Metamath Proof Explorer


Definition df-lsm

Description: Define subgroup sum (inner direct product of subgroups). (Contributed by NM, 28-Jan-2014)

Ref Expression
Assertion df-lsm LSSum = ( 𝑤 ∈ V ↦ ( 𝑡 ∈ 𝒫 ( Base ‘ 𝑤 ) , 𝑢 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ ran ( 𝑥𝑡 , 𝑦𝑢 ↦ ( 𝑥 ( +g𝑤 ) 𝑦 ) ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 clsm LSSum
1 vw 𝑤
2 cvv V
3 vt 𝑡
4 cbs Base
5 1 cv 𝑤
6 5 4 cfv ( Base ‘ 𝑤 )
7 6 cpw 𝒫 ( Base ‘ 𝑤 )
8 vu 𝑢
9 vx 𝑥
10 3 cv 𝑡
11 vy 𝑦
12 8 cv 𝑢
13 9 cv 𝑥
14 cplusg +g
15 5 14 cfv ( +g𝑤 )
16 11 cv 𝑦
17 13 16 15 co ( 𝑥 ( +g𝑤 ) 𝑦 )
18 9 11 10 12 17 cmpo ( 𝑥𝑡 , 𝑦𝑢 ↦ ( 𝑥 ( +g𝑤 ) 𝑦 ) )
19 18 crn ran ( 𝑥𝑡 , 𝑦𝑢 ↦ ( 𝑥 ( +g𝑤 ) 𝑦 ) )
20 3 8 7 7 19 cmpo ( 𝑡 ∈ 𝒫 ( Base ‘ 𝑤 ) , 𝑢 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ ran ( 𝑥𝑡 , 𝑦𝑢 ↦ ( 𝑥 ( +g𝑤 ) 𝑦 ) ) )
21 1 2 20 cmpt ( 𝑤 ∈ V ↦ ( 𝑡 ∈ 𝒫 ( Base ‘ 𝑤 ) , 𝑢 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ ran ( 𝑥𝑡 , 𝑦𝑢 ↦ ( 𝑥 ( +g𝑤 ) 𝑦 ) ) ) )
22 0 21 wceq LSSum = ( 𝑤 ∈ V ↦ ( 𝑡 ∈ 𝒫 ( Base ‘ 𝑤 ) , 𝑢 ∈ 𝒫 ( Base ‘ 𝑤 ) ↦ ran ( 𝑥𝑡 , 𝑦𝑢 ↦ ( 𝑥 ( +g𝑤 ) 𝑦 ) ) ) )