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Metamath Proof Explorer

Theorem eluzsubi

Description: Membership in an earlier upper set of integers. (Contributed by Paul Chapman, 22-Nov-2007) (Proof shortened by SN, 7-Feb-2025)

Ref Expression
Hypotheses eluzsubi.1 ⊢ 𝑀 ∈ ℤ
eluzsubi.2 ⊢ 𝐾 ∈ ℤ
Assertion eluzsubi ( 𝑁 ∈ ( ℤ≥ ‘ ( 𝑀 + 𝐾 ) ) → ( 𝑁 − 𝐾 ) ∈ ( ℤ≥ ‘ 𝑀 ) )

Proof

Step Hyp Ref Expression
1 eluzsubi.1 ⊢ 𝑀 ∈ ℤ
2 eluzsubi.2 ⊢ 𝐾 ∈ ℤ
3 eluzsub ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ ( ℤ≥ ‘ ( 𝑀 + 𝐾 ) ) ) → ( 𝑁 − 𝐾 ) ∈ ( ℤ≥ ‘ 𝑀 ) )
4 1 2 3 mp3an12 ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ ( 𝑀 + 𝐾 ) ) → ( 𝑁 − 𝐾 ) ∈ ( ℤ≥ ‘ 𝑀 ) )