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 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑀 ... ( 𝑁 + 1 ) ) = ( ( 𝑀 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) )

Proof

Step	Hyp	Ref	Expression
1		peano2uz	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) )
2		eluzfz2	⊢ ( ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑁 + 1 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) )
3	1 2	syl	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑁 + 1 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) )
4		peano2fzr	⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( 𝑁 + 1 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) → 𝑁 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) )
5	3 4	mpdan	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) )
6		fzsplit	⊢ ( 𝑁 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) → ( 𝑀 ... ( 𝑁 + 1 ) ) = ( ( 𝑀 ... 𝑁 ) ∪ ( ( 𝑁 + 1 ) ... ( 𝑁 + 1 ) ) ) )
7	5 6	syl	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑀 ... ( 𝑁 + 1 ) ) = ( ( 𝑀 ... 𝑁 ) ∪ ( ( 𝑁 + 1 ) ... ( 𝑁 + 1 ) ) ) )
8		eluzelz	⊢ ( ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑁 + 1 ) ∈ ℤ )
9		fzsn	⊢ ( ( 𝑁 + 1 ) ∈ ℤ → ( ( 𝑁 + 1 ) ... ( 𝑁 + 1 ) ) = { ( 𝑁 + 1 ) } )
10	1 8 9	3syl	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑁 + 1 ) ... ( 𝑁 + 1 ) ) = { ( 𝑁 + 1 ) } )
11	10	uneq2d	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑀 ... 𝑁 ) ∪ ( ( 𝑁 + 1 ) ... ( 𝑁 + 1 ) ) ) = ( ( 𝑀 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) )
12	7 11	eqtrd	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑀 ... ( 𝑁 + 1 ) ) = ( ( 𝑀 ... 𝑁 ) ∪ { ( 𝑁 + 1 ) } ) )