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Metamath Proof Explorer

Theorem elfzom1b

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 Mario Carneiro, 27-Sep-2015)

Ref Expression
Assertion elfzom1b 
|- ( ( K e. ZZ /\ N e. ZZ ) -> ( K e. ( 1 ..^ N ) <-> ( K - 1 ) e. ( 0 ..^ ( N - 1 ) ) ) )

Proof

Step Hyp Ref Expression
1 peano2zm 
 |-  ( N e. ZZ -> ( N - 1 ) e. ZZ )
2 elfzm1b 
 |-  ( ( K e. ZZ /\ ( N - 1 ) e. ZZ ) -> ( K e. ( 1 ... ( N - 1 ) ) <-> ( K - 1 ) e. ( 0 ... ( ( N - 1 ) - 1 ) ) ) )
3 1 2 sylan2 
 |-  ( ( K e. ZZ /\ N e. ZZ ) -> ( K e. ( 1 ... ( N - 1 ) ) <-> ( K - 1 ) e. ( 0 ... ( ( N - 1 ) - 1 ) ) ) )
4 fzoval 
 |-  ( N e. ZZ -> ( 1 ..^ N ) = ( 1 ... ( N - 1 ) ) )
5 4 adantl 
 |-  ( ( K e. ZZ /\ N e. ZZ ) -> ( 1 ..^ N ) = ( 1 ... ( N - 1 ) ) )
6 5 eleq2d 
 |-  ( ( K e. ZZ /\ N e. ZZ ) -> ( K e. ( 1 ..^ N ) <-> K e. ( 1 ... ( N - 1 ) ) ) )
7 1 adantl 
 |-  ( ( K e. ZZ /\ N e. ZZ ) -> ( N - 1 ) e. ZZ )
8 fzoval 
 |-  ( ( N - 1 ) e. ZZ -> ( 0 ..^ ( N - 1 ) ) = ( 0 ... ( ( N - 1 ) - 1 ) ) )
9 7 8 syl 
 |-  ( ( K e. ZZ /\ N e. ZZ ) -> ( 0 ..^ ( N - 1 ) ) = ( 0 ... ( ( N - 1 ) - 1 ) ) )
10 9 eleq2d 
 |-  ( ( K e. ZZ /\ N e. ZZ ) -> ( ( K - 1 ) e. ( 0 ..^ ( N - 1 ) ) <-> ( K - 1 ) e. ( 0 ... ( ( N - 1 ) - 1 ) ) ) )
11 3 6 10 3bitr4d 
 |-  ( ( K e. ZZ /\ N e. ZZ ) -> ( K e. ( 1 ..^ N ) <-> ( K - 1 ) e. ( 0 ..^ ( N - 1 ) ) ) )