MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ne0p Structured version   Unicode version

Theorem ne0p 22786
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 22260 . . . 4  |-  ( A  e.  CC  ->  (
0p `  A
)  =  0 )
2 fveq1 5802 . . . . 5  |-  ( F  =  0p  -> 
( F `  A
)  =  ( 0p `  A ) )
32eqeq1d 2402 . . . 4  |-  ( F  =  0p  -> 
( ( F `  A )  =  0  <-> 
( 0p `  A )  =  0 ) )
41, 3syl5ibrcom 222 . . 3  |-  ( A  e.  CC  ->  ( F  =  0p 
->  ( F `  A
)  =  0 ) )
54necon3d 2625 . 2  |-  ( A  e.  CC  ->  (
( F `  A
)  =/=  0  ->  F  =/=  0p ) )
65imp 427 1  |-  ( ( A  e.  CC  /\  ( F `  A )  =/=  0 )  ->  F  =/=  0p )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840    =/= wne 2596   ` cfv 5523   CCcc 9438   0cc0 9440   0pc0p 22258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-mulcl 9502  ax-i2m1 9508
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-fv 5531  df-0p 22259
This theorem is referenced by:  dgrmulc  22850  qaa  22901  iaa  22903  aareccl  22904  dchrfi  23801
  Copyright terms: Public domain W3C validator