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Metamath Proof Explorer

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) )