Step |
Hyp |
Ref |
Expression |
1 |
|
gsumsubg.1 |
⊢ 𝐻 = ( 𝐺 ↾s 𝐵 ) |
2 |
|
gsumsubg.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
3 |
|
gsumsubg.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
4 |
|
gsumsubg.b |
⊢ ( 𝜑 → 𝐵 ∈ ( SubGrp ‘ 𝐺 ) ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
6 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
7 |
4
|
elfvexd |
⊢ ( 𝜑 → 𝐺 ∈ V ) |
8 |
5
|
subgss |
⊢ ( 𝐵 ∈ ( SubGrp ‘ 𝐺 ) → 𝐵 ⊆ ( Base ‘ 𝐺 ) ) |
9 |
4 8
|
syl |
⊢ ( 𝜑 → 𝐵 ⊆ ( Base ‘ 𝐺 ) ) |
10 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
11 |
10
|
subg0cl |
⊢ ( 𝐵 ∈ ( SubGrp ‘ 𝐺 ) → ( 0g ‘ 𝐺 ) ∈ 𝐵 ) |
12 |
4 11
|
syl |
⊢ ( 𝜑 → ( 0g ‘ 𝐺 ) ∈ 𝐵 ) |
13 |
|
subgrcl |
⊢ ( 𝐵 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp ) |
14 |
4 13
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
15 |
5 6 10
|
grplid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) 𝑥 ) = 𝑥 ) |
16 |
5 6 10
|
grprid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑥 ( +g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) = 𝑥 ) |
17 |
15 16
|
jca |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) = 𝑥 ) ) |
18 |
14 17
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ( ( 0g ‘ 𝐺 ) ( +g ‘ 𝐺 ) 𝑥 ) = 𝑥 ∧ ( 𝑥 ( +g ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) = 𝑥 ) ) |
19 |
5 6 1 7 2 9 3 12 18
|
gsumress |
⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) = ( 𝐻 Σg 𝐹 ) ) |