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Theorem dgrnznn 27009
Description: A nonzero polynomial with a root has positive degree. TODO: use in aaliou2 20124. (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 5670 . . . . . . . . . . . 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 5682 . . . . . . . . . . . . . 14  |-  ( P `
 0 )  e. 
_V
54fvconst2 5886 . . . . . . . . . . . . 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 2428 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( P `  0
)  =  0 )
87sneqd 3770 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  ->  { ( P ` 
0 ) }  =  { 0 } )
98xpeq2d 4842 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( CC  X.  {
( P `  0
) } )  =  ( CC  X.  {
0 } ) )
101, 9eqtrd 2419 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  ->  P  =  ( CC  X.  { 0 } ) )
11 df-0p 19429 . . . . . . . 8  |-  0 p  =  ( CC  X.  { 0 } )
1210, 11syl6eqr 2437 . . . . . . 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 2586 . . . . 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 20032 . . . 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 20019 . . . 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 10155 . . 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 1717    =/= wne 2550   {csn 3757    X. cxp 4816   ` cfv 5394   CCcc 8921   0cc0 8923   NNcn 9932   NN0cn0 10153   0 pc0p 19428  Polycply 19970  degcdgr 19973
This theorem is referenced by:  dgraalem  27019  dgraaub  27022
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-fzo 11066  df-fl 11129  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-rlim 12210  df-sum 12407  df-0p 19429  df-ply 19974  df-coe 19976  df-dgr 19977
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