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Theorem dgraa0p 27457
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 960 . . . . . 6  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  (deg `  P
)  <  (degAA `  A
) )
2 simpl2 959 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  P  e.  (Poly `  QQ ) )
3 dgrcl 19631 . . . . . . . . 9  |-  ( P  e.  (Poly `  QQ )  ->  (deg `  P
)  e.  NN0 )
42, 3syl 15 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  (deg `  P
)  e.  NN0 )
54nn0red 10035 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  (deg `  P
)  e.  RR )
6 simpl1 958 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  A  e.  AA )
7 dgraacl 27454 . . . . . . . . 9  |-  ( A  e.  AA  ->  (degAA `  A )  e.  NN )
86, 7syl 15 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  (degAA `  A )  e.  NN )
98nnred 9777 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  (degAA `  A )  e.  RR )
105, 9ltnled 8982 . . . . . 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 201 . . . . 5  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  -.  (degAA `  A
)  <_  (deg `  P
) )
12 simpl2 959 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0 p  /\  ( P `  A )  =  0 ) )  ->  P  e.  (Poly `  QQ )
)
13 simprl 732 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0 p  /\  ( P `  A )  =  0 ) )  ->  P  =/=  0 p )
14 simpl1 958 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0 p  /\  ( P `  A )  =  0 ) )  ->  A  e.  AA )
15 aacn 19713 . . . . . . . 8  |-  ( A  e.  AA  ->  A  e.  CC )
1614, 15syl 15 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0 p  /\  ( P `  A )  =  0 ) )  ->  A  e.  CC )
17 simprr 733 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0 p  /\  ( P `  A )  =  0 ) )  ->  ( P `  A )  =  0 )
18 dgraaub 27456 . . . . . . 7  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (degAA `  A
)  <_  (deg `  P
) )
1912, 13, 16, 17, 18syl22anc 1183 . . . . . 6  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0 p  /\  ( P `  A )  =  0 ) )  ->  (degAA `  A )  <_  (deg `  P ) )
2019expr 598 . . . . 5  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  ( ( P `
 A )  =  0  ->  (degAA `  A
)  <_  (deg `  P
) ) )
2111, 20mtod 168 . . . 4  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0 p )  ->  -.  ( P `  A )  =  0 )
2221ex 423 . . 3  |-  ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P
)  <  (degAA `  A
) )  ->  ( P  =/=  0 p  ->  -.  ( P `  A
)  =  0 ) )
2322necon4ad 2520 . 2  |-  ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P
)  <  (degAA `  A
) )  ->  (
( P `  A
)  =  0  ->  P  =  0 p
) )
24 0pval 19042 . . . . 5  |-  ( A  e.  CC  ->  (
0 p `  A
)  =  0 )
2515, 24syl 15 . . . 4  |-  ( A  e.  AA  ->  (
0 p `  A
)  =  0 )
26 fveq1 5540 . . . . 5  |-  ( P  =  0 p  -> 
( P `  A
)  =  ( 0 p `  A ) )
2726eqeq1d 2304 . . . 4  |-  ( P  =  0 p  -> 
( ( P `  A )  =  0  <-> 
( 0 p `  A )  =  0 ) )
2825, 27syl5ibrcom 213 . . 3  |-  ( A  e.  AA  ->  ( P  =  0 p  ->  ( P `  A
)  =  0 ) )
29283ad2ant1 976 . 2  |-  ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P
)  <  (degAA `  A
) )  ->  ( P  =  0 p  ->  ( P `  A
)  =  0 ) )
3023, 29impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271   CCcc 8751   0cc0 8753    < clt 8883    <_ cle 8884   NNcn 9762   NN0cn0 9981   QQcq 10332   0 pc0p 19040  Polycply 19582  degcdgr 19585   AAcaa 19710  degAAcdgraa 27448
This theorem is referenced by:  mpaaeu  27458
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-coe 19588  df-dgr 19589  df-aa 19711  df-dgraa 27450
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