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Theorem qaa 22901
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 11157 . 2  |-  ( A  e.  QQ  ->  A  e.  CC )
2 qsscn 11154 . . . . . . 7  |-  QQ  C_  CC
3 1z 10853 . . . . . . . 8  |-  1  e.  ZZ
4 zq 11149 . . . . . . . 8  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
53, 4ax-mp 5 . . . . . . 7  |-  1  e.  QQ
6 plyid 22788 . . . . . . 7  |-  ( ( QQ  C_  CC  /\  1  e.  QQ )  ->  Xp  e.  (Poly `  QQ ) )
72, 5, 6mp2an 670 . . . . . 6  |-  Xp  e.  (Poly `  QQ )
87a1i 11 . . . . 5  |-  ( A  e.  QQ  ->  Xp  e.  (Poly `  QQ ) )
9 plyconst 22785 . . . . . 6  |-  ( ( QQ  C_  CC  /\  A  e.  QQ )  ->  ( CC  X.  { A }
)  e.  (Poly `  QQ ) )
102, 9mpan 668 . . . . 5  |-  ( A  e.  QQ  ->  ( CC  X.  { A }
)  e.  (Poly `  QQ ) )
11 qaddcl 11159 . . . . . 6  |-  ( ( x  e.  QQ  /\  y  e.  QQ )  ->  ( x  +  y )  e.  QQ )
1211adantl 464 . . . . 5  |-  ( ( A  e.  QQ  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  ( x  +  y )  e.  QQ )
13 qmulcl 11161 . . . . . 6  |-  ( ( x  e.  QQ  /\  y  e.  QQ )  ->  ( x  x.  y
)  e.  QQ )
1413adantl 464 . . . . 5  |-  ( ( A  e.  QQ  /\  ( x  e.  QQ  /\  y  e.  QQ ) )  ->  ( x  x.  y )  e.  QQ )
15 qnegcl 11160 . . . . . . 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 22798 . . . 4  |-  ( A  e.  QQ  ->  (
Xp  oF  -  ( CC  X.  { A } ) )  e.  (Poly `  QQ ) )
19 peano2cn 9704 . . . . . 6  |-  ( A  e.  CC  ->  ( A  +  1 )  e.  CC )
201, 19syl 17 . . . . 5  |-  ( A  e.  QQ  ->  ( A  +  1 )  e.  CC )
21 fnresi 5633 . . . . . . . . . . 11  |-  (  _I  |`  CC )  Fn  CC
22 df-idp 22768 . . . . . . . . . . . 12  |-  Xp  =  (  _I  |`  CC )
2322fneq1i 5610 . . . . . . . . . . 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 5710 . . . . . . . . 9  |-  ( A  e.  QQ  ->  ( CC  X.  { A }
)  Fn  CC )
27 cnex 9521 . . . . . . . . . 10  |-  CC  e.  _V
2827a1i 11 . . . . . . . . 9  |-  ( A  e.  QQ  ->  CC  e.  _V )
29 inidm 3645 . . . . . . . . 9  |-  ( CC 
i^i  CC )  =  CC
3022fveq1i 5804 . . . . . . . . . . 11  |-  ( Xp `  ( A  +  1 ) )  =  ( (  _I  |`  CC ) `  ( A  +  1 ) )
31 fvresi 6031 . . . . . . . . . . 11  |-  ( ( A  +  1 )  e.  CC  ->  (
(  _I  |`  CC ) `
 ( A  + 
1 ) )  =  ( A  +  1 ) )
3230, 31syl5eq 2453 . . . . . . . . . 10  |-  ( ( A  +  1 )  e.  CC  ->  (
Xp `  ( A  +  1 ) )  =  ( A  +  1 ) )
3332adantl 464 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  ( A  +  1
)  e.  CC )  ->  ( Xp `  ( A  + 
1 ) )  =  ( A  +  1 ) )
34 fvconst2g 6059 . . . . . . . . 9  |-  ( ( A  e.  QQ  /\  ( A  +  1
)  e.  CC )  ->  ( ( CC 
X.  { A }
) `  ( A  +  1 ) )  =  A )
3525, 26, 28, 28, 29, 33, 34ofval 6484 . . . . . . . 8  |-  ( ( A  e.  QQ  /\  ( A  +  1
)  e.  CC )  ->  ( ( Xp  oF  -  ( CC  X.  { A } ) ) `  ( A  +  1
) )  =  ( ( A  +  1 )  -  A ) )
3620, 35mpdan 666 . . . . . . 7  |-  ( A  e.  QQ  ->  (
( Xp  oF  -  ( CC 
X.  { A }
) ) `  ( A  +  1 ) )  =  ( ( A  +  1 )  -  A ) )
37 ax-1cn 9498 . . . . . . . 8  |-  1  e.  CC
38 pncan2 9781 . . . . . . . 8  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  A
)  =  1 )
391, 37, 38sylancl 660 . . . . . . 7  |-  ( A  e.  QQ  ->  (
( A  +  1 )  -  A )  =  1 )
4036, 39eqtrd 2441 . . . . . 6  |-  ( A  e.  QQ  ->  (
( Xp  oF  -  ( CC 
X.  { A }
) ) `  ( A  +  1 ) )  =  1 )
41 ax-1ne0 9509 . . . . . . 7  |-  1  =/=  0
4241a1i 11 . . . . . 6  |-  ( A  e.  QQ  ->  1  =/=  0 )
4340, 42eqnetrd 2694 . . . . 5  |-  ( A  e.  QQ  ->  (
( Xp  oF  -  ( CC 
X.  { A }
) ) `  ( A  +  1 ) )  =/=  0 )
44 ne0p 22786 . . . . 5  |-  ( ( ( A  +  1 )  e.  CC  /\  ( ( Xp  oF  -  ( CC  X.  { A }
) ) `  ( A  +  1 ) )  =/=  0 )  ->  ( Xp  oF  -  ( CC  X.  { A }
) )  =/=  0p )
4520, 43, 44syl2anc 659 . . . 4  |-  ( A  e.  QQ  ->  (
Xp  oF  -  ( CC  X.  { A } ) )  =/=  0p )
46 eldifsn 4094 . . . 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 662 . . 3  |-  ( A  e.  QQ  ->  (
Xp  oF  -  ( CC  X.  { A } ) )  e.  ( (Poly `  QQ )  \  { 0p } ) )
4822fveq1i 5804 . . . . . . . 8  |-  ( Xp `  A )  =  ( (  _I  |`  CC ) `  A
)
49 fvresi 6031 . . . . . . . 8  |-  ( A  e.  CC  ->  (
(  _I  |`  CC ) `
 A )  =  A )
5048, 49syl5eq 2453 . . . . . . 7  |-  ( A  e.  CC  ->  (
Xp `  A
)  =  A )
5150adantl 464 . . . . . 6  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( Xp `  A )  =  A )
52 fvconst2g 6059 . . . . . 6  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( ( CC  X.  { A } ) `  A )  =  A )
5325, 26, 28, 28, 29, 51, 52ofval 6484 . . . . 5  |-  ( ( A  e.  QQ  /\  A  e.  CC )  ->  ( ( Xp  oF  -  ( CC  X.  { A }
) ) `  A
)  =  ( A  -  A ) )
541, 53mpdan 666 . . . 4  |-  ( A  e.  QQ  ->  (
( Xp  oF  -  ( CC 
X.  { A }
) ) `  A
)  =  ( A  -  A ) )
551subidd 9873 . . . 4  |-  ( A  e.  QQ  ->  ( A  -  A )  =  0 )
5654, 55eqtrd 2441 . . 3  |-  ( A  e.  QQ  ->  (
( Xp  oF  -  ( CC 
X.  { A }
) ) `  A
)  =  0 )
57 fveq1 5802 . . . . 5  |-  ( f  =  ( Xp  oF  -  ( CC  X.  { A }
) )  ->  (
f `  A )  =  ( ( Xp  oF  -  ( CC  X.  { A } ) ) `  A ) )
5857eqeq1d 2402 . . . 4  |-  ( f  =  ( Xp  oF  -  ( CC  X.  { A }
) )  ->  (
( f `  A
)  =  0  <->  (
( Xp  oF  -  ( CC 
X.  { A }
) ) `  A
)  =  0 ) )
5958rspcev 3157 . . 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 659 . 2  |-  ( A  e.  QQ  ->  E. f  e.  ( (Poly `  QQ )  \  { 0p } ) ( f `
 A )  =  0 )
61 elqaa 22900 . 2  |-  ( A  e.  AA  <->  ( A  e.  CC  /\  E. f  e.  ( (Poly `  QQ )  \  { 0p } ) ( f `
 A )  =  0 ) )
621, 60, 61sylanbrc 662 1  |-  ( A  e.  QQ  ->  A  e.  AA )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840    =/= wne 2596   E.wrex 2752   _Vcvv 3056    \ cdif 3408    C_ wss 3411   {csn 3969    _I cid 4730    X. cxp 4938    |` cres 4942    Fn wfn 5518   ` cfv 5523  (class class class)co 6232    oFcof 6473   CCcc 9438   0cc0 9440   1c1 9441    + caddc 9443    x. cmul 9445    - cmin 9759   -ucneg 9760   ZZcz 10823   QQcq 11143   0pc0p 22258  Polycply 22763   Xpcidp 22764   AAcaa 22892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518  ax-addf 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-of 6475  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-map 7377  df-pm 7378  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-sup 7853  df-oi 7887  df-card 8270  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-n0 10755  df-z 10824  df-uz 11044  df-q 11144  df-rp 11182  df-fz 11642  df-fzo 11766  df-fl 11877  df-mod 11946  df-seq 12060  df-exp 12119  df-hash 12358  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-clim 13365  df-rlim 13366  df-sum 13563  df-0p 22259  df-ply 22767  df-idp 22768  df-coe 22769  df-dgr 22770  df-aa 22893
This theorem is referenced by:  qssaa  22902
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