Theorem lsmfval

Description: The subgroup sum function (for a group or vector space). (Contributed by NM, 28-Jan-2014) (Revised by Mario Carneiro, 19-Apr-2016)

	Ref	Expression
Hypotheses	lsmfval.v	⊢ 𝐵 = ( Base ‘ 𝐺 )
	lsmfval.a	⊢ + = ( +g ‘ 𝐺 )
	lsmfval.s	⊢ ⊕ = ( LSSum ‘ 𝐺 )
Assertion	lsmfval	⊢ ( 𝐺 ∈ 𝑉 → ⊕ = ( 𝑡 ∈ 𝒫 𝐵 , 𝑢 ∈ 𝒫 𝐵 ↦ ran ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 + 𝑦 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		lsmfval.v	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		lsmfval.a	⊢ + = ( +g ‘ 𝐺 )
3		lsmfval.s	⊢ ⊕ = ( LSSum ‘ 𝐺 )
4		elex	⊢ ( 𝐺 ∈ 𝑉 → 𝐺 ∈ V )
5		fveq2	⊢ ( 𝑤 = 𝐺 → ( Base ‘ 𝑤 ) = ( Base ‘ 𝐺 ) )
6	5 1	eqtr4di	⊢ ( 𝑤 = 𝐺 → ( Base ‘ 𝑤 ) = 𝐵 )
7	6	pweqd	⊢ ( 𝑤 = 𝐺 → 𝒫 ( Base ‘ 𝑤 ) = 𝒫 𝐵 )
8		fveq2	⊢ ( 𝑤 = 𝐺 → ( +g ‘ 𝑤 ) = ( +g ‘ 𝐺 ) )
9	8 2	eqtr4di	⊢ ( 𝑤 = 𝐺 → ( +g ‘ 𝑤 ) = + )
10	9	oveqd	⊢ ( 𝑤 = 𝐺 → ( 𝑥 ( +g ‘ 𝑤 ) 𝑦 ) = ( 𝑥 + 𝑦 ) )
11	10	mpoeq3dv	⊢ ( 𝑤 = 𝐺 → ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 ( +g ‘ 𝑤 ) 𝑦 ) ) = ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 + 𝑦 ) ) )
12	11	rneqd	⊢ ( 𝑤 = 𝐺 → ran ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 ( +g ‘ 𝑤 ) 𝑦 ) ) = ran ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 + 𝑦 ) ) )
13	7 7 12	mpoeq123dv	⊢ ( 𝑤 = 𝐺 → ( 𝑡 ∈ 𝒫 ( Base ‘ 𝑤 ) , 𝑢 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ ran ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 ( +g ‘ 𝑤 ) 𝑦 ) ) ) = ( 𝑡 ∈ 𝒫 𝐵 , 𝑢 ∈ 𝒫 𝐵 ↦ ran ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 + 𝑦 ) ) ) )
14		df-lsm	⊢ LSSum = ( 𝑤 ∈ V ↦ ( 𝑡 ∈ 𝒫 ( Base ‘ 𝑤 ) , 𝑢 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ ran ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 ( +g ‘ 𝑤 ) 𝑦 ) ) ) )
15	1	fvexi	⊢ 𝐵 ∈ V
16	15	pwex	⊢ 𝒫 𝐵 ∈ V
17	16 16	mpoex	⊢ ( 𝑡 ∈ 𝒫 𝐵 , 𝑢 ∈ 𝒫 𝐵 ↦ ran ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 + 𝑦 ) ) ) ∈ V
18	13 14 17	fvmpt	⊢ ( 𝐺 ∈ V → ( LSSum ‘ 𝐺 ) = ( 𝑡 ∈ 𝒫 𝐵 , 𝑢 ∈ 𝒫 𝐵 ↦ ran ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 + 𝑦 ) ) ) )
19	4 18	syl	⊢ ( 𝐺 ∈ 𝑉 → ( LSSum ‘ 𝐺 ) = ( 𝑡 ∈ 𝒫 𝐵 , 𝑢 ∈ 𝒫 𝐵 ↦ ran ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 + 𝑦 ) ) ) )
20	3 19	eqtrid	⊢ ( 𝐺 ∈ 𝑉 → ⊕ = ( 𝑡 ∈ 𝒫 𝐵 , 𝑢 ∈ 𝒫 𝐵 ↦ ran ( 𝑥 ∈ 𝑡 , 𝑦 ∈ 𝑢 ↦ ( 𝑥 + 𝑦 ) ) ) )