Theorem rerecne0d

Description: The reciprocal of a nonzero number is nonzero. (Contributed by SN, 4-Apr-2026)

	Ref	Expression
Hypotheses	sn-rereccld.a	|- ( ph -> A e. RR )
	sn-rereccld.z	|- ( ph -> A =/= 0 )
Assertion	rerecne0d	|- ( ph -> ( 1 /R A ) =/= 0 )

Proof

Step	Hyp	Ref	Expression
1		sn-rereccld.a	|- ( ph -> A e. RR )
2		sn-rereccld.z	|- ( ph -> A =/= 0 )
3		ax-1ne0	|- 1 =/= 0
4		1red	|- ( ph -> 1 e. RR )
5	4 1 2	redivne0bd	|- ( ph -> ( 1 =/= 0 <-> ( 1 /R A ) =/= 0 ) )
6	3 5	mpbii	|- ( ph -> ( 1 /R A ) =/= 0 )