Step |
Hyp |
Ref |
Expression |
1 |
|
0subg.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
2 |
|
grpmnd |
⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd ) |
3 |
1
|
0subm |
⊢ ( 𝐺 ∈ Mnd → { 0 } ∈ ( SubMnd ‘ 𝐺 ) ) |
4 |
2 3
|
syl |
⊢ ( 𝐺 ∈ Grp → { 0 } ∈ ( SubMnd ‘ 𝐺 ) ) |
5 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
6 |
1 5
|
grpinvid |
⊢ ( 𝐺 ∈ Grp → ( ( invg ‘ 𝐺 ) ‘ 0 ) = 0 ) |
7 |
|
fvex |
⊢ ( ( invg ‘ 𝐺 ) ‘ 0 ) ∈ V |
8 |
7
|
elsn |
⊢ ( ( ( invg ‘ 𝐺 ) ‘ 0 ) ∈ { 0 } ↔ ( ( invg ‘ 𝐺 ) ‘ 0 ) = 0 ) |
9 |
6 8
|
sylibr |
⊢ ( 𝐺 ∈ Grp → ( ( invg ‘ 𝐺 ) ‘ 0 ) ∈ { 0 } ) |
10 |
1
|
fvexi |
⊢ 0 ∈ V |
11 |
|
fveq2 |
⊢ ( 𝑎 = 0 → ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) = ( ( invg ‘ 𝐺 ) ‘ 0 ) ) |
12 |
11
|
eleq1d |
⊢ ( 𝑎 = 0 → ( ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ∈ { 0 } ↔ ( ( invg ‘ 𝐺 ) ‘ 0 ) ∈ { 0 } ) ) |
13 |
10 12
|
ralsn |
⊢ ( ∀ 𝑎 ∈ { 0 } ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ∈ { 0 } ↔ ( ( invg ‘ 𝐺 ) ‘ 0 ) ∈ { 0 } ) |
14 |
9 13
|
sylibr |
⊢ ( 𝐺 ∈ Grp → ∀ 𝑎 ∈ { 0 } ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ∈ { 0 } ) |
15 |
5
|
issubg3 |
⊢ ( 𝐺 ∈ Grp → ( { 0 } ∈ ( SubGrp ‘ 𝐺 ) ↔ ( { 0 } ∈ ( SubMnd ‘ 𝐺 ) ∧ ∀ 𝑎 ∈ { 0 } ( ( invg ‘ 𝐺 ) ‘ 𝑎 ) ∈ { 0 } ) ) ) |
16 |
4 14 15
|
mpbir2and |
⊢ ( 𝐺 ∈ Grp → { 0 } ∈ ( SubGrp ‘ 𝐺 ) ) |