Theorem gsumsplit2

Description: Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014) (Revised by AV, 5-Jun-2019)

	Ref	Expression
Hypotheses	gsumsplit2.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	gsumsplit2.z	⊢ 0 = ( 0g ‘ 𝐺 )
	gsumsplit2.p	⊢ + = ( +g ‘ 𝐺 )
	gsumsplit2.g	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
	gsumsplit2.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	gsumsplit2.f	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
	gsumsplit2.w	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) finSupp 0 )
	gsumsplit2.i	⊢ ( 𝜑 → ( 𝐶 ∩ 𝐷 ) = ∅ )
	gsumsplit2.u	⊢ ( 𝜑 → 𝐴 = ( 𝐶 ∪ 𝐷 ) )
Assertion	gsumsplit2	⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) = ( ( 𝐺 Σg ( 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) + ( 𝐺 Σg ( 𝑘 ∈ 𝐷 ↦ 𝑋 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		gsumsplit2.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		gsumsplit2.z	⊢ 0 = ( 0g ‘ 𝐺 )
3		gsumsplit2.p	⊢ + = ( +g ‘ 𝐺 )
4		gsumsplit2.g	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
5		gsumsplit2.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
6		gsumsplit2.f	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
7		gsumsplit2.w	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) finSupp 0 )
8		gsumsplit2.i	⊢ ( 𝜑 → ( 𝐶 ∩ 𝐷 ) = ∅ )
9		gsumsplit2.u	⊢ ( 𝜑 → 𝐴 = ( 𝐶 ∪ 𝐷 ) )
10	6	fmpttd	⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) : 𝐴 ⟶ 𝐵 )
11	1 2 3 4 5 10 7 8 9	gsumsplit	⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) = ( ( 𝐺 Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ↾ 𝐶 ) ) + ( 𝐺 Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ↾ 𝐷 ) ) ) )
12		ssun1	⊢ 𝐶 ⊆ ( 𝐶 ∪ 𝐷 )
13	12 9	sseqtrrid	⊢ ( 𝜑 → 𝐶 ⊆ 𝐴 )
14	13	resmptd	⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ↾ 𝐶 ) = ( 𝑘 ∈ 𝐶 ↦ 𝑋 ) )
15	14	oveq2d	⊢ ( 𝜑 → ( 𝐺 Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ↾ 𝐶 ) ) = ( 𝐺 Σg ( 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) )
16		ssun2	⊢ 𝐷 ⊆ ( 𝐶 ∪ 𝐷 )
17	16 9	sseqtrrid	⊢ ( 𝜑 → 𝐷 ⊆ 𝐴 )
18	17	resmptd	⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ↾ 𝐷 ) = ( 𝑘 ∈ 𝐷 ↦ 𝑋 ) )
19	18	oveq2d	⊢ ( 𝜑 → ( 𝐺 Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ↾ 𝐷 ) ) = ( 𝐺 Σg ( 𝑘 ∈ 𝐷 ↦ 𝑋 ) ) )
20	15 19	oveq12d	⊢ ( 𝜑 → ( ( 𝐺 Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ↾ 𝐶 ) ) + ( 𝐺 Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ↾ 𝐷 ) ) ) = ( ( 𝐺 Σg ( 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) + ( 𝐺 Σg ( 𝑘 ∈ 𝐷 ↦ 𝑋 ) ) ) )
21	11 20	eqtrd	⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) = ( ( 𝐺 Σg ( 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) + ( 𝐺 Σg ( 𝑘 ∈ 𝐷 ↦ 𝑋 ) ) ) )