Step |
Hyp |
Ref |
Expression |
1 |
|
grpinveu.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpinveu.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
grpinveu.o |
⊢ 0 = ( 0g ‘ 𝐺 ) |
4 |
|
eqcom |
⊢ ( 0 = 𝑋 ↔ 𝑋 = 0 ) |
5 |
1 3
|
grpidcl |
⊢ ( 𝐺 ∈ Grp → 0 ∈ 𝐵 ) |
6 |
1 2
|
grprcan |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑋 ) = ( 0 + 𝑋 ) ↔ 𝑋 = 0 ) ) |
7 |
6
|
3exp2 |
⊢ ( 𝐺 ∈ Grp → ( 𝑋 ∈ 𝐵 → ( 0 ∈ 𝐵 → ( 𝑋 ∈ 𝐵 → ( ( 𝑋 + 𝑋 ) = ( 0 + 𝑋 ) ↔ 𝑋 = 0 ) ) ) ) ) |
8 |
5 7
|
mpid |
⊢ ( 𝐺 ∈ Grp → ( 𝑋 ∈ 𝐵 → ( 𝑋 ∈ 𝐵 → ( ( 𝑋 + 𝑋 ) = ( 0 + 𝑋 ) ↔ 𝑋 = 0 ) ) ) ) |
9 |
8
|
pm2.43d |
⊢ ( 𝐺 ∈ Grp → ( 𝑋 ∈ 𝐵 → ( ( 𝑋 + 𝑋 ) = ( 0 + 𝑋 ) ↔ 𝑋 = 0 ) ) ) |
10 |
9
|
imp |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑋 + 𝑋 ) = ( 0 + 𝑋 ) ↔ 𝑋 = 0 ) ) |
11 |
1 2 3
|
grplid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 0 + 𝑋 ) = 𝑋 ) |
12 |
11
|
eqeq2d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑋 + 𝑋 ) = ( 0 + 𝑋 ) ↔ ( 𝑋 + 𝑋 ) = 𝑋 ) ) |
13 |
10 12
|
bitr3d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 = 0 ↔ ( 𝑋 + 𝑋 ) = 𝑋 ) ) |
14 |
4 13
|
bitr2id |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑋 + 𝑋 ) = 𝑋 ↔ 0 = 𝑋 ) ) |