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Metamath Proof Explorer

Theorem fzsuc

Description: Join a successor to the end of a finite set of sequential integers. (Contributed by NM, 19-Jul-2008) (Revised by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion fzsuc 
|- ( N e. ( ZZ>= ` M ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )

Proof

Step Hyp Ref Expression
1 peano2uz 
 |-  ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) )
2 eluzfz2 
 |-  ( ( N + 1 ) e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( M ... ( N + 1 ) ) )
3 1 2 syl 
 |-  ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( M ... ( N + 1 ) ) )
4 peano2fzr 
 |-  ( ( N e. ( ZZ>= ` M ) /\ ( N + 1 ) e. ( M ... ( N + 1 ) ) ) -> N e. ( M ... ( N + 1 ) ) )
5 3 4 mpdan 
 |-  ( N e. ( ZZ>= ` M ) -> N e. ( M ... ( N + 1 ) ) )
6 fzsplit 
 |-  ( N e. ( M ... ( N + 1 ) ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. ( ( N + 1 ) ... ( N + 1 ) ) ) )
7 5 6 syl 
 |-  ( N e. ( ZZ>= ` M ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. ( ( N + 1 ) ... ( N + 1 ) ) ) )
8 eluzelz 
 |-  ( ( N + 1 ) e. ( ZZ>= ` M ) -> ( N + 1 ) e. ZZ )
9 fzsn 
 |-  ( ( N + 1 ) e. ZZ -> ( ( N + 1 ) ... ( N + 1 ) ) = { ( N + 1 ) } )
10 1 8 9 3syl 
 |-  ( N e. ( ZZ>= ` M ) -> ( ( N + 1 ) ... ( N + 1 ) ) = { ( N + 1 ) } )
11 10 uneq2d 
 |-  ( N e. ( ZZ>= ` M ) -> ( ( M ... N ) u. ( ( N + 1 ) ... ( N + 1 ) ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )
12 7 11 eqtrd 
 |-  ( N e. ( ZZ>= ` M ) -> ( M ... ( N + 1 ) ) = ( ( M ... N ) u. { ( N + 1 ) } ) )