Theorem abvne0

Description: The absolute value of a nonzero number is nonzero. (Contributed by Mario Carneiro, 8-Sep-2014)

	Ref	Expression
Hypotheses	abvf.a	|- A = ( AbsVal ` R )
	abvf.b	|- B = ( Base ` R )
	abveq0.z	|- .0. = ( 0g ` R )
Assertion	abvne0	|- ( ( F e. A /\ X e. B /\ X =/= .0. ) -> ( F ` X ) =/= 0 )

Proof

Step	Hyp	Ref	Expression
1		abvf.a	|- A = ( AbsVal ` R )
2		abvf.b	|- B = ( Base ` R )
3		abveq0.z	|- .0. = ( 0g ` R )
4	1 2 3	abveq0	|- ( ( F e. A /\ X e. B ) -> ( ( F ` X ) = 0 <-> X = .0. ) )
5	4	necon3bid	|- ( ( F e. A /\ X e. B ) -> ( ( F ` X ) =/= 0 <-> X =/= .0. ) )
6	5	biimp3ar	|- ( ( F e. A /\ X e. B /\ X =/= .0. ) -> ( F ` X ) =/= 0 )