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Theorem ne0p 19589
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 19026 . . . 4  |-  ( A  e.  CC  ->  (
0 p `  A
)  =  0 )
2 fveq1 5524 . . . . 5  |-  ( F  =  0 p  -> 
( F `  A
)  =  ( 0 p `  A ) )
32eqeq1d 2291 . . . 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 2484 . 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 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255   CCcc 8735   0cc0 8737   0 pc0p 19024
This theorem is referenced by:  dgrmulc  19652  qaa  19703  iaa  19705  aareccl  19706  dchrfi  20494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-mulcl 8799  ax-i2m1 8805
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-0p 19025
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