Step |
Hyp |
Ref |
Expression |
1 |
|
grplcan.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grplcan.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
grpasscan1.n |
⊢ 𝑁 = ( invg ‘ 𝐺 ) |
4 |
|
simp1 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐺 ∈ Grp ) |
5 |
|
simp2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
6 |
1 3
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 ) |
7 |
6
|
3adant2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 ) |
8 |
|
simp3 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 ) |
9 |
1 2
|
grpass |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ ( 𝑁 ‘ 𝑌 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = ( 𝑋 + ( ( 𝑁 ‘ 𝑌 ) + 𝑌 ) ) ) |
10 |
4 5 7 8 9
|
syl13anc |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = ( 𝑋 + ( ( 𝑁 ‘ 𝑌 ) + 𝑌 ) ) ) |
11 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
12 |
1 2 11 3
|
grplinv |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑌 ) + 𝑌 ) = ( 0g ‘ 𝐺 ) ) |
13 |
12
|
3adant2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑁 ‘ 𝑌 ) + 𝑌 ) = ( 0g ‘ 𝐺 ) ) |
14 |
13
|
oveq2d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + ( ( 𝑁 ‘ 𝑌 ) + 𝑌 ) ) = ( 𝑋 + ( 0g ‘ 𝐺 ) ) ) |
15 |
1 2 11
|
grprid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 + ( 0g ‘ 𝐺 ) ) = 𝑋 ) |
16 |
15
|
3adant3 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + ( 0g ‘ 𝐺 ) ) = 𝑋 ) |
17 |
10 14 16
|
3eqtrd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = 𝑋 ) |