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 ) } ) )