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Metamath Proof Explorer

Theorem fzo0addel

Description: Translate membership in a 0-based half-open integer range. (Contributed by AV, 30-Apr-2020)

Ref Expression
Assertion fzo0addel 
|- ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> ( A + D ) e. ( D ..^ ( C + D ) ) )

Proof

Step Hyp Ref Expression
1 fzoaddel 
 |-  ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> ( A + D ) e. ( ( 0 + D ) ..^ ( C + D ) ) )
2 zcn 
 |-  ( D e. ZZ -> D e. CC )
3 addlid 
 |-  ( D e. CC -> ( 0 + D ) = D )
4 3 eqcomd 
 |-  ( D e. CC -> D = ( 0 + D ) )
5 2 4 syl 
 |-  ( D e. ZZ -> D = ( 0 + D ) )
6 5 adantl 
 |-  ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> D = ( 0 + D ) )
7 6 oveq1d 
 |-  ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> ( D ..^ ( C + D ) ) = ( ( 0 + D ) ..^ ( C + D ) ) )
8 1 7 eleqtrrd 
 |-  ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> ( A + D ) e. ( D ..^ ( C + D ) ) )