Theorem ne0p 23963

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

Step	Hyp	Ref	Expression
1		0pval 23438	. . . 4 |- ( A e. CC -> ( 0p ` A ) = 0 )
2		fveq1 6190	. . . . 5 |- ( F = 0p -> ( F ` A ) = ( 0p ` A ) )
3	2	eqeq1d 2624	. . . 4 |- ( F = 0p -> ( ( F ` A ) = 0 <-> ( 0p ` A ) = 0 ) )
4	1, 3	syl5ibrcom 237	. . 3 |- ( A e. CC -> ( F = 0p -> ( F ` A ) = 0 ) )
5	4	necon3d 2815	. 2 |- ( A e. CC -> ( ( F ` A ) =/= 0 -> F =/= 0p ) )
6	5	imp 445	1 |- ( ( A e. CC /\ ( F ` A ) =/= 0 ) -> F =/= 0p )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888   CCcc 9934   0cc0 9936   0pc0p 23436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-0p 23437

This theorem is referenced by:  dgrmulc  24027  qaa  24078  iaa  24080  aareccl  24081  dchrfi  24980