Step |
Hyp |
Ref |
Expression |
1 |
|
grpidrcan.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpidrcan.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
grpidrcan.o |
⊢ 0 = ( 0g ‘ 𝐺 ) |
4 |
1 2 3
|
grplid |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ) → ( 0 + 𝑋 ) = 𝑋 ) |
5 |
4
|
3adant3 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 0 + 𝑋 ) = 𝑋 ) |
6 |
5
|
eqeq2d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝑍 + 𝑋 ) = ( 0 + 𝑋 ) ↔ ( 𝑍 + 𝑋 ) = 𝑋 ) ) |
7 |
|
simp1 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → 𝐺 ∈ Grp ) |
8 |
|
simp3 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → 𝑍 ∈ 𝐵 ) |
9 |
1 3
|
grpidcl |
⊢ ( 𝐺 ∈ Grp → 0 ∈ 𝐵 ) |
10 |
9
|
3ad2ant1 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → 0 ∈ 𝐵 ) |
11 |
|
simp2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 ) |
12 |
1 2
|
grprcan |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑍 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑍 + 𝑋 ) = ( 0 + 𝑋 ) ↔ 𝑍 = 0 ) ) |
13 |
7 8 10 11 12
|
syl13anc |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝑍 + 𝑋 ) = ( 0 + 𝑋 ) ↔ 𝑍 = 0 ) ) |
14 |
6 13
|
bitr3d |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝑍 + 𝑋 ) = 𝑋 ↔ 𝑍 = 0 ) ) |