Step |
Hyp |
Ref |
Expression |
1 |
|
grpidd.b |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐺 ) ) |
2 |
|
grpidd.p |
⊢ ( 𝜑 → + = ( +g ‘ 𝐺 ) ) |
3 |
|
grpidd.z |
⊢ ( 𝜑 → 0 ∈ 𝐵 ) |
4 |
|
grpidd.i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 0 + 𝑥 ) = 𝑥 ) |
5 |
|
grpidd.j |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 + 0 ) = 𝑥 ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
7 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
8 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
9 |
3 1
|
eleqtrd |
⊢ ( 𝜑 → 0 ∈ ( Base ‘ 𝐺 ) ) |
10 |
1
|
eleq2d |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↔ 𝑥 ∈ ( Base ‘ 𝐺 ) ) ) |
11 |
10
|
biimpar |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → 𝑥 ∈ 𝐵 ) |
12 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → + = ( +g ‘ 𝐺 ) ) |
13 |
12
|
oveqd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 0 + 𝑥 ) = ( 0 ( +g ‘ 𝐺 ) 𝑥 ) ) |
14 |
13 4
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 0 ( +g ‘ 𝐺 ) 𝑥 ) = 𝑥 ) |
15 |
11 14
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( 0 ( +g ‘ 𝐺 ) 𝑥 ) = 𝑥 ) |
16 |
12
|
oveqd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 + 0 ) = ( 𝑥 ( +g ‘ 𝐺 ) 0 ) ) |
17 |
16 5
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 ( +g ‘ 𝐺 ) 0 ) = 𝑥 ) |
18 |
11 17
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 0 ) = 𝑥 ) |
19 |
6 7 8 9 15 18
|
ismgmid2 |
⊢ ( 𝜑 → 0 = ( 0g ‘ 𝐺 ) ) |