Metamath Proof Explorer

Theorem fzo0addelr

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

Ref Expression
Assertion fzo0addelr 
|- ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> ( A + D ) e. ( D ..^ ( D + C ) ) )

Proof

Step Hyp Ref Expression
1 fzo0addel 
 |-  ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> ( A + D ) e. ( D ..^ ( C + D ) ) )
2 zcn 
 |-  ( D e. ZZ -> D e. CC )
3 elfzoel2 
 |-  ( A e. ( 0 ..^ C ) -> C e. ZZ )
4 3 zcnd 
 |-  ( A e. ( 0 ..^ C ) -> C e. CC )
5 addcom 
 |-  ( ( D e. CC /\ C e. CC ) -> ( D + C ) = ( C + D ) )
6 2 4 5 syl2anr 
 |-  ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> ( D + C ) = ( C + D ) )
7 6 oveq2d 
 |-  ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> ( D ..^ ( D + C ) ) = ( D ..^ ( C + D ) ) )
8 1 7 eleqtrrd 
 |-  ( ( A e. ( 0 ..^ C ) /\ D e. ZZ ) -> ( A + D ) e. ( D ..^ ( D + C ) ) )