Step |
Hyp |
Ref |
Expression |
1 |
|
mgmidsssn0.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
mgmidsssn0.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
mgmidsssn0.p |
⊢ + = ( +g ‘ 𝐺 ) |
4 |
|
mgmidsssn0.o |
⊢ 𝑂 = { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) } |
5 |
|
simpr |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) ) ) → ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) ) ) |
6 |
|
oveq1 |
⊢ ( 𝑧 = 𝑥 → ( 𝑧 + 𝑦 ) = ( 𝑥 + 𝑦 ) ) |
7 |
6
|
eqeq1d |
⊢ ( 𝑧 = 𝑥 → ( ( 𝑧 + 𝑦 ) = 𝑦 ↔ ( 𝑥 + 𝑦 ) = 𝑦 ) ) |
8 |
7
|
ovanraleqv |
⊢ ( 𝑧 = 𝑥 → ( ∀ 𝑦 ∈ 𝐵 ( ( 𝑧 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑧 ) = 𝑦 ) ↔ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) ) ) |
9 |
8
|
rspcev |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) ) → ∃ 𝑧 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑧 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑧 ) = 𝑦 ) ) |
10 |
9
|
adantl |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) ) ) → ∃ 𝑧 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝑧 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑧 ) = 𝑦 ) ) |
11 |
1 2 3 10
|
ismgmid |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) ) ) → ( ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) ) ↔ 0 = 𝑥 ) ) |
12 |
5 11
|
mpbid |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) ) ) → 0 = 𝑥 ) |
13 |
12
|
eqcomd |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) ) ) → 𝑥 = 0 ) |
14 |
|
velsn |
⊢ ( 𝑥 ∈ { 0 } ↔ 𝑥 = 0 ) |
15 |
13 14
|
sylibr |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) ) ) → 𝑥 ∈ { 0 } ) |
16 |
15
|
expr |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ) → ( ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) → 𝑥 ∈ { 0 } ) ) |
17 |
16
|
ralrimiva |
⊢ ( 𝐺 ∈ 𝑉 → ∀ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) → 𝑥 ∈ { 0 } ) ) |
18 |
|
rabss |
⊢ ( { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) } ⊆ { 0 } ↔ ∀ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) → 𝑥 ∈ { 0 } ) ) |
19 |
17 18
|
sylibr |
⊢ ( 𝐺 ∈ 𝑉 → { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) } ⊆ { 0 } ) |
20 |
4 19
|
eqsstrid |
⊢ ( 𝐺 ∈ 𝑉 → 𝑂 ⊆ { 0 } ) |