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Metamath Proof Explorer

Theorem elfz1uz

Description: Membership in a 1-based finite set of sequential integers with an upper integer. (Contributed by AV, 23-Jan-2022)

Ref Expression
Assertion elfz1uz 
|- ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> N e. ( 1 ... M ) )

Proof

Step Hyp Ref Expression
1 simpl 
 |-  ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> N e. NN )
2 eluzle 
 |-  ( M e. ( ZZ>= ` N ) -> N <_ M )
3 2 adantl 
 |-  ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> N <_ M )
4 eluzelz 
 |-  ( M e. ( ZZ>= ` N ) -> M e. ZZ )
5 4 adantl 
 |-  ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> M e. ZZ )
6 fznn 
 |-  ( M e. ZZ -> ( N e. ( 1 ... M ) <-> ( N e. NN /\ N <_ M ) ) )
7 5 6 syl 
 |-  ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> ( N e. ( 1 ... M ) <-> ( N e. NN /\ N <_ M ) ) )
8 1 3 7 mpbir2and 
 |-  ( ( N e. NN /\ M e. ( ZZ>= ` N ) ) -> N e. ( 1 ... M ) )