Step |
Hyp |
Ref |
Expression |
1 |
|
grpsubinv.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpsubinv.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
grpsubinv.m |
⊢ − = ( -g ‘ 𝐺 ) |
4 |
|
grpsubinv.n |
⊢ 𝑁 = ( invg ‘ 𝐺 ) |
5 |
|
grpsubinv.g |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
6 |
|
grpsubinv.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
7 |
|
grpsubinv.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
8 |
1 4
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 ) |
9 |
5 7 8
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 ) |
10 |
1 2 4 3
|
grpsubval |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝑋 − ( 𝑁 ‘ 𝑌 ) ) = ( 𝑋 + ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) ) ) |
11 |
6 9 10
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 − ( 𝑁 ‘ 𝑌 ) ) = ( 𝑋 + ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) ) ) |
12 |
1 4
|
grpinvinv |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) = 𝑌 ) |
13 |
5 7 12
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) = 𝑌 ) |
14 |
13
|
oveq2d |
⊢ ( 𝜑 → ( 𝑋 + ( 𝑁 ‘ ( 𝑁 ‘ 𝑌 ) ) ) = ( 𝑋 + 𝑌 ) ) |
15 |
11 14
|
eqtrd |
⊢ ( 𝜑 → ( 𝑋 − ( 𝑁 ‘ 𝑌 ) ) = ( 𝑋 + 𝑌 ) ) |