  ->  (
f `  A )  =  ( ( X p  o F  -  ( CC  X.  { A } ) ) `  A ) )
5857eqeq1d 2395 . . . 4  |-  ( f  =  ( X p  o F  -  ( CC  X.  { A }
) )  ->  (
( f `  A
)  =  0  <->  (
( X p  o F  -  ( CC  X.  { A } ) ) `  A )  =  0 ) )
5958rspcev 2995 . . 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 20106 . 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 1649    e. wcel 1717    =/= wne 2550   E.wrex 2650   _Vcvv 2899    \ cdif 3260    C_ wss 3263   {csn 3757    _I cid 4434    X. cxp 4816    |` cres 4820    Fn wfn 5389   ` cfv 5394  (class class class)co 6020    o Fcof 6242   CCcc 8921   0cc0 8923   1c1 8924    + caddc 8926    x. cmul 8928    - cmin 9223   -ucneg 9224   ZZcz 10214   QQcq 10506   0 pc0p 19428  Polycply 19970   X pcidp 19971   AAcaa 20098
This theorem is referenced by:  qssaa  20108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-q 10507  df-rp 10545  df-fz 10976  df-fzo 11066  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-rlim 12210  df-sum 12407  df-0p 19429  df-ply 19974  df-idp 19975  df-coe 19976  df-dgr 19977  df-aa 20099
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