Theorem gsumunsn

Description: Append an element to a finite group sum. (Contributed by Mario Carneiro, 19-Dec-2014) (Proof shortened by AV, 8-Mar-2019)

	Ref	Expression
Hypotheses	gsumunsn.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	gsumunsn.p	⊢ + = ( +g ‘ 𝐺 )
	gsumunsn.g	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
	gsumunsn.a	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	gsumunsn.f	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
	gsumunsn.m	⊢ ( 𝜑 → 𝑀 ∈ 𝑉 )
	gsumunsn.d	⊢ ( 𝜑 → ¬ 𝑀 ∈ 𝐴 )
	gsumunsn.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	gsumunsn.s	⊢ ( 𝑘 = 𝑀 → 𝑋 = 𝑌 )
Assertion	gsumunsn	⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 ∪ { 𝑀 } ) ↦ 𝑋 ) ) = ( ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) + 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		gsumunsn.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		gsumunsn.p	⊢ + = ( +g ‘ 𝐺 )
3		gsumunsn.g	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
4		gsumunsn.a	⊢ ( 𝜑 → 𝐴 ∈ Fin )
5		gsumunsn.f	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 )
6		gsumunsn.m	⊢ ( 𝜑 → 𝑀 ∈ 𝑉 )
7		gsumunsn.d	⊢ ( 𝜑 → ¬ 𝑀 ∈ 𝐴 )
8		gsumunsn.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
9		gsumunsn.s	⊢ ( 𝑘 = 𝑀 → 𝑋 = 𝑌 )
10	9	adantl	⊢ ( ( 𝜑 ∧ 𝑘 = 𝑀 ) → 𝑋 = 𝑌 )
11	1 2 3 4 5 6 7 8 10	gsumunsnd	⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 ∪ { 𝑀 } ) ↦ 𝑋 ) ) = ( ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) + 𝑌 ) )