Step |
Hyp |
Ref |
Expression |
1 |
|
grpsubcl.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpsubcl.m |
⊢ − = ( -g ‘ 𝐺 ) |
3 |
|
grpinvsub.n |
⊢ 𝑁 = ( invg ‘ 𝐺 ) |
4 |
1 3
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 ) |
5 |
4
|
3adant2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 ) |
6 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
7 |
1 6 3
|
grpinvadd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑋 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑌 ) ) ) = ( ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) ) |
8 |
5 7
|
syld3an3 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑋 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑌 ) ) ) = ( ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) ) |
9 |
1 3
|
grpinvinv |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) = 𝑌 ) |
10 |
9
|
3adant2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) = 𝑌 ) |
11 |
10
|
oveq1d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) = ( 𝑌 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) ) |
12 |
8 11
|
eqtrd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑋 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑌 ) ) ) = ( 𝑌 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) ) |
13 |
1 6 3 2
|
grpsubval |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑌 ) ) ) |
14 |
13
|
3adant1 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 − 𝑌 ) = ( 𝑋 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑌 ) ) ) |
15 |
14
|
fveq2d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑋 − 𝑌 ) ) = ( 𝑁 ‘ ( 𝑋 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑌 ) ) ) ) |
16 |
1 6 3 2
|
grpsubval |
⊢ ( ( 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 − 𝑋 ) = ( 𝑌 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) ) |
17 |
16
|
ancoms |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 − 𝑋 ) = ( 𝑌 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) ) |
18 |
17
|
3adant1 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 − 𝑋 ) = ( 𝑌 ( +g ‘ 𝐺 ) ( 𝑁 ‘ 𝑋 ) ) ) |
19 |
12 15 18
|
3eqtr4d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑋 − 𝑌 ) ) = ( 𝑌 − 𝑋 ) ) |