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Metamath Proof Explorer

Theorem gsumsn

Description: Group sum of a singleton. (Contributed by Mario Carneiro, 19-Dec-2014) (Revised by Mario Carneiro, 24-Apr-2016) (Proof shortened by AV, 11-Dec-2019)

	Ref	Expression
Hypotheses	gsumsn.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	gsumsn.s	⊢ ( 𝑘 = 𝑀 → 𝐴 = 𝐶 )
Assertion	gsumsn	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑀 ∈ 𝑉 ∧ 𝐶 ∈ 𝐵 ) → ( 𝐺 Σ_g ( 𝑘 ∈ { 𝑀 } ↦ 𝐴 ) ) = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		gsumsn.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		gsumsn.s	⊢ ( 𝑘 = 𝑀 → 𝐴 = 𝐶 )
3		nfcv	⊢ Ⅎ 𝑘 𝐶
4	3 1 2	gsumsnf	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑀 ∈ 𝑉 ∧ 𝐶 ∈ 𝐵 ) → ( 𝐺 Σ_g ( 𝑘 ∈ { 𝑀 } ↦ 𝐴 ) ) = 𝐶 )