Metamath Proof Explorer

Theorem zpnn0elfzo

Description: Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018)

Ref Expression
Assertion zpnn0elfzo 
|- ( ( Z e. ZZ /\ N e. NN0 ) -> ( Z + N ) e. ( Z ..^ ( ( Z + N ) + 1 ) ) )

Proof

Step Hyp Ref Expression
1 uzid 
 |-  ( Z e. ZZ -> Z e. ( ZZ>= ` Z ) )
2 1 anim1i 
 |-  ( ( Z e. ZZ /\ N e. NN0 ) -> ( Z e. ( ZZ>= ` Z ) /\ N e. NN0 ) )
3 nn0z 
 |-  ( N e. NN0 -> N e. ZZ )
4 zaddcl 
 |-  ( ( Z e. ZZ /\ N e. ZZ ) -> ( Z + N ) e. ZZ )
5 3 4 sylan2 
 |-  ( ( Z e. ZZ /\ N e. NN0 ) -> ( Z + N ) e. ZZ )
6 elfzomin 
 |-  ( ( Z + N ) e. ZZ -> ( Z + N ) e. ( ( Z + N ) ..^ ( ( Z + N ) + 1 ) ) )
7 5 6 syl 
 |-  ( ( Z e. ZZ /\ N e. NN0 ) -> ( Z + N ) e. ( ( Z + N ) ..^ ( ( Z + N ) + 1 ) ) )
8 uzaddcl 
 |-  ( ( Z e. ( ZZ>= ` Z ) /\ N e. NN0 ) -> ( Z + N ) e. ( ZZ>= ` Z ) )
9 fzoss1 
 |-  ( ( Z + N ) e. ( ZZ>= ` Z ) -> ( ( Z + N ) ..^ ( ( Z + N ) + 1 ) ) C_ ( Z ..^ ( ( Z + N ) + 1 ) ) )
10 8 9 syl 
 |-  ( ( Z e. ( ZZ>= ` Z ) /\ N e. NN0 ) -> ( ( Z + N ) ..^ ( ( Z + N ) + 1 ) ) C_ ( Z ..^ ( ( Z + N ) + 1 ) ) )
11 10 sselda 
 |-  ( ( ( Z e. ( ZZ>= ` Z ) /\ N e. NN0 ) /\ ( Z + N ) e. ( ( Z + N ) ..^ ( ( Z + N ) + 1 ) ) ) -> ( Z + N ) e. ( Z ..^ ( ( Z + N ) + 1 ) ) )
12 2 7 11 syl2anc 
 |-  ( ( Z e. ZZ /\ N e. NN0 ) -> ( Z + N ) e. ( Z ..^ ( ( Z + N ) + 1 ) ) )