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Metamath Proof Explorer

Theorem dpjeq

Description: Decompose a group sum into projections. (Contributed by Mario Carneiro, 26-Apr-2016) (Revised by AV, 14-Jul-2019)

Ref Expression
Hypotheses dpjfval.1 ⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 )
dpjfval.2 ⊢ ( 𝜑 → dom 𝑆 = 𝐼 )
dpjfval.p ⊢ 𝑃 = ( 𝐺 dProj 𝑆 )
dpjidcl.3 ⊢ ( 𝜑 → 𝐴 ∈ ( 𝐺 DProd 𝑆 ) )
dpjidcl.0 ⊢ 0 = ( 0g ‘ 𝐺 )
dpjidcl.w ⊢ 𝑊 = { ℎ ∈ X 𝑖 ∈ 𝐼 ( 𝑆 ‘ 𝑖 ) ∣ ℎ finSupp 0 }
dpjeq.c ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ 𝐶 ) ∈ 𝑊 )
Assertion dpjeq ( 𝜑 → ( 𝐴 = ( 𝐺 Σg ( 𝑥 ∈ 𝐼 ↦ 𝐶 ) ) ↔ ∀ 𝑥 ∈ 𝐼 ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) = 𝐶 ) )

Proof

Step Hyp Ref Expression
1 dpjfval.1 ⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 )
2 dpjfval.2 ⊢ ( 𝜑 → dom 𝑆 = 𝐼 )
3 dpjfval.p ⊢ 𝑃 = ( 𝐺 dProj 𝑆 )
4 dpjidcl.3 ⊢ ( 𝜑 → 𝐴 ∈ ( 𝐺 DProd 𝑆 ) )
5 dpjidcl.0 ⊢ 0 = ( 0g ‘ 𝐺 )
6 dpjidcl.w ⊢ 𝑊 = { ℎ ∈ X 𝑖 ∈ 𝐼 ( 𝑆 ‘ 𝑖 ) ∣ ℎ finSupp 0 }
7 dpjeq.c ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ 𝐶 ) ∈ 𝑊 )
8 1 2 3 4 5 6 dpjidcl ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) ) ∈ 𝑊 ∧ 𝐴 = ( 𝐺 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) ) ) ) )
9 8 simprd ⊢ ( 𝜑 → 𝐴 = ( 𝐺 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) ) ) )
10 9 eqeq1d ⊢ ( 𝜑 → ( 𝐴 = ( 𝐺 Σg ( 𝑥 ∈ 𝐼 ↦ 𝐶 ) ) ↔ ( 𝐺 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) ) ) = ( 𝐺 Σg ( 𝑥 ∈ 𝐼 ↦ 𝐶 ) ) ) )
11 8 simpld ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) ) ∈ 𝑊 )
12 5 6 1 2 11 7 dprdf11 ⊢ ( 𝜑 → ( ( 𝐺 Σg ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) ) ) = ( 𝐺 Σg ( 𝑥 ∈ 𝐼 ↦ 𝐶 ) ) ↔ ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) ) = ( 𝑥 ∈ 𝐼 ↦ 𝐶 ) ) )
13 fvex ⊢ ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) ∈ V
14 13 rgenw ⊢ ∀ 𝑥 ∈ 𝐼 ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) ∈ V
15 mpteqb ⊢ ( ∀ 𝑥 ∈ 𝐼 ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) ∈ V → ( ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) ) = ( 𝑥 ∈ 𝐼 ↦ 𝐶 ) ↔ ∀ 𝑥 ∈ 𝐼 ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) = 𝐶 ) )
16 14 15 mp1i ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) ) = ( 𝑥 ∈ 𝐼 ↦ 𝐶 ) ↔ ∀ 𝑥 ∈ 𝐼 ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) = 𝐶 ) )
17 10 12 16 3bitrd ⊢ ( 𝜑 → ( 𝐴 = ( 𝐺 Σg ( 𝑥 ∈ 𝐼 ↦ 𝐶 ) ) ↔ ∀ 𝑥 ∈ 𝐼 ( ( 𝑃 ‘ 𝑥 ) ‘ 𝐴 ) = 𝐶 ) )