Metamath Proof Explorer

Theorem elfzm1b

Description: An integer is a member of a 1-based finite set of sequential integers iff its predecessor is a member of the corresponding 0-based set. (Contributed by Paul Chapman, 22-Jun-2011)

Ref Expression
Assertion elfzm1b 
|- ( ( K e. ZZ /\ N e. ZZ ) -> ( K e. ( 1 ... N ) <-> ( K - 1 ) e. ( 0 ... ( N - 1 ) ) ) )

Proof

Step Hyp Ref Expression
1 1z 
 |-  1 e. ZZ
2 fzsubel 
 |-  ( ( ( 1 e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ 1 e. ZZ ) ) -> ( K e. ( 1 ... N ) <-> ( K - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) ) )
3 1 2 mpanl1 
 |-  ( ( N e. ZZ /\ ( K e. ZZ /\ 1 e. ZZ ) ) -> ( K e. ( 1 ... N ) <-> ( K - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) ) )
4 1 3 mpanr2 
 |-  ( ( N e. ZZ /\ K e. ZZ ) -> ( K e. ( 1 ... N ) <-> ( K - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) ) )
5 1m1e0 
 |-  ( 1 - 1 ) = 0
6 5 oveq1i 
 |-  ( ( 1 - 1 ) ... ( N - 1 ) ) = ( 0 ... ( N - 1 ) )
7 6 eleq2i 
 |-  ( ( K - 1 ) e. ( ( 1 - 1 ) ... ( N - 1 ) ) <-> ( K - 1 ) e. ( 0 ... ( N - 1 ) ) )
8 4 7 bitrdi 
 |-  ( ( N e. ZZ /\ K e. ZZ ) -> ( K e. ( 1 ... N ) <-> ( K - 1 ) e. ( 0 ... ( N - 1 ) ) ) )
9 8 ancoms 
 |-  ( ( K e. ZZ /\ N e. ZZ ) -> ( K e. ( 1 ... N ) <-> ( K - 1 ) e. ( 0 ... ( N - 1 ) ) ) )