Step |
Hyp |
Ref |
Expression |
1 |
|
mulginvcom.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
mulginvcom.t |
⊢ · = ( .g ‘ 𝐺 ) |
3 |
|
mulginvcom.i |
⊢ 𝐼 = ( invg ‘ 𝐺 ) |
4 |
1 3
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 ) |
5 |
4
|
3adant2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 ) |
6 |
1 2 3
|
mulginvcom |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 ) → ( 𝑁 · ( 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) ) = ( 𝐼 ‘ ( 𝑁 · ( 𝐼 ‘ 𝑋 ) ) ) ) |
7 |
5 6
|
syld3an3 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · ( 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) ) = ( 𝐼 ‘ ( 𝑁 · ( 𝐼 ‘ 𝑋 ) ) ) ) |
8 |
1 3
|
grpinvinv |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) = 𝑋 ) |
9 |
8
|
3adant2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) = 𝑋 ) |
10 |
9
|
oveq2d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · ( 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) ) = ( 𝑁 · 𝑋 ) ) |
11 |
7 10
|
eqtr3d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ) → ( 𝐼 ‘ ( 𝑁 · ( 𝐼 ‘ 𝑋 ) ) ) = ( 𝑁 · 𝑋 ) ) |