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Theorem dgrnznn 29497
Description: A nonzero polynomial with a root has positive degree. TODO: use in aaliou2 21811. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Assertion
Ref Expression
dgrnznn  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (deg `  P
)  e.  NN )

Proof of Theorem dgrnznn
StepHypRef Expression
1 simpr 461 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  ->  P  =  ( CC  X.  { ( P ` 
0 ) } ) )
21fveq1d 5698 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( P `  A
)  =  ( ( CC  X.  { ( P `  0 ) } ) `  A
) )
3 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( P `  A
)  =  0 )
4 fvex 5706 . . . . . . . . . . . . . 14  |-  ( P `
 0 )  e. 
_V
54fvconst2 5938 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
( CC  X.  {
( P `  0
) } ) `  A )  =  ( P `  0 ) )
65ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( ( CC  X.  { ( P ` 
0 ) } ) `
 A )  =  ( P `  0
) )
72, 3, 63eqtr3rd 2484 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( P `  0
)  =  0 )
87sneqd 3894 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  ->  { ( P ` 
0 ) }  =  { 0 } )
98xpeq2d 4869 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( CC  X.  {
( P `  0
) } )  =  ( CC  X.  {
0 } ) )
101, 9eqtrd 2475 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  ->  P  =  ( CC  X.  { 0 } ) )
11 df-0p 21153 . . . . . . . 8  |-  0p  =  ( CC  X.  { 0 } )
1210, 11syl6eqr 2493 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  ->  P  =  0p
)
1312ex 434 . . . . . 6  |-  ( ( A  e.  CC  /\  ( P `  A )  =  0 )  -> 
( P  =  ( CC  X.  { ( P `  0 ) } )  ->  P  =  0p ) )
1413necon3ad 2649 . . . . 5  |-  ( ( A  e.  CC  /\  ( P `  A )  =  0 )  -> 
( P  =/=  0p  ->  -.  P  =  ( CC  X.  { ( P `  0 ) } ) ) )
1514impcom 430 . . . 4  |-  ( ( P  =/=  0p  /\  ( A  e.  CC  /\  ( P `
 A )  =  0 ) )  ->  -.  P  =  ( CC  X.  { ( P `
 0 ) } ) )
1615adantll 713 . . 3  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  -.  P  =  ( CC  X.  { ( P ` 
0 ) } ) )
17 0dgrb 21719 . . . 4  |-  ( P  e.  (Poly `  S
)  ->  ( (deg `  P )  =  0  <-> 
P  =  ( CC 
X.  { ( P `
 0 ) } ) ) )
1817ad2antrr 725 . . 3  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  ( (deg `  P )  =  0  <-> 
P  =  ( CC 
X.  { ( P `
 0 ) } ) ) )
1916, 18mtbird 301 . 2  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  -.  (deg `  P )  =  0 )
20 dgrcl 21706 . . . 4  |-  ( P  e.  (Poly `  S
)  ->  (deg `  P
)  e.  NN0 )
2120ad2antrr 725 . . 3  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (deg `  P
)  e.  NN0 )
22 elnn0 10586 . . 3  |-  ( (deg
`  P )  e. 
NN0 
<->  ( (deg `  P
)  e.  NN  \/  (deg `  P )  =  0 ) )
2321, 22sylib 196 . 2  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  ( (deg `  P )  e.  NN  \/  (deg `  P )  =  0 ) )
24 orel2 383 . 2  |-  ( -.  (deg `  P )  =  0  ->  (
( (deg `  P
)  e.  NN  \/  (deg `  P )  =  0 )  ->  (deg `  P )  e.  NN ) )
2519, 23, 24sylc 60 1  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (deg `  P
)  e.  NN )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   {csn 3882    X. cxp 4843   ` cfv 5423   CCcc 9285   0cc0 9287   NNcn 10327   NN0cn0 10584   0pc0p 21152  Polycply 21657  degcdgr 21660
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-fzo 11554  df-fl 11647  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-rlim 12972  df-sum 13169  df-0p 21153  df-ply 21661  df-coe 21663  df-dgr 21664
This theorem is referenced by:  dgraalem  29507  dgraaub  29510
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