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Metamath Proof Explorer

Theorem 1fzopredsuc

Description: Join 0 and a successor to the beginning and the end of an open integer interval starting at 1. (Contributed by AV, 14-Jul-2020)

Ref Expression
Assertion 1fzopredsuc 
|- ( N e. NN0 -> ( 0 ... N ) = ( ( { 0 } u. ( 1 ..^ N ) ) u. { N } ) )

Proof

Step Hyp Ref Expression
1 elnn0uz 
 |-  ( N e. NN0 <-> N e. ( ZZ>= ` 0 ) )
2 fzopredsuc 
 |-  ( N e. ( ZZ>= ` 0 ) -> ( 0 ... N ) = ( ( { 0 } u. ( ( 0 + 1 ) ..^ N ) ) u. { N } ) )
3 0p1e1 
 |-  ( 0 + 1 ) = 1
4 3 oveq1i 
 |-  ( ( 0 + 1 ) ..^ N ) = ( 1 ..^ N )
5 4 uneq2i 
 |-  ( { 0 } u. ( ( 0 + 1 ) ..^ N ) ) = ( { 0 } u. ( 1 ..^ N ) )
6 5 uneq1i 
 |-  ( ( { 0 } u. ( ( 0 + 1 ) ..^ N ) ) u. { N } ) = ( ( { 0 } u. ( 1 ..^ N ) ) u. { N } )
7 2 6 eqtrdi 
 |-  ( N e. ( ZZ>= ` 0 ) -> ( 0 ... N ) = ( ( { 0 } u. ( 1 ..^ N ) ) u. { N } ) )
8 1 7 sylbi 
 |-  ( N e. NN0 -> ( 0 ... N ) = ( ( { 0 } u. ( 1 ..^ N ) ) u. { N } ) )