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Theorem dgraa0p 29504
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  =  0p ) )

Proof of Theorem dgraa0p
StepHypRef Expression
1 simpl3 993 . . . . . 6  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0p )  ->  (deg `  P
)  <  (degAA `  A
) )
2 simpl2 992 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0p )  ->  P  e.  (Poly `  QQ ) )
3 dgrcl 21700 . . . . . . . . 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  =/=  0p )  ->  (deg `  P
)  e.  NN0 )
54nn0red 10636 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0p )  ->  (deg `  P
)  e.  RR )
6 simpl1 991 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0p )  ->  A  e.  AA )
7 dgraacl 29501 . . . . . . . . 9  |-  ( A  e.  AA  ->  (degAA `  A )  e.  NN )
86, 7syl 16 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0p )  ->  (degAA `  A )  e.  NN )
98nnred 10336 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0p )  ->  (degAA `  A )  e.  RR )
105, 9ltnled 9520 . . . . . 6  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0p )  ->  ( (deg `  P )  <  (degAA `  A )  <->  -.  (degAA `  A )  <_  (deg `  P ) ) )
111, 10mpbid 210 . . . . 5  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0p )  ->  -.  (degAA `  A
)  <_  (deg `  P
) )
12 simpl2 992 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0p  /\  ( P `  A )  =  0 ) )  ->  P  e.  (Poly `  QQ )
)
13 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0p  /\  ( P `  A )  =  0 ) )  ->  P  =/=  0p )
14 simpl1 991 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0p  /\  ( P `  A )  =  0 ) )  ->  A  e.  AA )
15 aacn 21782 . . . . . . . 8  |-  ( A  e.  AA  ->  A  e.  CC )
1614, 15syl 16 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0p  /\  ( P `  A )  =  0 ) )  ->  A  e.  CC )
17 simprr 756 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0p  /\  ( P `  A )  =  0 ) )  ->  ( P `  A )  =  0 )
18 dgraaub 29503 . . . . . . 7  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (degAA `  A
)  <_  (deg `  P
) )
1912, 13, 16, 17, 18syl22anc 1219 . . . . . 6  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0p  /\  ( P `  A )  =  0 ) )  ->  (degAA `  A )  <_  (deg `  P ) )
2019expr 615 . . . . 5  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0p )  ->  ( ( P `
 A )  =  0  ->  (degAA `  A
)  <_  (deg `  P
) ) )
2111, 20mtod 177 . . . 4  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0p )  ->  -.  ( P `  A )  =  0 )
2221ex 434 . . 3  |-  ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P
)  <  (degAA `  A
) )  ->  ( P  =/=  0p  ->  -.  ( P `  A
)  =  0 ) )
2322necon4ad 2671 . 2  |-  ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P
)  <  (degAA `  A
) )  ->  (
( P `  A
)  =  0  ->  P  =  0p
) )
24 0pval 21148 . . . . 5  |-  ( A  e.  CC  ->  (
0p `  A
)  =  0 )
2515, 24syl 16 . . . 4  |-  ( A  e.  AA  ->  (
0p `  A
)  =  0 )
26 fveq1 5689 . . . . 5  |-  ( P  =  0p  -> 
( P `  A
)  =  ( 0p `  A ) )
2726eqeq1d 2450 . . . 4  |-  ( P  =  0p  -> 
( ( P `  A )  =  0  <-> 
( 0p `  A )  =  0 ) )
2825, 27syl5ibrcom 222 . . 3  |-  ( A  e.  AA  ->  ( P  =  0p 
->  ( P `  A
)  =  0 ) )
29283ad2ant1 1009 . 2  |-  ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P
)  <  (degAA `  A
) )  ->  ( P  =  0p 
->  ( P `  A
)  =  0 ) )
3023, 29impbid 191 1  |-  ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P
)  <  (degAA `  A
) )  ->  (
( P `  A
)  =  0  <->  P  =  0p ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2605   class class class wbr 4291   ` cfv 5417   CCcc 9279   0cc0 9281    < clt 9417    <_ cle 9418   NNcn 10321   NN0cn0 10578   QQcq 10952   0pc0p 21146  Polycply 21651  degcdgr 21654   AAcaa 21779  degAAcdgraa 29495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-inf2 7846  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-oi 7723  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-q 10953  df-rp 10991  df-fz 11437  df-fzo 11548  df-fl 11641  df-mod 11708  df-seq 11806  df-exp 11865  df-hash 12103  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-clim 12965  df-rlim 12966  df-sum 13163  df-0p 21147  df-ply 21655  df-coe 21657  df-dgr 21658  df-aa 21780  df-dgraa 29497
This theorem is referenced by:  mpaaeu  29505
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