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Metamath Proof Explorer

Theorem nnaddm1cl

Description: Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003) (Proof shortened by Mario Carneiro, 16-May-2014)

Ref Expression
Assertion nnaddm1cl ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) − 1 ) ∈ ℕ )

Proof

Step Hyp Ref Expression
1 nncn ⊢ ( 𝐴 ∈ ℕ → 𝐴 ∈ ℂ )
2 nncn ⊢ ( 𝐵 ∈ ℕ → 𝐵 ∈ ℂ )
3 ax-1cn ⊢ 1 ∈ ℂ
4 addsub ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − 1 ) = ( ( 𝐴 − 1 ) + 𝐵 ) )
5 3 4 mp3an3 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − 1 ) = ( ( 𝐴 − 1 ) + 𝐵 ) )
6 1 2 5 syl2an ⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) − 1 ) = ( ( 𝐴 − 1 ) + 𝐵 ) )
7 nnm1nn0 ⊢ ( 𝐴 ∈ ℕ → ( 𝐴 − 1 ) ∈ ℕ0 )
8 nn0nnaddcl ⊢ ( ( ( 𝐴 − 1 ) ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 − 1 ) + 𝐵 ) ∈ ℕ )
9 7 8 sylan ⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 − 1 ) + 𝐵 ) ∈ ℕ )
10 6 9 eqeltrd ⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ) → ( ( 𝐴 + 𝐵 ) − 1 ) ∈ ℕ )