Step |
Hyp |
Ref |
Expression |
1 |
|
grpsubid.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpsubid.o |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
grpsubid.m |
⊢ − = ( -g ‘ 𝐺 ) |
4 |
|
id |
⊢ ( 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵 ) |
5 |
1 2
|
grpidcl |
⊢ ( 𝐺 ∈ Grp → 0 ∈ 𝐵 ) |
6 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
7 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
8 |
1 6 7 3
|
grpsubval |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → ( 𝑋 − 0 ) = ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 0 ) ) ) |
9 |
4 5 8
|
syl2anr |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 − 0 ) = ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 0 ) ) ) |
10 |
2 7
|
grpinvid |
⊢ ( 𝐺 ∈ Grp → ( ( invg ‘ 𝐺 ) ‘ 0 ) = 0 ) |
11 |
10
|
adantr |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 0 ) = 0 ) |
12 |
11
|
oveq2d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ( +g ‘ 𝐺 ) ( ( invg ‘ 𝐺 ) ‘ 0 ) ) = ( 𝑋 ( +g ‘ 𝐺 ) 0 ) ) |
13 |
1 6 2
|
grprid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ( +g ‘ 𝐺 ) 0 ) = 𝑋 ) |
14 |
9 12 13
|
3eqtrd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 − 0 ) = 𝑋 ) |