Metamath Proof Explorer


Theorem dprdff

Description: A finitely supported function in S is a function into the base. (Contributed by Mario Carneiro, 25-Apr-2016) (Revised by AV, 11-Jul-2019)

Ref Expression
Hypotheses dprdff.w 𝑊 = { X 𝑖𝐼 ( 𝑆𝑖 ) ∣ finSupp 0 }
dprdff.1 ( 𝜑𝐺 dom DProd 𝑆 )
dprdff.2 ( 𝜑 → dom 𝑆 = 𝐼 )
dprdff.3 ( 𝜑𝐹𝑊 )
dprdff.b 𝐵 = ( Base ‘ 𝐺 )
Assertion dprdff ( 𝜑𝐹 : 𝐼𝐵 )

Proof

Step Hyp Ref Expression
1 dprdff.w 𝑊 = { X 𝑖𝐼 ( 𝑆𝑖 ) ∣ finSupp 0 }
2 dprdff.1 ( 𝜑𝐺 dom DProd 𝑆 )
3 dprdff.2 ( 𝜑 → dom 𝑆 = 𝐼 )
4 dprdff.3 ( 𝜑𝐹𝑊 )
5 dprdff.b 𝐵 = ( Base ‘ 𝐺 )
6 1 2 3 dprdw ( 𝜑 → ( 𝐹𝑊 ↔ ( 𝐹 Fn 𝐼 ∧ ∀ 𝑥𝐼 ( 𝐹𝑥 ) ∈ ( 𝑆𝑥 ) ∧ 𝐹 finSupp 0 ) ) )
7 4 6 mpbid ( 𝜑 → ( 𝐹 Fn 𝐼 ∧ ∀ 𝑥𝐼 ( 𝐹𝑥 ) ∈ ( 𝑆𝑥 ) ∧ 𝐹 finSupp 0 ) )
8 7 simp1d ( 𝜑𝐹 Fn 𝐼 )
9 7 simp2d ( 𝜑 → ∀ 𝑥𝐼 ( 𝐹𝑥 ) ∈ ( 𝑆𝑥 ) )
10 2 3 dprdf2 ( 𝜑𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) )
11 10 ffvelrnda ( ( 𝜑𝑥𝐼 ) → ( 𝑆𝑥 ) ∈ ( SubGrp ‘ 𝐺 ) )
12 5 subgss ( ( 𝑆𝑥 ) ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑆𝑥 ) ⊆ 𝐵 )
13 11 12 syl ( ( 𝜑𝑥𝐼 ) → ( 𝑆𝑥 ) ⊆ 𝐵 )
14 13 sseld ( ( 𝜑𝑥𝐼 ) → ( ( 𝐹𝑥 ) ∈ ( 𝑆𝑥 ) → ( 𝐹𝑥 ) ∈ 𝐵 ) )
15 14 ralimdva ( 𝜑 → ( ∀ 𝑥𝐼 ( 𝐹𝑥 ) ∈ ( 𝑆𝑥 ) → ∀ 𝑥𝐼 ( 𝐹𝑥 ) ∈ 𝐵 ) )
16 9 15 mpd ( 𝜑 → ∀ 𝑥𝐼 ( 𝐹𝑥 ) ∈ 𝐵 )
17 ffnfv ( 𝐹 : 𝐼𝐵 ↔ ( 𝐹 Fn 𝐼 ∧ ∀ 𝑥𝐼 ( 𝐹𝑥 ) ∈ 𝐵 ) )
18 8 16 17 sylanbrc ( 𝜑𝐹 : 𝐼𝐵 )