Theorem infxrrnmptcl

Description: The infimum of an arbitrary indexed set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypotheses	infxrrnmptcl.1	|- F/ x ph
	infxrrnmptcl.2	|- ( ( ph /\ x e. A ) -> B e. RR* )
Assertion	infxrrnmptcl	|- ( ph -> inf ( ran ( x e. A |-> B ) , RR* , < ) e. RR* )

Proof

Step	Hyp	Ref	Expression
1		infxrrnmptcl.1	|- F/ x ph
2		infxrrnmptcl.2	|- ( ( ph /\ x e. A ) -> B e. RR* )
3		eqid	|- ( x e. A |-> B ) = ( x e. A |-> B )
4	1 3 2	rnmptssd	|- ( ph -> ran ( x e. A |-> B ) C_ RR* )
5	4	infxrcld	|- ( ph -> inf ( ran ( x e. A |-> B ) , RR* , < ) e. RR* )