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Metamath Proof Explorer

Theorem rrxsuppss

Description: Support of Euclidean vectors. (Contributed by Thierry Arnoux, 7-Jul-2019)

Ref Expression
Hypotheses rrxmval.1 ⊢ 𝑋 = { ℎ ∈ ( ℝ ↑m 𝐼 ) ∣ ℎ finSupp 0 }
rrxf.1 ⊢ ( 𝜑 → 𝐹 ∈ 𝑋 )
Assertion rrxsuppss ( 𝜑 → ( 𝐹 supp 0 ) ⊆ 𝐼 )

Proof

Step Hyp Ref Expression
1 rrxmval.1 ⊢ 𝑋 = { ℎ ∈ ( ℝ ↑m 𝐼 ) ∣ ℎ finSupp 0 }
2 rrxf.1 ⊢ ( 𝜑 → 𝐹 ∈ 𝑋 )
3 suppssdm ⊢ ( 𝐹 supp 0 ) ⊆ dom 𝐹
4 1 2 rrxf ⊢ ( 𝜑 → 𝐹 : 𝐼 ⟶ ℝ )
5 3 4 fssdm ⊢ ( 𝜑 → ( 𝐹 supp 0 ) ⊆ 𝐼 )