Step |
Hyp |
Ref |
Expression |
1 |
|
eqg0subg.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
2 |
|
eqg0subg.s |
⊢ 𝑆 = { 0 } |
3 |
|
eqg0subg.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
4 |
|
eqg0subg.r |
⊢ 𝑅 = ( 𝐺 ~QG 𝑆 ) |
5 |
|
df-ec |
⊢ [ 𝑋 ] 𝑅 = ( 𝑅 “ { 𝑋 } ) |
6 |
1 2 3 4
|
eqg0subg |
⊢ ( 𝐺 ∈ Grp → 𝑅 = ( I ↾ 𝐵 ) ) |
7 |
6
|
adantr |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → 𝑅 = ( I ↾ 𝐵 ) ) |
8 |
7
|
imaeq1d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑅 “ { 𝑋 } ) = ( ( I ↾ 𝐵 ) “ { 𝑋 } ) ) |
9 |
|
snssi |
⊢ ( 𝑋 ∈ 𝐵 → { 𝑋 } ⊆ 𝐵 ) |
10 |
9
|
adantl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → { 𝑋 } ⊆ 𝐵 ) |
11 |
|
resima2 |
⊢ ( { 𝑋 } ⊆ 𝐵 → ( ( I ↾ 𝐵 ) “ { 𝑋 } ) = ( I “ { 𝑋 } ) ) |
12 |
10 11
|
syl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( ( I ↾ 𝐵 ) “ { 𝑋 } ) = ( I “ { 𝑋 } ) ) |
13 |
|
imai |
⊢ ( I “ { 𝑋 } ) = { 𝑋 } |
14 |
12 13
|
eqtrdi |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( ( I ↾ 𝐵 ) “ { 𝑋 } ) = { 𝑋 } ) |
15 |
8 14
|
eqtrd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑅 “ { 𝑋 } ) = { 𝑋 } ) |
16 |
5 15
|
eqtrid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → [ 𝑋 ] 𝑅 = { 𝑋 } ) |