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Theorem dgraa0p 31018
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 1001 . . . . . 6  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0p )  ->  (deg `  P
)  <  (degAA `  A
) )
2 simpl2 1000 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0p )  ->  P  e.  (Poly `  QQ ) )
3 dgrcl 22498 . . . . . . . . 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 10865 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0p )  ->  (deg `  P
)  e.  RR )
6 simpl1 999 . . . . . . . . 9  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0p )  ->  A  e.  AA )
7 dgraacl 31015 . . . . . . . . 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 10563 . . . . . . 7  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  P  =/=  0p )  ->  (degAA `  A )  e.  RR )
105, 9ltnled 9743 . . . . . 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 1000 . . . . . . 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 999 . . . . . . . 8  |-  ( ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P )  <  (degAA `  A ) )  /\  ( P  =/=  0p  /\  ( P `  A )  =  0 ) )  ->  A  e.  AA )
15 aacn 22580 . . . . . . . 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 31017 . . . . . . 7  |-  ( ( ( P  e.  (Poly `  QQ )  /\  P  =/=  0p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (degAA `  A
)  <_  (deg `  P
) )
1912, 13, 16, 17, 18syl22anc 1229 . . . . . 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 2687 . 2  |-  ( ( A  e.  AA  /\  P  e.  (Poly `  QQ )  /\  (deg `  P
)  <  (degAA `  A
) )  ->  (
( P `  A
)  =  0  ->  P  =  0p
) )
24 0pval 21946 . . . . 5  |-  ( A  e.  CC  ->  (
0p `  A
)  =  0 )
2515, 24syl 16 . . . 4  |-  ( A  e.  AA  ->  (
0p `  A
)  =  0 )
26 fveq1 5871 . . . . 5  |-  ( P  =  0p  -> 
( P `  A
)  =  ( 0p `  A ) )
2726eqeq1d 2469 . . . 4  |-  ( P  =  0p  -> 
( ( P `  A )  =  0  <-> 
( 0p `  A )  =  0 ) )
2825, 27syl5ibrcom 222 . . 3  |-  ( A  e.  AA  ->  ( P  =  0p 
->  ( P `  A
)  =  0 ) )
29283ad2ant1 1017 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594   CCcc 9502   0cc0 9504    < clt 9640    <_ cle 9641   NNcn 10548   NN0cn0 10807   QQcq 11194   0pc0p 21944  Polycply 22449  degcdgr 22452   AAcaa 22577  degAAcdgraa 31009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-rlim 13292  df-sum 13489  df-0p 21945  df-ply 22453  df-coe 22455  df-dgr 22456  df-aa 22578  df-dgraa 31011
This theorem is referenced by:  mpaaeu  31019
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