Metamath Proof Explorer
Description: An associative cancellation law for groups. (Contributed by SN, 29-Jan-2025)
|
|
Ref |
Expression |
|
Hypotheses |
grpasscan2d.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
|
|
grpasscan2d.p |
⊢ + = ( +g ‘ 𝐺 ) |
|
|
grpasscan2d.n |
⊢ 𝑁 = ( invg ‘ 𝐺 ) |
|
|
grpasscan2d.g |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
|
|
grpasscan2d.1 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
|
|
grpasscan2d.2 |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
|
Assertion |
grpasscan2d |
⊢ ( 𝜑 → ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = 𝑋 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
grpasscan2d.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpasscan2d.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
grpasscan2d.n |
⊢ 𝑁 = ( invg ‘ 𝐺 ) |
4 |
|
grpasscan2d.g |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
5 |
|
grpasscan2d.1 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
6 |
|
grpasscan2d.2 |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
7 |
1 2 3
|
grpasscan2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = 𝑋 ) |
8 |
4 5 6 7
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑋 + ( 𝑁 ‘ 𝑌 ) ) + 𝑌 ) = 𝑋 ) |