This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 ∈ ( 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 ) } ) )