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Theorem ne0p 22581
Description: A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.)
Assertion
Ref Expression
ne0p  |-  ( ( A  e.  CC  /\  ( F `  A )  =/=  0 )  ->  F  =/=  0p )

Proof of Theorem ne0p
StepHypRef Expression
1 0pval 22055 . . . 4  |-  ( A  e.  CC  ->  (
0p `  A
)  =  0 )
2 fveq1 5855 . . . . 5  |-  ( F  =  0p  -> 
( F `  A
)  =  ( 0p `  A ) )
32eqeq1d 2445 . . . 4  |-  ( F  =  0p  -> 
( ( F `  A )  =  0  <-> 
( 0p `  A )  =  0 ) )
41, 3syl5ibrcom 222 . . 3  |-  ( A  e.  CC  ->  ( F  =  0p 
->  ( F `  A
)  =  0 ) )
54necon3d 2667 . 2  |-  ( A  e.  CC  ->  (
( F `  A
)  =/=  0  ->  F  =/=  0p ) )
65imp 429 1  |-  ( ( A  e.  CC  /\  ( F `  A )  =/=  0 )  ->  F  =/=  0p )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   ` cfv 5578   CCcc 9493   0cc0 9495   0pc0p 22053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-mulcl 9557  ax-i2m1 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-0p 22054
This theorem is referenced by:  dgrmulc  22644  qaa  22695  iaa  22697  aareccl  22698  dchrfi  23506
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