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Metamath Proof Explorer

Theorem fzdif2

Description: Split the last element of a finite set of sequential integers. More generic than fzsuc . (Contributed by Thierry Arnoux, 22-Aug-2020)

Ref Expression
Assertion fzdif2 ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑀 ... 𝑁 ) ∖ { 𝑁 } ) = ( 𝑀 ... ( 𝑁 − 1 ) ) )

Proof

Step Hyp Ref Expression
1 fzspl ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑀 ... 𝑁 ) = ( ( 𝑀 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) )
2 1 difeq1d ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑀 ... 𝑁 ) ∖ { 𝑁 } ) = ( ( ( 𝑀 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ∖ { 𝑁 } ) )
3 difun2 ⊢ ( ( ( 𝑀 ... ( 𝑁 − 1 ) ) ∪ { 𝑁 } ) ∖ { 𝑁 } ) = ( ( 𝑀 ... ( 𝑁 − 1 ) ) ∖ { 𝑁 } )
4 2 3 eqtrdi ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑀 ... 𝑁 ) ∖ { 𝑁 } ) = ( ( 𝑀 ... ( 𝑁 − 1 ) ) ∖ { 𝑁 } ) )
5 eluzelz ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ℤ )
6 uzid ⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ( ℤ≥ ‘ 𝑁 ) )
7 uznfz ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑁 ) → ¬ 𝑁 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) )
8 5 6 7 3syl ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ¬ 𝑁 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) )
9 disjsn ⊢ ( ( ( 𝑀 ... ( 𝑁 − 1 ) ) ∩ { 𝑁 } ) = ∅ ↔ ¬ 𝑁 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) )
10 8 9 sylibr ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑀 ... ( 𝑁 − 1 ) ) ∩ { 𝑁 } ) = ∅ )
11 disjdif2 ⊢ ( ( ( 𝑀 ... ( 𝑁 − 1 ) ) ∩ { 𝑁 } ) = ∅ → ( ( 𝑀 ... ( 𝑁 − 1 ) ) ∖ { 𝑁 } ) = ( 𝑀 ... ( 𝑁 − 1 ) ) )
12 10 11 syl ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑀 ... ( 𝑁 − 1 ) ) ∖ { 𝑁 } ) = ( 𝑀 ... ( 𝑁 − 1 ) ) )
13 4 12 eqtrd ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑀 ... 𝑁 ) ∖ { 𝑁 } ) = ( 𝑀 ... ( 𝑁 − 1 ) ) )