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Theorem dgraa0p 27322
Description: A rational polynomial of degree less than an algebraic number cannot be zero at that number unless it is the zero polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Assertion
Ref Expression
dgraa0p  |-  ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P
)  <  (degAA `  A
) )  ->  (
( P `  A
)  =  0  <->  P  =  0 p ) )

Proof of Theorem dgraa0p
StepHypRef Expression
1 simpl3 962 . . . . . 6  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  (deg `  P
)  <  (degAA `  A
) )
2 simpl2 961 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  P  e.  (Poly `  QQ ) )
3 dgrcl 20144 . . . . . . . . 9  |-  ( P  e.  (Poly `  QQ )  ->  (deg `  P
)  e.  NN0 )
42, 3syl 16 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  (deg `  P
)  e.  NN0 )
54nn0red 10267 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  (deg `  P
)  e.  RR )
6 simpl1 960 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  A  e.  AA )
7 dgraacl 27319 . . . . . . . . 9  |-  ( A  e.  AA  ->  (degAA `  A )  e.  NN )
86, 7syl 16 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  (degAA `  A )  e.  NN )
98nnred 10007 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  (degAA `  A )  e.  RR )
105, 9ltnled 9212 . . . . . 6  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  ( (deg `  P )  <  (degAA `  A )  <->  -.  (degAA `  A )  <_  (deg `  P ) ) )
111, 10mpbid 202 . . . . 5  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  -.  (degAA `  A
)  <_  (deg `  P
) )
12 simpl2 961 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0 p  /\  ( P `  A )  =  0 ) )  ->  P  e.  (Poly `  QQ )
)
13 simprl 733 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0 p  /\  ( P `  A )  =  0 ) )  ->  P  =/=  0 p )
14 simpl1 960 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0 p  /\  ( P `  A )  =  0 ) )  ->  A  e.  AA )
15 aacn 20226 . . . . . . . 8  |-  ( A  e.  AA  ->  A  e.  CC )
1614, 15syl 16 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0 p  /\  ( P `  A )  =  0 ) )  ->  A  e.  CC )
17 simprr 734 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0 p  /\  ( P `  A )  =  0 ) )  ->  ( P `  A )  =  0 )
18 dgraaub 27321 . . . . . . 7  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (degAA `  A
)  <_  (deg `  P
) )
1912, 13, 16, 17, 18syl22anc 1185 . . . . . 6  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0 p  /\  ( P `  A )  =  0 ) )  ->  (degAA `  A )  <_  (deg `  P ) )
2019expr 599 . . . . 5  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  ( ( P `
 A )  =  0  ->  (degAA `  A
)  <_  (deg `  P
) ) )
2111, 20mtod 170 . . . 4  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  -.  ( P `  A )  =  0 )
2221ex 424 . . 3  |-  ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P
)  <  (degAA `  A
) )  ->  ( P  =/=  0 p  ->  -.  ( P `  A
)  =  0 ) )
2322necon4ad 2659 . 2  |-  ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P
)  <  (degAA `  A
) )  ->  (
( P `  A
)  =  0  ->  P  =  0 p
) )
24 0pval 19555 . . . . 5  |-  ( A  e.  CC  ->  (
0 p `  A
)  =  0 )
2515, 24syl 16 . . . 4  |-  ( A  e.  AA  ->  (
0 p `  A
)  =  0 )
26 fveq1 5719 . . . . 5  |-  ( P  =  0 p  -> 
( P `  A
)  =  ( 0 p `  A ) )
2726eqeq1d 2443 . . . 4  |-  ( P  =  0 p  -> 
( ( P `  A )  =  0  <-> 
( 0 p `  A )  =  0 ) )
2825, 27syl5ibrcom 214 . . 3  |-  ( A  e.  AA  ->  ( P  =  0 p  ->  ( P `  A
)  =  0 ) )
29283ad2ant1 978 . 2  |-  ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P
)  <  (degAA `  A
) )  ->  ( P  =  0 p  ->  ( P `  A
)  =  0 ) )
3023, 29impbid 184 1  |-  ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P
)  <  (degAA `  A
) )  ->  (
( P `  A
)  =  0  <->  P  =  0 p ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446   CCcc 8980   0cc0 8982    < clt 9112    <_ cle 9113   NNcn 9992   NN0cn0 10213   QQcq 10566   0 pc0p 19553  Polycply 20095  degcdgr 20098   AAcaa 20223  degAAcdgraa 27313
This theorem is referenced by:  mpaaeu  27323
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-coe 20101  df-dgr 20102  df-aa 20224  df-dgraa 27315
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