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Theorem dgrnznn 23138
Description: A nonzero polynomial with a root has positive degree. (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 462 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  ->  P  =  ( CC  X.  { ( P ` 
0 ) } ) )
21fveq1d 5822 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( P `  A
)  =  ( ( CC  X.  { ( P `  0 ) } ) `  A
) )
3 simplr 760 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( P `  A
)  =  0 )
4 fvex 5830 . . . . . . . . . . . . . 14  |-  ( P `
 0 )  e. 
_V
54fvconst2 6074 . . . . . . . . . . . . 13  |-  ( A  e.  CC  ->  (
( CC  X.  {
( P `  0
) } ) `  A )  =  ( P `  0 ) )
65ad2antrr 730 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( ( CC  X.  { ( P ` 
0 ) } ) `
 A )  =  ( P `  0
) )
72, 3, 63eqtr3rd 2466 . . . . . . . . . . 11  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( P `  0
)  =  0 )
87sneqd 3948 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  ->  { ( P ` 
0 ) }  =  { 0 } )
98xpeq2d 4815 . . . . . . . . 9  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  -> 
( CC  X.  {
( P `  0
) } )  =  ( CC  X.  {
0 } ) )
101, 9eqtrd 2457 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  ->  P  =  ( CC  X.  { 0 } ) )
11 df-0p 22565 . . . . . . . 8  |-  0p  =  ( CC  X.  { 0 } )
1210, 11syl6eqr 2475 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  ( P `  A
)  =  0 )  /\  P  =  ( CC  X.  { ( P `  0 ) } ) )  ->  P  =  0p
)
1312ex 435 . . . . . 6  |-  ( ( A  e.  CC  /\  ( P `  A )  =  0 )  -> 
( P  =  ( CC  X.  { ( P `  0 ) } )  ->  P  =  0p ) )
1413necon3ad 2609 . . . . 5  |-  ( ( A  e.  CC  /\  ( P `  A )  =  0 )  -> 
( P  =/=  0p  ->  -.  P  =  ( CC  X.  { ( P `  0 ) } ) ) )
1514impcom 431 . . . 4  |-  ( ( P  =/=  0p  /\  ( A  e.  CC  /\  ( P `
 A )  =  0 ) )  ->  -.  P  =  ( CC  X.  { ( P `
 0 ) } ) )
1615adantll 718 . . 3  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  -.  P  =  ( CC  X.  { ( P ` 
0 ) } ) )
17 0dgrb 23137 . . . 4  |-  ( P  e.  (Poly `  S
)  ->  ( (deg `  P )  =  0  <-> 
P  =  ( CC 
X.  { ( P `
 0 ) } ) ) )
1817ad2antrr 730 . . 3  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  ( (deg `  P )  =  0  <-> 
P  =  ( CC 
X.  { ( P `
 0 ) } ) ) )
1916, 18mtbird 302 . 2  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  -.  (deg `  P )  =  0 )
20 dgrcl 23124 . . . 4  |-  ( P  e.  (Poly `  S
)  ->  (deg `  P
)  e.  NN0 )
2120ad2antrr 730 . . 3  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  (deg `  P
)  e.  NN0 )
22 elnn0 10817 . . 3  |-  ( (deg
`  P )  e. 
NN0 
<->  ( (deg `  P
)  e.  NN  \/  (deg `  P )  =  0 ) )
2321, 22sylib 199 . 2  |-  ( ( ( P  e.  (Poly `  S )  /\  P  =/=  0p )  /\  ( A  e.  CC  /\  ( P `  A
)  =  0 ) )  ->  ( (deg `  P )  e.  NN  \/  (deg `  P )  =  0 ) )
24 orel2 384 . 2  |-  ( -.  (deg `  P )  =  0  ->  (
( (deg `  P
)  e.  NN  \/  (deg `  P )  =  0 )  ->  (deg `  P )  e.  NN ) )
2519, 23, 24sylc 62 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 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2594   {csn 3936    X. cxp 4789   ` cfv 5539   CCcc 9483   0cc0 9485   NNcn 10555   NN0cn0 10815   0pc0p 22564  Polycply 23075  degcdgr 23078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-inf2 8094  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562  ax-pre-sup 9563  ax-addf 9564
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-int 4194  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-se 4751  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-isom 5548  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-of 6484  df-om 6646  df-1st 6746  df-2nd 6747  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7520  df-dom 7521  df-sdom 7522  df-fin 7523  df-sup 7904  df-inf 7905  df-oi 7973  df-card 8320  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-div 10216  df-nn 10556  df-2 10614  df-3 10615  df-n0 10816  df-z 10884  df-uz 11106  df-rp 11249  df-fz 11731  df-fzo 11862  df-fl 11973  df-seq 12159  df-exp 12218  df-hash 12461  df-cj 13101  df-re 13102  df-im 13103  df-sqrt 13237  df-abs 13238  df-clim 13490  df-rlim 13491  df-sum 13691  df-0p 22565  df-ply 23079  df-coe 23081  df-dgr 23082
This theorem is referenced by:  dgraalem  35920  dgraalemOLD  35921  dgraaub  35926  dgraaubOLD  35927  etransclem47  38029
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