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Metamath Proof Explorer

Theorem fsuppmapnn0ub

Description: If a function over the nonnegative integers is finitely supported, then there is an upper bound for the arguments resulting in nonzero values. (Contributed by AV, 6-Oct-2019)

Ref Expression
Assertion fsuppmapnn0ub ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ( 𝐹 finSupp 𝑍 → ∃ 𝑚 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) )

Proof

Step Hyp Ref Expression
1 simpr ⊢ ( ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) ∧ 𝐹 finSupp 𝑍 ) → 𝐹 finSupp 𝑍 )
2 1 fsuppimpd ⊢ ( ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) ∧ 𝐹 finSupp 𝑍 ) → ( 𝐹 supp 𝑍 ) ∈ Fin )
3 2 ex ⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ( 𝐹 finSupp 𝑍 → ( 𝐹 supp 𝑍 ) ∈ Fin ) )
4 elmapfn ⊢ ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) → 𝐹 Fn ℕ0 )
5 4 adantr ⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → 𝐹 Fn ℕ0 )
6 nn0ex ⊢ ℕ0 ∈ V
7 6 a1i ⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ℕ0 ∈ V )
8 simpr ⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → 𝑍 ∈ 𝑉 )
9 suppvalfn ⊢ ( ( 𝐹 Fn ℕ0 ∧ ℕ0 ∈ V ∧ 𝑍 ∈ 𝑉 ) → ( 𝐹 supp 𝑍 ) = { 𝑥 ∈ ℕ0 ∣ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 } )
10 5 7 8 9 syl3anc ⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ( 𝐹 supp 𝑍 ) = { 𝑥 ∈ ℕ0 ∣ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 } )
11 10 eleq1d ⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ( ( 𝐹 supp 𝑍 ) ∈ Fin ↔ { 𝑥 ∈ ℕ0 ∣ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 } ∈ Fin ) )
12 rabssnn0fi ⊢ ( { 𝑥 ∈ ℕ0 ∣ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 } ∈ Fin ↔ ∃ 𝑚 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ¬ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ) )
13 nne ⊢ ( ¬ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ↔ ( 𝐹 ‘ 𝑥 ) = 𝑍 )
14 13 imbi2i ⊢ ( ( 𝑚 < 𝑥 → ¬ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ) ↔ ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) )
15 14 ralbii ⊢ ( ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ¬ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ) ↔ ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) )
16 15 rexbii ⊢ ( ∃ 𝑚 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ¬ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 ) ↔ ∃ 𝑚 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) )
17 12 16 sylbb ⊢ ( { 𝑥 ∈ ℕ0 ∣ ( 𝐹 ‘ 𝑥 ) ≠ 𝑍 } ∈ Fin → ∃ 𝑚 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) )
18 11 17 biimtrdi ⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ( ( 𝐹 supp 𝑍 ) ∈ Fin → ∃ 𝑚 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) )
19 3 18 syld ⊢ ( ( 𝐹 ∈ ( 𝑅 ↑m ℕ0 ) ∧ 𝑍 ∈ 𝑉 ) → ( 𝐹 finSupp 𝑍 → ∃ 𝑚 ∈ ℕ0 ∀ 𝑥 ∈ ℕ0 ( 𝑚 < 𝑥 → ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) )