Database
BASIC ALGEBRAIC STRUCTURES
Groups
Abelian groups
Internal direct products
dprdffsupp
Metamath Proof Explorer
Description: A finitely supported function in S is a finitely supported function.
(Contributed by Mario Carneiro , 25-Apr-2016) (Revised by AV , 11-Jul-2019)
Ref
Expression
Hypotheses
dprdff.w
⊢ 𝑊 = { ℎ ∈ X 𝑖 ∈ 𝐼 ( 𝑆 ‘ 𝑖 ) ∣ ℎ finSupp 0 }
dprdff.1
⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 )
dprdff.2
⊢ ( 𝜑 → dom 𝑆 = 𝐼 )
dprdff.3
⊢ ( 𝜑 → 𝐹 ∈ 𝑊 )
Assertion
dprdffsupp
⊢ ( 𝜑 → 𝐹 finSupp 0 )
Proof
Step
Hyp
Ref
Expression
1
dprdff.w
⊢ 𝑊 = { ℎ ∈ X 𝑖 ∈ 𝐼 ( 𝑆 ‘ 𝑖 ) ∣ ℎ finSupp 0 }
2
dprdff.1
⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 )
3
dprdff.2
⊢ ( 𝜑 → dom 𝑆 = 𝐼 )
4
dprdff.3
⊢ ( 𝜑 → 𝐹 ∈ 𝑊 )
5
1 2 3
dprdw
⊢ ( 𝜑 → ( 𝐹 ∈ 𝑊 ↔ ( 𝐹 Fn 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ∈ ( 𝑆 ‘ 𝑥 ) ∧ 𝐹 finSupp 0 ) ) )
6
4 5
mpbid
⊢ ( 𝜑 → ( 𝐹 Fn 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ∈ ( 𝑆 ‘ 𝑥 ) ∧ 𝐹 finSupp 0 ) )
7
6
simp3d
⊢ ( 𝜑 → 𝐹 finSupp 0 )