Theorem o1eq

Description: Two functions that are eventually equal to one another are eventually bounded if one of them is. (Contributed by Mario Carneiro, 26-May-2016)

	Ref	Expression
Hypotheses	rlimeq.1	|- ( ( ph /\ x e. A ) -> B e. CC )
	rlimeq.2	|- ( ( ph /\ x e. A ) -> C e. CC )
	rlimeq.3	|- ( ph -> D e. RR )
	rlimeq.4	|- ( ( ph /\ ( x e. A /\ D <_ x ) ) -> B = C )
Assertion	o1eq	|- ( ph -> ( ( x e. A |-> B ) e. O(1) <-> ( x e. A |-> C ) e. O(1) ) )

Proof

Step	Hyp	Ref	Expression
1		rlimeq.1	|- ( ( ph /\ x e. A ) -> B e. CC )
2		rlimeq.2	|- ( ( ph /\ x e. A ) -> C e. CC )
3		rlimeq.3	|- ( ph -> D e. RR )
4		rlimeq.4	|- ( ( ph /\ ( x e. A /\ D <_ x ) ) -> B = C )
5	1	abscld	|- ( ( ph /\ x e. A ) -> ( abs ` B ) e. RR )
6	2	abscld	|- ( ( ph /\ x e. A ) -> ( abs ` C ) e. RR )
7	4	fveq2d	|- ( ( ph /\ ( x e. A /\ D <_ x ) ) -> ( abs ` B ) = ( abs ` C ) )
8	5 6 3 7	lo1eq	|- ( ph -> ( ( x e. A |-> ( abs ` B ) ) e. <_O(1) <-> ( x e. A |-> ( abs ` C ) ) e. <_O(1) ) )
9	1	lo1o12	|- ( ph -> ( ( x e. A |-> B ) e. O(1) <-> ( x e. A |-> ( abs ` B ) ) e. <_O(1) ) )
10	2	lo1o12	|- ( ph -> ( ( x e. A |-> C ) e. O(1) <-> ( x e. A |-> ( abs ` C ) ) e. <_O(1) ) )
11	8 9 10	3bitr4d	|- ( ph -> ( ( x e. A |-> B ) e. O(1) <-> ( x e. A |-> C ) e. O(1) ) )