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 )