Step |
Hyp |
Ref |
Expression |
1 |
|
gaass.1 |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
2 |
|
gaass.2 |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
4 |
1 2 3
|
isga |
⊢ ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) ↔ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ V ) ∧ ( ⊕ : ( 𝑋 × 𝑌 ) ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝑌 ( ( ( 0g ‘ 𝐺 ) ⊕ 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑦 + 𝑧 ) ⊕ 𝑥 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) ) ) ) ) |
5 |
4
|
simprbi |
⊢ ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) → ( ⊕ : ( 𝑋 × 𝑌 ) ⟶ 𝑌 ∧ ∀ 𝑥 ∈ 𝑌 ( ( ( 0g ‘ 𝐺 ) ⊕ 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑦 + 𝑧 ) ⊕ 𝑥 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) ) ) ) |
6 |
|
simpr |
⊢ ( ( ( ( 0g ‘ 𝐺 ) ⊕ 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑦 + 𝑧 ) ⊕ 𝑥 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) ) → ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑦 + 𝑧 ) ⊕ 𝑥 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) ) |
7 |
6
|
ralimi |
⊢ ( ∀ 𝑥 ∈ 𝑌 ( ( ( 0g ‘ 𝐺 ) ⊕ 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑦 + 𝑧 ) ⊕ 𝑥 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) ) → ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑦 + 𝑧 ) ⊕ 𝑥 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) ) |
8 |
5 7
|
simpl2im |
⊢ ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) → ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑦 + 𝑧 ) ⊕ 𝑥 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) ) |
9 |
|
oveq2 |
⊢ ( 𝑥 = 𝐶 → ( ( 𝑦 + 𝑧 ) ⊕ 𝑥 ) = ( ( 𝑦 + 𝑧 ) ⊕ 𝐶 ) ) |
10 |
|
oveq2 |
⊢ ( 𝑥 = 𝐶 → ( 𝑧 ⊕ 𝑥 ) = ( 𝑧 ⊕ 𝐶 ) ) |
11 |
10
|
oveq2d |
⊢ ( 𝑥 = 𝐶 → ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝐶 ) ) ) |
12 |
9 11
|
eqeq12d |
⊢ ( 𝑥 = 𝐶 → ( ( ( 𝑦 + 𝑧 ) ⊕ 𝑥 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) ↔ ( ( 𝑦 + 𝑧 ) ⊕ 𝐶 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝐶 ) ) ) ) |
13 |
|
oveq1 |
⊢ ( 𝑦 = 𝐴 → ( 𝑦 + 𝑧 ) = ( 𝐴 + 𝑧 ) ) |
14 |
13
|
oveq1d |
⊢ ( 𝑦 = 𝐴 → ( ( 𝑦 + 𝑧 ) ⊕ 𝐶 ) = ( ( 𝐴 + 𝑧 ) ⊕ 𝐶 ) ) |
15 |
|
oveq1 |
⊢ ( 𝑦 = 𝐴 → ( 𝑦 ⊕ ( 𝑧 ⊕ 𝐶 ) ) = ( 𝐴 ⊕ ( 𝑧 ⊕ 𝐶 ) ) ) |
16 |
14 15
|
eqeq12d |
⊢ ( 𝑦 = 𝐴 → ( ( ( 𝑦 + 𝑧 ) ⊕ 𝐶 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝐶 ) ) ↔ ( ( 𝐴 + 𝑧 ) ⊕ 𝐶 ) = ( 𝐴 ⊕ ( 𝑧 ⊕ 𝐶 ) ) ) ) |
17 |
|
oveq2 |
⊢ ( 𝑧 = 𝐵 → ( 𝐴 + 𝑧 ) = ( 𝐴 + 𝐵 ) ) |
18 |
17
|
oveq1d |
⊢ ( 𝑧 = 𝐵 → ( ( 𝐴 + 𝑧 ) ⊕ 𝐶 ) = ( ( 𝐴 + 𝐵 ) ⊕ 𝐶 ) ) |
19 |
|
oveq1 |
⊢ ( 𝑧 = 𝐵 → ( 𝑧 ⊕ 𝐶 ) = ( 𝐵 ⊕ 𝐶 ) ) |
20 |
19
|
oveq2d |
⊢ ( 𝑧 = 𝐵 → ( 𝐴 ⊕ ( 𝑧 ⊕ 𝐶 ) ) = ( 𝐴 ⊕ ( 𝐵 ⊕ 𝐶 ) ) ) |
21 |
18 20
|
eqeq12d |
⊢ ( 𝑧 = 𝐵 → ( ( ( 𝐴 + 𝑧 ) ⊕ 𝐶 ) = ( 𝐴 ⊕ ( 𝑧 ⊕ 𝐶 ) ) ↔ ( ( 𝐴 + 𝐵 ) ⊕ 𝐶 ) = ( 𝐴 ⊕ ( 𝐵 ⊕ 𝐶 ) ) ) ) |
22 |
12 16 21
|
rspc3v |
⊢ ( ( 𝐶 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑋 ∀ 𝑧 ∈ 𝑋 ( ( 𝑦 + 𝑧 ) ⊕ 𝑥 ) = ( 𝑦 ⊕ ( 𝑧 ⊕ 𝑥 ) ) → ( ( 𝐴 + 𝐵 ) ⊕ 𝐶 ) = ( 𝐴 ⊕ ( 𝐵 ⊕ 𝐶 ) ) ) ) |
23 |
8 22
|
syl5 |
⊢ ( ( 𝐶 ∈ 𝑌 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) → ( ( 𝐴 + 𝐵 ) ⊕ 𝐶 ) = ( 𝐴 ⊕ ( 𝐵 ⊕ 𝐶 ) ) ) ) |
24 |
23
|
3coml |
⊢ ( ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌 ) → ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) → ( ( 𝐴 + 𝐵 ) ⊕ 𝐶 ) = ( 𝐴 ⊕ ( 𝐵 ⊕ 𝐶 ) ) ) ) |
25 |
24
|
impcom |
⊢ ( ( ⊕ ∈ ( 𝐺 GrpAct 𝑌 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌 ) ) → ( ( 𝐴 + 𝐵 ) ⊕ 𝐶 ) = ( 𝐴 ⊕ ( 𝐵 ⊕ 𝐶 ) ) ) |