Step |
Hyp |
Ref |
Expression |
1 |
|
grpinvinv.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpinvinv.n |
⊢ 𝑁 = ( invg ‘ 𝐺 ) |
3 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑥 ) ) = ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑥 ) ) |
4 |
1 2
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑥 ) ∈ 𝐵 ) |
5 |
1 2
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑦 ) ∈ 𝐵 ) |
6 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
7 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
8 |
1 6 7 2
|
grpinvid1 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑦 ) = 𝑥 ↔ ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) ) |
9 |
8
|
3com23 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑦 ) = 𝑥 ↔ ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) ) |
10 |
1 6 7 2
|
grpinvid2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑥 ) = 𝑦 ↔ ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) = ( 0g ‘ 𝐺 ) ) ) |
11 |
9 10
|
bitr4d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑦 ) = 𝑥 ↔ ( 𝑁 ‘ 𝑥 ) = 𝑦 ) ) |
12 |
11
|
3expb |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑁 ‘ 𝑦 ) = 𝑥 ↔ ( 𝑁 ‘ 𝑥 ) = 𝑦 ) ) |
13 |
|
eqcom |
⊢ ( 𝑥 = ( 𝑁 ‘ 𝑦 ) ↔ ( 𝑁 ‘ 𝑦 ) = 𝑥 ) |
14 |
|
eqcom |
⊢ ( 𝑦 = ( 𝑁 ‘ 𝑥 ) ↔ ( 𝑁 ‘ 𝑥 ) = 𝑦 ) |
15 |
12 13 14
|
3bitr4g |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 = ( 𝑁 ‘ 𝑦 ) ↔ 𝑦 = ( 𝑁 ‘ 𝑥 ) ) ) |
16 |
3 4 5 15
|
f1ocnv2d |
⊢ ( 𝐺 ∈ Grp → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑥 ) ) : 𝐵 –1-1-onto→ 𝐵 ∧ ◡ ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑥 ) ) = ( 𝑦 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑦 ) ) ) ) |
17 |
16
|
simprd |
⊢ ( 𝐺 ∈ Grp → ◡ ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑥 ) ) = ( 𝑦 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑦 ) ) ) |
18 |
1 2
|
grpinvf |
⊢ ( 𝐺 ∈ Grp → 𝑁 : 𝐵 ⟶ 𝐵 ) |
19 |
18
|
feqmptd |
⊢ ( 𝐺 ∈ Grp → 𝑁 = ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑥 ) ) ) |
20 |
19
|
cnveqd |
⊢ ( 𝐺 ∈ Grp → ◡ 𝑁 = ◡ ( 𝑥 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑥 ) ) ) |
21 |
18
|
feqmptd |
⊢ ( 𝐺 ∈ Grp → 𝑁 = ( 𝑦 ∈ 𝐵 ↦ ( 𝑁 ‘ 𝑦 ) ) ) |
22 |
17 20 21
|
3eqtr4d |
⊢ ( 𝐺 ∈ Grp → ◡ 𝑁 = 𝑁 ) |