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 ) ) |