Step |
Hyp |
Ref |
Expression |
1 |
|
grpidinv.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpidinv.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
4 |
1 3
|
grpidcl |
⊢ ( 𝐺 ∈ Grp → ( 0g ‘ 𝐺 ) ∈ 𝐵 ) |
5 |
|
oveq1 |
⊢ ( 𝑢 = ( 0g ‘ 𝐺 ) → ( 𝑢 + 𝑥 ) = ( ( 0g ‘ 𝐺 ) + 𝑥 ) ) |
6 |
5
|
eqeq1d |
⊢ ( 𝑢 = ( 0g ‘ 𝐺 ) → ( ( 𝑢 + 𝑥 ) = 𝑥 ↔ ( ( 0g ‘ 𝐺 ) + 𝑥 ) = 𝑥 ) ) |
7 |
|
oveq2 |
⊢ ( 𝑢 = ( 0g ‘ 𝐺 ) → ( 𝑥 + 𝑢 ) = ( 𝑥 + ( 0g ‘ 𝐺 ) ) ) |
8 |
7
|
eqeq1d |
⊢ ( 𝑢 = ( 0g ‘ 𝐺 ) → ( ( 𝑥 + 𝑢 ) = 𝑥 ↔ ( 𝑥 + ( 0g ‘ 𝐺 ) ) = 𝑥 ) ) |
9 |
6 8
|
anbi12d |
⊢ ( 𝑢 = ( 0g ‘ 𝐺 ) → ( ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) ↔ ( ( ( 0g ‘ 𝐺 ) + 𝑥 ) = 𝑥 ∧ ( 𝑥 + ( 0g ‘ 𝐺 ) ) = 𝑥 ) ) ) |
10 |
|
eqeq2 |
⊢ ( 𝑢 = ( 0g ‘ 𝐺 ) → ( ( 𝑦 + 𝑥 ) = 𝑢 ↔ ( 𝑦 + 𝑥 ) = ( 0g ‘ 𝐺 ) ) ) |
11 |
|
eqeq2 |
⊢ ( 𝑢 = ( 0g ‘ 𝐺 ) → ( ( 𝑥 + 𝑦 ) = 𝑢 ↔ ( 𝑥 + 𝑦 ) = ( 0g ‘ 𝐺 ) ) ) |
12 |
10 11
|
anbi12d |
⊢ ( 𝑢 = ( 0g ‘ 𝐺 ) → ( ( ( 𝑦 + 𝑥 ) = 𝑢 ∧ ( 𝑥 + 𝑦 ) = 𝑢 ) ↔ ( ( 𝑦 + 𝑥 ) = ( 0g ‘ 𝐺 ) ∧ ( 𝑥 + 𝑦 ) = ( 0g ‘ 𝐺 ) ) ) ) |
13 |
12
|
rexbidv |
⊢ ( 𝑢 = ( 0g ‘ 𝐺 ) → ( ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑢 ∧ ( 𝑥 + 𝑦 ) = 𝑢 ) ↔ ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = ( 0g ‘ 𝐺 ) ∧ ( 𝑥 + 𝑦 ) = ( 0g ‘ 𝐺 ) ) ) ) |
14 |
9 13
|
anbi12d |
⊢ ( 𝑢 = ( 0g ‘ 𝐺 ) → ( ( ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) ∧ ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑢 ∧ ( 𝑥 + 𝑦 ) = 𝑢 ) ) ↔ ( ( ( ( 0g ‘ 𝐺 ) + 𝑥 ) = 𝑥 ∧ ( 𝑥 + ( 0g ‘ 𝐺 ) ) = 𝑥 ) ∧ ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = ( 0g ‘ 𝐺 ) ∧ ( 𝑥 + 𝑦 ) = ( 0g ‘ 𝐺 ) ) ) ) ) |
15 |
14
|
ralbidv |
⊢ ( 𝑢 = ( 0g ‘ 𝐺 ) → ( ∀ 𝑥 ∈ 𝐵 ( ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) ∧ ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑢 ∧ ( 𝑥 + 𝑦 ) = 𝑢 ) ) ↔ ∀ 𝑥 ∈ 𝐵 ( ( ( ( 0g ‘ 𝐺 ) + 𝑥 ) = 𝑥 ∧ ( 𝑥 + ( 0g ‘ 𝐺 ) ) = 𝑥 ) ∧ ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = ( 0g ‘ 𝐺 ) ∧ ( 𝑥 + 𝑦 ) = ( 0g ‘ 𝐺 ) ) ) ) ) |
16 |
15
|
adantl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑢 = ( 0g ‘ 𝐺 ) ) → ( ∀ 𝑥 ∈ 𝐵 ( ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) ∧ ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑢 ∧ ( 𝑥 + 𝑦 ) = 𝑢 ) ) ↔ ∀ 𝑥 ∈ 𝐵 ( ( ( ( 0g ‘ 𝐺 ) + 𝑥 ) = 𝑥 ∧ ( 𝑥 + ( 0g ‘ 𝐺 ) ) = 𝑥 ) ∧ ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = ( 0g ‘ 𝐺 ) ∧ ( 𝑥 + 𝑦 ) = ( 0g ‘ 𝐺 ) ) ) ) ) |
17 |
1 2 3
|
grpidinv2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ) → ( ( ( ( 0g ‘ 𝐺 ) + 𝑥 ) = 𝑥 ∧ ( 𝑥 + ( 0g ‘ 𝐺 ) ) = 𝑥 ) ∧ ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = ( 0g ‘ 𝐺 ) ∧ ( 𝑥 + 𝑦 ) = ( 0g ‘ 𝐺 ) ) ) ) |
18 |
17
|
ralrimiva |
⊢ ( 𝐺 ∈ Grp → ∀ 𝑥 ∈ 𝐵 ( ( ( ( 0g ‘ 𝐺 ) + 𝑥 ) = 𝑥 ∧ ( 𝑥 + ( 0g ‘ 𝐺 ) ) = 𝑥 ) ∧ ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = ( 0g ‘ 𝐺 ) ∧ ( 𝑥 + 𝑦 ) = ( 0g ‘ 𝐺 ) ) ) ) |
19 |
4 16 18
|
rspcedvd |
⊢ ( 𝐺 ∈ Grp → ∃ 𝑢 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ( ( ( 𝑢 + 𝑥 ) = 𝑥 ∧ ( 𝑥 + 𝑢 ) = 𝑥 ) ∧ ∃ 𝑦 ∈ 𝐵 ( ( 𝑦 + 𝑥 ) = 𝑢 ∧ ( 𝑥 + 𝑦 ) = 𝑢 ) ) ) |