This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: A group operation is associative. (Contributed by NM, 14-Aug-2011)
|
|
Ref |
Expression |
|
Hypotheses |
grpcl.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
|
|
grpcl.p |
⊢ + = ( +g ‘ 𝐺 ) |
|
Assertion |
grpass |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
grpcl.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
| 2 |
|
grpcl.p |
⊢ + = ( +g ‘ 𝐺 ) |
| 3 |
|
grpmnd |
⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd ) |
| 4 |
1 2
|
mndass |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) ) |
| 5 |
3 4
|
sylan |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) ) |