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

Proof

Step Hyp Ref Expression
1 fzosplitsn ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑀 ..^ ( 𝑁 + 1 ) ) = ( ( 𝑀 ..^ 𝑁 ) ∪ { 𝑁 } ) )
2 1 difeq1d ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑀 ..^ ( 𝑁 + 1 ) ) ∖ { 𝑁 } ) = ( ( ( 𝑀 ..^ 𝑁 ) ∪ { 𝑁 } ) ∖ { 𝑁 } ) )
3 difun2 ⊢ ( ( ( 𝑀 ..^ 𝑁 ) ∪ { 𝑁 } ) ∖ { 𝑁 } ) = ( ( 𝑀 ..^ 𝑁 ) ∖ { 𝑁 } )
4 2 3 eqtrdi ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑀 ..^ ( 𝑁 + 1 ) ) ∖ { 𝑁 } ) = ( ( 𝑀 ..^ 𝑁 ) ∖ { 𝑁 } ) )
5 fzonel ⊢ ¬ 𝑁 ∈ ( 𝑀 ..^ 𝑁 )
6 disjsn ⊢ ( ( ( 𝑀 ..^ 𝑁 ) ∩ { 𝑁 } ) = ∅ ↔ ¬ 𝑁 ∈ ( 𝑀 ..^ 𝑁 ) )
7 5 6 mpbir ⊢ ( ( 𝑀 ..^ 𝑁 ) ∩ { 𝑁 } ) = ∅
8 disjdif2 ⊢ ( ( ( 𝑀 ..^ 𝑁 ) ∩ { 𝑁 } ) = ∅ → ( ( 𝑀 ..^ 𝑁 ) ∖ { 𝑁 } ) = ( 𝑀 ..^ 𝑁 ) )
9 7 8 mp1i ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑀 ..^ 𝑁 ) ∖ { 𝑁 } ) = ( 𝑀 ..^ 𝑁 ) )
10 4 9 eqtrd ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑀 ..^ ( 𝑁 + 1 ) ) ∖ { 𝑁 } ) = ( 𝑀 ..^ 𝑁 ) )