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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 ) )