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Metamath Proof Explorer

Theorem gsummptfsadd

Description: The sum of two group sums expressed as mappings. (Contributed by AV, 4-Apr-2019) (Revised by AV, 9-Jul-2019)

Ref Expression
Hypotheses gsummptfsadd.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
gsummptfsadd.z ⊢ 0 = ( 0g ‘ 𝐺 )
gsummptfsadd.p ⊢ + = ( +g ‘ 𝐺 )
gsummptfsadd.g ⊢ ( 𝜑 → 𝐺 ∈ CMnd )
gsummptfsadd.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
gsummptfsadd.c ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝐵 )
gsummptfsadd.d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐷 ∈ 𝐵 )
gsummptfsadd.f ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) )
gsummptfsadd.h ⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝐴 ↦ 𝐷 ) )
gsummptfsadd.w ⊢ ( 𝜑 → 𝐹 finSupp 0 )
gsummptfsadd.v ⊢ ( 𝜑 → 𝐻 finSupp 0 )
Assertion gsummptfsadd ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ ( 𝐶 + 𝐷 ) ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) )

Proof

Step Hyp Ref Expression
1 gsummptfsadd.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 gsummptfsadd.z ⊢ 0 = ( 0g ‘ 𝐺 )
3 gsummptfsadd.p ⊢ + = ( +g ‘ 𝐺 )
4 gsummptfsadd.g ⊢ ( 𝜑 → 𝐺 ∈ CMnd )
5 gsummptfsadd.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
6 gsummptfsadd.c ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝐵 )
7 gsummptfsadd.d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐷 ∈ 𝐵 )
8 gsummptfsadd.f ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) )
9 gsummptfsadd.h ⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝐴 ↦ 𝐷 ) )
10 gsummptfsadd.w ⊢ ( 𝜑 → 𝐹 finSupp 0 )
11 gsummptfsadd.v ⊢ ( 𝜑 → 𝐻 finSupp 0 )
12 5 6 7 8 9 offval2 ⊢ ( 𝜑 → ( 𝐹 ∘f + 𝐻 ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐶 + 𝐷 ) ) )
13 12 eqcomd ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( 𝐶 + 𝐷 ) ) = ( 𝐹 ∘f + 𝐻 ) )
14 13 oveq2d ⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ ( 𝐶 + 𝐷 ) ) ) = ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) )
15 8 6 fmpt3d ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
16 9 7 fmpt3d ⊢ ( 𝜑 → 𝐻 : 𝐴 ⟶ 𝐵 )
17 1 2 3 4 5 15 16 10 11 gsumadd ⊢ ( 𝜑 → ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) )
18 14 17 eqtrd ⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ ( 𝐶 + 𝐷 ) ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) )