Theorem o1le

Description: Transfer eventual boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 25-Sep-2014) (Proof shortened by Mario Carneiro, 26-May-2016)

	Ref	Expression
Hypotheses	o1le.1	|- ( ph -> M e. RR )
	o1le.2	|- ( ph -> ( x e. A |-> B ) e. O(1) )
	o1le.3	|- ( ( ph /\ x e. A ) -> B e. V )
	o1le.4	|- ( ( ph /\ x e. A ) -> C e. CC )
	o1le.5	|- ( ( ph /\ ( x e. A /\ M <_ x ) ) -> ( abs ` C ) <_ ( abs ` B ) )
Assertion	o1le	|- ( ph -> ( x e. A |-> C ) e. O(1) )

Proof

Step	Hyp	Ref	Expression
1		o1le.1	|- ( ph -> M e. RR )
2		o1le.2	|- ( ph -> ( x e. A |-> B ) e. O(1) )
3		o1le.3	|- ( ( ph /\ x e. A ) -> B e. V )
4		o1le.4	|- ( ( ph /\ x e. A ) -> C e. CC )
5		o1le.5	|- ( ( ph /\ ( x e. A /\ M <_ x ) ) -> ( abs ` C ) <_ ( abs ` B ) )
6	3 2	o1mptrcl	|- ( ( ph /\ x e. A ) -> B e. CC )
7	6	lo1o12	|- ( ph -> ( ( x e. A |-> B ) e. O(1) <-> ( x e. A |-> ( abs ` B ) ) e. <_O(1) ) )
8	2 7	mpbid	|- ( ph -> ( x e. A |-> ( abs ` B ) ) e. <_O(1) )
9		fvexd	|- ( ( ph /\ x e. A ) -> ( abs ` B ) e. _V )
10	4	abscld	|- ( ( ph /\ x e. A ) -> ( abs ` C ) e. RR )
11	1 8 9 10 5	lo1le	|- ( ph -> ( x e. A |-> ( abs ` C ) ) e. <_O(1) )
12	4	lo1o12	|- ( ph -> ( ( x e. A |-> C ) e. O(1) <-> ( x e. A |-> ( abs ` C ) ) e. <_O(1) ) )
13	11 12	mpbird	|- ( ph -> ( x e. A |-> C ) e. O(1) )