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Metamath Proof Explorer

Theorem fzopred

Description: Join a predecessor to the beginning of an open integer interval. Generalization of fzo0sn0fzo1 . (Contributed by AV, 14-Jul-2020)

Ref Expression
Assertion fzopred ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) → ( 𝑀 ..^ 𝑁 ) = ( { 𝑀 } ∪ ( ( 𝑀 + 1 ) ..^ 𝑁 ) ) )

Proof

Step Hyp Ref Expression
1 fzolb ⊢ ( 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) )
2 fzofzp1 ⊢ ( 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝑀 + 1 ) ∈ ( 𝑀 ... 𝑁 ) )
3 1 2 sylbir ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) → ( 𝑀 + 1 ) ∈ ( 𝑀 ... 𝑁 ) )
4 fzosplit ⊢ ( ( 𝑀 + 1 ) ∈ ( 𝑀 ... 𝑁 ) → ( 𝑀 ..^ 𝑁 ) = ( ( 𝑀 ..^ ( 𝑀 + 1 ) ) ∪ ( ( 𝑀 + 1 ) ..^ 𝑁 ) ) )
5 3 4 syl ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) → ( 𝑀 ..^ 𝑁 ) = ( ( 𝑀 ..^ ( 𝑀 + 1 ) ) ∪ ( ( 𝑀 + 1 ) ..^ 𝑁 ) ) )
6 fzosn ⊢ ( 𝑀 ∈ ℤ → ( 𝑀 ..^ ( 𝑀 + 1 ) ) = { 𝑀 } )
7 6 3ad2ant1 ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) → ( 𝑀 ..^ ( 𝑀 + 1 ) ) = { 𝑀 } )
8 7 uneq1d ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) → ( ( 𝑀 ..^ ( 𝑀 + 1 ) ) ∪ ( ( 𝑀 + 1 ) ..^ 𝑁 ) ) = ( { 𝑀 } ∪ ( ( 𝑀 + 1 ) ..^ 𝑁 ) ) )
9 5 8 eqtrd ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) → ( 𝑀 ..^ 𝑁 ) = ( { 𝑀 } ∪ ( ( 𝑀 + 1 ) ..^ 𝑁 ) ) )