Theorem gsumsubm

Description: Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014)

	Ref	Expression
Hypotheses	gsumsubm.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	gsumsubm.s	⊢ ( 𝜑 → 𝑆 ∈ ( SubMnd ‘ 𝐺 ) )
	gsumsubm.f	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 )
	gsumsubm.h	⊢ 𝐻 = ( 𝐺 ↾s 𝑆 )
Assertion	gsumsubm	⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) = ( 𝐻 Σg 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		gsumsubm.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		gsumsubm.s	⊢ ( 𝜑 → 𝑆 ∈ ( SubMnd ‘ 𝐺 ) )
3		gsumsubm.f	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 )
4		gsumsubm.h	⊢ 𝐻 = ( 𝐺 ↾s 𝑆 )
5		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
6		eqid	⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
7		submrcl	⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) → 𝐺 ∈ Mnd )
8	2 7	syl	⊢ ( 𝜑 → 𝐺 ∈ Mnd )
9	5	submss	⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) → 𝑆 ⊆ ( Base ‘ 𝐺 ) )
10	2 9	syl	⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝐺 ) )
11		eqid	⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
12	11	subm0cl	⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) → ( 0g ‘ 𝐺 ) ∈ 𝑆 )
13	2 12	syl	⊢ ( 𝜑 → ( 0g ‘ 𝐺 ) ∈ 𝑆 )
14	5 6 11	mndlrid	⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) = 𝑥 ) )
15	8 14	sylan	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) = 𝑥 ) )
16	5 6 4 8 1 10 3 13 15	gsumress	⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) = ( 𝐻 Σg 𝐹 ) )