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Metamath Proof Explorer

Theorem elo1mpt2

Description: Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 12-May-2016) (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 )
Assertion elo1mpt2 
|- ( ph -> ( ( x e. A |-> B ) e. O(1) <-> E. y e. ( C [,) +oo ) E. m e. RR A. x e. A ( y <_ x -> ( abs ` B ) <_ m ) ) )

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 2 lo1o12 
 |-  ( ph -> ( ( x e. A |-> B ) e. O(1) <-> ( x e. A |-> ( abs ` B ) ) e. <_O(1) ) )
5 2 abscld 
 |-  ( ( ph /\ x e. A ) -> ( abs ` B ) e. RR )
6 1 5 3 ello1mpt2 
 |-  ( ph -> ( ( x e. A |-> ( abs ` B ) ) e. <_O(1) <-> E. y e. ( C [,) +oo ) E. m e. RR A. x e. A ( y <_ x -> ( abs ` B ) <_ m ) ) )
7 4 6 bitrd 
 |-  ( ph -> ( ( x e. A |-> B ) e. O(1) <-> E. y e. ( C [,) +oo ) E. m e. RR A. x e. A ( y <_ x -> ( abs ` B ) <_ m ) ) )