Theorem vdwap0

Description: Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014)

	Ref	Expression
Assertion	vdwap0	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝐴 ( AP ‘ 0 ) 𝐷 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		noel	⊢ ¬ 𝑚 ∈ ∅
2	1	pm2.21i	⊢ ( 𝑚 ∈ ∅ → ¬ 𝑥 = ( 𝐴 + ( 𝑚 · 𝐷 ) ) )
3		risefall0lem	⊢ ( 0 ... ( 0 − 1 ) ) = ∅
4	2 3	eleq2s	⊢ ( 𝑚 ∈ ( 0 ... ( 0 − 1 ) ) → ¬ 𝑥 = ( 𝐴 + ( 𝑚 · 𝐷 ) ) )
5	4	nrex	⊢ ¬ ∃ 𝑚 ∈ ( 0 ... ( 0 − 1 ) ) 𝑥 = ( 𝐴 + ( 𝑚 · 𝐷 ) )
6		0nn0	⊢ 0 ∈ ℕ0
7		vdwapval	⊢ ( ( 0 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝑥 ∈ ( 𝐴 ( AP ‘ 0 ) 𝐷 ) ↔ ∃ 𝑚 ∈ ( 0 ... ( 0 − 1 ) ) 𝑥 = ( 𝐴 + ( 𝑚 · 𝐷 ) ) ) )
8	6 7	mp3an1	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝑥 ∈ ( 𝐴 ( AP ‘ 0 ) 𝐷 ) ↔ ∃ 𝑚 ∈ ( 0 ... ( 0 − 1 ) ) 𝑥 = ( 𝐴 + ( 𝑚 · 𝐷 ) ) ) )
9	5 8	mtbiri	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ¬ 𝑥 ∈ ( 𝐴 ( AP ‘ 0 ) 𝐷 ) )
10	9	eq0rdv	⊢ ( ( 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ ) → ( 𝐴 ( AP ‘ 0 ) 𝐷 ) = ∅ )