Metamath Proof Explorer

Theorem fzrev2i

Description: Reversal of start and end of a finite set of sequential integers. (Contributed by NM, 25-Nov-2005)

Ref Expression
Assertion fzrev2i 
|- ( ( J e. ZZ /\ K e. ( M ... N ) ) -> ( J - K ) e. ( ( J - N ) ... ( J - M ) ) )

Proof

Step Hyp Ref Expression
1 simpr 
 |-  ( ( J e. ZZ /\ K e. ( M ... N ) ) -> K e. ( M ... N ) )
2 elfzel1 
 |-  ( K e. ( M ... N ) -> M e. ZZ )
3 2 adantl 
 |-  ( ( J e. ZZ /\ K e. ( M ... N ) ) -> M e. ZZ )
4 elfzel2 
 |-  ( K e. ( M ... N ) -> N e. ZZ )
5 4 adantl 
 |-  ( ( J e. ZZ /\ K e. ( M ... N ) ) -> N e. ZZ )
6 simpl 
 |-  ( ( J e. ZZ /\ K e. ( M ... N ) ) -> J e. ZZ )
7 elfzelz 
 |-  ( K e. ( M ... N ) -> K e. ZZ )
8 7 adantl 
 |-  ( ( J e. ZZ /\ K e. ( M ... N ) ) -> K e. ZZ )
9 fzrev2 
 |-  ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( J e. ZZ /\ K e. ZZ ) ) -> ( K e. ( M ... N ) <-> ( J - K ) e. ( ( J - N ) ... ( J - M ) ) ) )
10 3 5 6 8 9 syl22anc 
 |-  ( ( J e. ZZ /\ K e. ( M ... N ) ) -> ( K e. ( M ... N ) <-> ( J - K ) e. ( ( J - N ) ... ( J - M ) ) ) )
11 1 10 mpbid 
 |-  ( ( J e. ZZ /\ K e. ( M ... N ) ) -> ( J - K ) e. ( ( J - N ) ... ( J - M ) ) )