This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Distributive law for the multiplication operation of a ring
(left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007)
|
|
Ref |
Expression |
|
Hypotheses |
ringdi.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
|
|
ringdi.p |
⊢ + = ( +g ‘ 𝑅 ) |
|
|
ringdi.t |
⊢ · = ( .r ‘ 𝑅 ) |
|
Assertion |
ringdi |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 · ( 𝑌 + 𝑍 ) ) = ( ( 𝑋 · 𝑌 ) + ( 𝑋 · 𝑍 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ringdi.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
| 2 |
|
ringdi.p |
⊢ + = ( +g ‘ 𝑅 ) |
| 3 |
|
ringdi.t |
⊢ · = ( .r ‘ 𝑅 ) |
| 4 |
1 2 3
|
ringdilem |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 · ( 𝑌 + 𝑍 ) ) = ( ( 𝑋 · 𝑌 ) + ( 𝑋 · 𝑍 ) ) ∧ ( ( 𝑋 + 𝑌 ) · 𝑍 ) = ( ( 𝑋 · 𝑍 ) + ( 𝑌 · 𝑍 ) ) ) ) |
| 5 |
4
|
simpld |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 · ( 𝑌 + 𝑍 ) ) = ( ( 𝑋 · 𝑌 ) + ( 𝑋 · 𝑍 ) ) ) |