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Metamath Proof Explorer

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 ) ) ) )