Theorem elo1d

Description: Sufficient condition for elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 21-Sep-2014) (Proof shortened by Mario Carneiro, 26-May-2016)

	Ref	Expression
Hypotheses	elo1mpt.1	|- ( ph -> A C_ RR )
	elo1mpt.2	|- ( ( ph /\ x e. A ) -> B e. CC )
	elo1d.3	|- ( ph -> C e. RR )
	elo1d.4	|- ( ph -> M e. RR )
	elo1d.5	|- ( ( ph /\ ( x e. A /\ C <_ x ) ) -> ( abs ` B ) <_ M )
Assertion	elo1d	|- ( ph -> ( x e. A |-> B ) e. O(1) )

Proof

Step	Hyp	Ref	Expression
1		elo1mpt.1	|- ( ph -> A C_ RR )
2		elo1mpt.2	|- ( ( ph /\ x e. A ) -> B e. CC )
3		elo1d.3	|- ( ph -> C e. RR )
4		elo1d.4	|- ( ph -> M e. RR )
5		elo1d.5	|- ( ( ph /\ ( x e. A /\ C <_ x ) ) -> ( abs ` B ) <_ M )
6	2	abscld	|- ( ( ph /\ x e. A ) -> ( abs ` B ) e. RR )
7	1 6 3 4 5	ello1d	|- ( ph -> ( x e. A |-> ( abs ` B ) ) e. <_O(1) )
8	2	lo1o12	|- ( ph -> ( ( x e. A |-> B ) e. O(1) <-> ( x e. A |-> ( abs ` B ) ) e. <_O(1) ) )
9	7 8	mpbird	|- ( ph -> ( x e. A |-> B ) e. O(1) )