Metamath Proof Explorer
Description: A group operation is associative. (Contributed by SN, 29-Jan-2025)
|
|
Ref |
Expression |
|
Hypotheses |
grpassd.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
|
|
grpassd.p |
⊢ + = ( +g ‘ 𝐺 ) |
|
|
grpassd.g |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
|
|
grpassd.1 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
|
|
grpassd.2 |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
|
|
grpassd.3 |
⊢ ( 𝜑 → 𝑍 ∈ 𝐵 ) |
|
Assertion |
grpassd |
⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
grpassd.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
grpassd.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
grpassd.g |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
4 |
|
grpassd.1 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
5 |
|
grpassd.2 |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
6 |
|
grpassd.3 |
⊢ ( 𝜑 → 𝑍 ∈ 𝐵 ) |
7 |
1 2
|
grpass |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) ) |
8 |
3 4 5 6 7
|
syl13anc |
⊢ ( 𝜑 → ( ( 𝑋 + 𝑌 ) + 𝑍 ) = ( 𝑋 + ( 𝑌 + 𝑍 ) ) ) |