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Theorem ne0p 20126
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 19563 . . . 4  |-  ( A  e.  CC  ->  (
0 p `  A
)  =  0 )
2 fveq1 5727 . . . . 5  |-  ( F  =  0 p  -> 
( F `  A
)  =  ( 0 p `  A ) )
32eqeq1d 2444 . . . 4  |-  ( F  =  0 p  -> 
( ( F `  A )  =  0  <-> 
( 0 p `  A )  =  0 ) )
41, 3syl5ibrcom 214 . . 3  |-  ( A  e.  CC  ->  ( F  =  0 p  ->  ( F `  A
)  =  0 ) )
54necon3d 2639 . 2  |-  ( A  e.  CC  ->  (
( F `  A
)  =/=  0  ->  F  =/=  0 p ) )
65imp 419 1  |-  ( ( A  e.  CC  /\  ( F `  A )  =/=  0 )  ->  F  =/=  0 p )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   ` cfv 5454   CCcc 8988   0cc0 8990   0 pc0p 19561
This theorem is referenced by:  dgrmulc  20189  qaa  20240  iaa  20242  aareccl  20243  dchrfi  21039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-mulcl 9052  ax-i2m1 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-0p 19562
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