Theorem gsummptun

Description: Group sum of a disjoint union, whereas sums are expressed as mappings. (Contributed by Thierry Arnoux, 28-Mar-2018) (Proof shortened by AV, 11-Dec-2019)

	Ref	Expression
Hypotheses	gsummptun.b	⊢ 𝐵 = ( Base ‘ 𝑊 )
	gsummptun.p	⊢ + = ( +g ‘ 𝑊 )
	gsummptun.w	⊢ ( 𝜑 → 𝑊 ∈ CMnd )
	gsummptun.a	⊢ ( 𝜑 → ( 𝐴 ∪ 𝐶 ) ∈ Fin )
	gsummptun.d	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) = ∅ )
	gsummptun.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∪ 𝐶 ) ) → 𝐷 ∈ 𝐵 )
Assertion	gsummptun	⊢ ( 𝜑 → ( 𝑊 Σg ( 𝑥 ∈ ( 𝐴 ∪ 𝐶 ) ↦ 𝐷 ) ) = ( ( 𝑊 Σg ( 𝑥 ∈ 𝐴 ↦ 𝐷 ) ) + ( 𝑊 Σg ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		gsummptun.b	⊢ 𝐵 = ( Base ‘ 𝑊 )
2		gsummptun.p	⊢ + = ( +g ‘ 𝑊 )
3		gsummptun.w	⊢ ( 𝜑 → 𝑊 ∈ CMnd )
4		gsummptun.a	⊢ ( 𝜑 → ( 𝐴 ∪ 𝐶 ) ∈ Fin )
5		gsummptun.d	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) = ∅ )
6		gsummptun.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∪ 𝐶 ) ) → 𝐷 ∈ 𝐵 )
7		eqidd	⊢ ( 𝜑 → ( 𝐴 ∪ 𝐶 ) = ( 𝐴 ∪ 𝐶 ) )
8	1 2 3 4 6 5 7	gsummptfidmsplit	⊢ ( 𝜑 → ( 𝑊 Σg ( 𝑥 ∈ ( 𝐴 ∪ 𝐶 ) ↦ 𝐷 ) ) = ( ( 𝑊 Σg ( 𝑥 ∈ 𝐴 ↦ 𝐷 ) ) + ( 𝑊 Σg ( 𝑥 ∈ 𝐶 ↦ 𝐷 ) ) ) )