Step |
Hyp |
Ref |
Expression |
1 |
|
grplrinv.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grplrinv.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
grplrinv.i |
⊢ 0 = ( 0g ‘ 𝐺 ) |
4 |
|
eqid |
⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 ) |
5 |
1 4
|
grpinvcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ∈ 𝐵 ) |
6 |
|
oveq1 |
⊢ ( 𝑦 = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) → ( 𝑦 + 𝑥 ) = ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) + 𝑥 ) ) |
7 |
6
|
eqeq1d |
⊢ ( 𝑦 = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) → ( ( 𝑦 + 𝑥 ) = 0 ↔ ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) + 𝑥 ) = 0 ) ) |
8 |
|
oveq2 |
⊢ ( 𝑦 = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) → ( 𝑥 + 𝑦 ) = ( 𝑥 + ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) ) |
9 |
8
|
eqeq1d |
⊢ ( 𝑦 = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) → ( ( 𝑥 + 𝑦 ) = 0 ↔ ( 𝑥 + ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = 0 ) ) |
10 |
7 9
|
anbi12d |
⊢ ( 𝑦 = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) → ( ( ( 𝑦 + 𝑥 ) = 0 ∧ ( 𝑥 + 𝑦 ) = 0 ) ↔ ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) + 𝑥 ) = 0 ∧ ( 𝑥 + ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = 0 ) ) ) |
11 |
10
|
adantl |
⊢ ( ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑦 = ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) → ( ( ( 𝑦 + 𝑥 ) = 0 ∧ ( 𝑥 + 𝑦 ) = 0 ) ↔ ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) + 𝑥 ) = 0 ∧ ( 𝑥 + ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = 0 ) ) ) |
12 |
1 2 3 4
|
grplinv |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) + 𝑥 ) = 0 ) |
13 |
1 2 3 4
|
grprinv |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 + ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = 0 ) |
14 |
12 13
|
jca |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( ( ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) + 𝑥 ) = 0 ∧ ( 𝑥 + ( ( invg ‘ 𝐺 ) ‘ 𝑥 ) ) = 0 ) ) |
15 |
5 11 14
|
rspcedvd |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 0 ∧ ( 𝑥 + 𝑦 ) = 0 ) ) |
16 |
15
|
ralrimiva |
⊢ ( 𝐺 ∈ Grp → ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 0 ∧ ( 𝑥 + 𝑦 ) = 0 ) ) |