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Theorem dgrnznn 27208
Description: A nonzero polynomial with a root has positive degree. TODO: use in aaliou2 20210. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Assertion
Ref Expression
dgrnznn  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (deg `  P
)  e.  NN )

Proof of Theorem dgrnznn
StepHypRef Expression
1 simpr 448 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  ->  P  =  ( CC  X.  { ( P ` 
0 ) } ) )
21fveq1d 5689 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( P `  A
)  =  ( ( CC  X.  { ( P `  0 ) } ) `  A
) )
3 simplr 732 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( P `  A
)  =  0 )
4 fvex 5701 . . . . . . . . . . . . . 14  |-  ( P `
 0 )  e. 
_V
54fvconst2 5906 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
( CC  X.  {
( P `  0
) } ) `  A )  =  ( P `  0 ) )
65ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( ( CC  X.  { ( P ` 
0 ) } ) `
 A )  =  ( P `  0
) )
72, 3, 63eqtr3rd 2445 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( P `  0
)  =  0 )
87sneqd 3787 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  ->  { ( P ` 
0 ) }  =  { 0 } )
98xpeq2d 4861 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( CC  X.  {
( P `  0
) } )  =  ( CC  X.  {
0 } ) )
101, 9eqtrd 2436 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  ->  P  =  ( CC  X.  { 0 } ) )
11 df-0p 19515 . . . . . . . 8  |-  0 p  =  ( CC  X.  { 0 } )
1210, 11syl6eqr 2454 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  ->  P  =  0 p
)
1312ex 424 . . . . . 6  |-  ( ( A  e.  CC  /\  ( P `  A )  =  0 )  -> 
( P  =  ( CC  X.  { ( P `  0 ) } )  ->  P  =  0 p ) )
1413necon3ad 2603 . . . . 5  |-  ( ( A  e.  CC  /\  ( P `  A )  =  0 )  -> 
( P  =/=  0 p  ->  -.  P  =  ( CC  X.  { ( P `  0 ) } ) ) )
1514impcom 420 . . . 4  |-  ( ( P  =/=  0 p  /\  ( A  e.  CC  /\  ( P `
 A )  =  0 ) )  ->  -.  P  =  ( CC  X.  { ( P `
 0 ) } ) )
1615adantll 695 . . 3  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  -.  P  =  ( CC  X.  { ( P ` 
0 ) } ) )
17 0dgrb 20118 . . . 4  |-  ( P  e.  (Poly `  S
)  ->  ( (deg `  P )  =  0  <-> 
P  =  ( CC 
X.  { ( P `
 0 ) } ) ) )
1817ad2antrr 707 . . 3  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  ( (deg `  P )  =  0  <-> 
P  =  ( CC 
X.  { ( P `
 0 ) } ) ) )
1916, 18mtbird 293 . 2  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  -.  (deg `  P )  =  0 )
20 dgrcl 20105 . . . 4  |-  ( P  e.  (Poly `  S
)  ->  (deg `  P
)  e.  NN0 )
2120ad2antrr 707 . . 3  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (deg `  P
)  e.  NN0 )
22 elnn0 10179 . . 3  |-  ( (deg
`  P )  e. 
NN0 
<->  ( (deg `  P
)  e.  NN  \/  (deg `  P )  =  0 ) )
2321, 22sylib 189 . 2  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  ( (deg `  P )  e.  NN  \/  (deg `  P )  =  0 ) )
24 orel2 373 . 2  |-  ( -.  (deg `  P )  =  0  ->  (
( (deg `  P
)  e.  NN  \/  (deg `  P )  =  0 )  ->  (deg `  P )  e.  NN ) )
2519, 23, 24sylc 58 1  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0 p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (deg `  P
)  e.  NN )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   {csn 3774    X. cxp 4835   ` cfv 5413   CCcc 8944   0cc0 8946   NNcn 9956   NN0cn0 10177   0 pc0p 19514  Polycply 20056  degcdgr 20059
This theorem is referenced by:  dgraalem  27218  dgraaub  27221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-coe 20062  df-dgr 20063
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