Theorem fsuppmapnn0fz

Description: If a function over the nonnegative integers is finitely supported, then there is an upper bound for a finite set of sequential integers containing the support of the function. (Contributed by AV, 30-Sep-2019) (Proof shortened by AV, 6-Oct-2019)

	Ref	Expression
Assertion	fsuppmapnn0fz	⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ( 𝐹 finSupp 𝑍 → ∃ 𝑚 ∈ ℕ0 ( 𝐹 supp 𝑍 ) ⊆ ( 0 ... 𝑚 ) ) )

Proof

Step	Hyp	Ref	Expression
1		fsuppmapnn0ub	⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ( 𝐹 finSupp 𝑍 → ∃ 𝑚 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) )
2		simpllr	⊢ ( ( ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑚 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) → 𝑍 ∈ 𝑉 )
3		simplll	⊢ ( ( ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑚 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) → 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) )
4		simplr	⊢ ( ( ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑚 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) → 𝑚 ∈ ℕ0 )
5		simpr	⊢ ( ( ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑚 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) → ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) )
6	2 3 4 5	suppssfz	⊢ ( ( ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑚 ∈ ℕ0 ) ∧ ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) → ( 𝐹 supp 𝑍 ) ⊆ ( 0 ... 𝑚 ) )
7	6	ex	⊢ ( ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) ∧ 𝑚 ∈ ℕ0 ) → ( ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) → ( 𝐹 supp 𝑍 ) ⊆ ( 0 ... 𝑚 ) ) )
8	7	reximdva	⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ( ∃ 𝑚 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) → ∃ 𝑚 ∈ ℕ0 ( 𝐹 supp 𝑍 ) ⊆ ( 0 ... 𝑚 ) ) )
9	1 8	syld	⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ( 𝐹 finSupp 𝑍 → ∃ 𝑚 ∈ ℕ0 ( 𝐹 supp 𝑍 ) ⊆ ( 0 ... 𝑚 ) ) )