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Theorem ne0p 19693
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  =/=  0 p )

Proof of Theorem ne0p
StepHypRef Expression
1 0pval 19130 . . . 4  |-  ( A  e.  CC  ->  (
0 p `  A
)  =  0 )
2 fveq1 5607 . . . . 5  |-  ( F  =  0 p  -> 
( F `  A
)  =  ( 0 p `  A ) )
32eqeq1d 2366 . . . 4  |-  ( F  =  0 p  -> 
( ( F `  A )  =  0  <-> 
( 0 p `  A )  =  0 ) )
41, 3syl5ibrcom 213 . . 3  |-  ( A  e.  CC  ->  ( F  =  0 p  ->  ( F `  A
)  =  0 ) )
54necon3d 2559 . 2  |-  ( A  e.  CC  ->  (
( F `  A
)  =/=  0  ->  F  =/=  0 p ) )
65imp 418 1  |-  ( ( A  e.  CC  /\  ( F `  A )  =/=  0 )  ->  F  =/=  0 p )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   ` cfv 5337   CCcc 8825   0cc0 8827   0 pc0p 19128
This theorem is referenced by:  dgrmulc  19756  qaa  19807  iaa  19809  aareccl  19810  dchrfi  20606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-mulcl 8889  ax-i2m1 8895
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fv 5345  df-0p 19129
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