Metamath Proof Explorer


Theorem fzodif2

Description: Split the last element of a half-open range of sequential integers. (Contributed by Thierry Arnoux, 5-Dec-2021)

Ref Expression
Assertion fzodif2
|- ( N e. ( ZZ>= ` M ) -> ( ( M ..^ ( N + 1 ) ) \ { N } ) = ( M ..^ N ) )

Proof

Step Hyp Ref Expression
1 fzosplitsn
 |-  ( N e. ( ZZ>= ` M ) -> ( M ..^ ( N + 1 ) ) = ( ( M ..^ N ) u. { N } ) )
2 1 difeq1d
 |-  ( N e. ( ZZ>= ` M ) -> ( ( M ..^ ( N + 1 ) ) \ { N } ) = ( ( ( M ..^ N ) u. { N } ) \ { N } ) )
3 difun2
 |-  ( ( ( M ..^ N ) u. { N } ) \ { N } ) = ( ( M ..^ N ) \ { N } )
4 2 3 eqtrdi
 |-  ( N e. ( ZZ>= ` M ) -> ( ( M ..^ ( N + 1 ) ) \ { N } ) = ( ( M ..^ N ) \ { N } ) )
5 fzonel
 |-  -. N e. ( M ..^ N )
6 disjsn
 |-  ( ( ( M ..^ N ) i^i { N } ) = (/) <-> -. N e. ( M ..^ N ) )
7 5 6 mpbir
 |-  ( ( M ..^ N ) i^i { N } ) = (/)
8 disjdif2
 |-  ( ( ( M ..^ N ) i^i { N } ) = (/) -> ( ( M ..^ N ) \ { N } ) = ( M ..^ N ) )
9 7 8 mp1i
 |-  ( N e. ( ZZ>= ` M ) -> ( ( M ..^ N ) \ { N } ) = ( M ..^ N ) )
10 4 9 eqtrd
 |-  ( N e. ( ZZ>= ` M ) -> ( ( M ..^ ( N + 1 ) ) \ { N } ) = ( M ..^ N ) )