Metamath Proof Explorer

Theorem fseqsupubi

Description: The values of a finite real sequence are bounded by their supremum. (Contributed by NM, 20-Sep-2005)

Ref Expression
Assertion fseqsupubi 
|- ( ( K e. ( M ... N ) /\ F : ( M ... N ) --> RR ) -> ( F ` K ) <_ sup ( ran F , RR , < ) )

Proof

Step Hyp Ref Expression
1 frn 
 |-  ( F : ( M ... N ) --> RR -> ran F C_ RR )
2 1 adantl 
 |-  ( ( K e. ( M ... N ) /\ F : ( M ... N ) --> RR ) -> ran F C_ RR )
3 fdm 
 |-  ( F : ( M ... N ) --> RR -> dom F = ( M ... N ) )
4 ne0i 
 |-  ( K e. ( M ... N ) -> ( M ... N ) =/= (/) )
5 dm0rn0 
 |-  ( dom F = (/) <-> ran F = (/) )
6 eqeq1 
 |-  ( dom F = ( M ... N ) -> ( dom F = (/) <-> ( M ... N ) = (/) ) )
7 6 biimpd 
 |-  ( dom F = ( M ... N ) -> ( dom F = (/) -> ( M ... N ) = (/) ) )
8 5 7 syl5bir 
 |-  ( dom F = ( M ... N ) -> ( ran F = (/) -> ( M ... N ) = (/) ) )
9 8 necon3d 
 |-  ( dom F = ( M ... N ) -> ( ( M ... N ) =/= (/) -> ran F =/= (/) ) )
10 4 9 mpan9 
 |-  ( ( K e. ( M ... N ) /\ dom F = ( M ... N ) ) -> ran F =/= (/) )
11 3 10 sylan2 
 |-  ( ( K e. ( M ... N ) /\ F : ( M ... N ) --> RR ) -> ran F =/= (/) )
12 fsequb2 
 |-  ( F : ( M ... N ) --> RR -> E. x e. RR A. y e. ran F y <_ x )
13 12 adantl 
 |-  ( ( K e. ( M ... N ) /\ F : ( M ... N ) --> RR ) -> E. x e. RR A. y e. ran F y <_ x )
14 ffn 
 |-  ( F : ( M ... N ) --> RR -> F Fn ( M ... N ) )
15 fnfvelrn 
 |-  ( ( F Fn ( M ... N ) /\ K e. ( M ... N ) ) -> ( F ` K ) e. ran F )
16 15 ancoms 
 |-  ( ( K e. ( M ... N ) /\ F Fn ( M ... N ) ) -> ( F ` K ) e. ran F )
17 14 16 sylan2 
 |-  ( ( K e. ( M ... N ) /\ F : ( M ... N ) --> RR ) -> ( F ` K ) e. ran F )
18 2 11 13 17 suprubd 
 |-  ( ( K e. ( M ... N ) /\ F : ( M ... N ) --> RR ) -> ( F ` K ) <_ sup ( ran F , RR , < ) )