Database
BASIC ALGEBRAIC STRUCTURES
Rings
Ring unit
Semirings
srgdi
Metamath Proof Explorer
Description: Distributive law for the multiplication operation of a semiring.
(Contributed by Steve Rodriguez , 9-Sep-2007) (Revised by Thierry
Arnoux , 1-Apr-2018)
Ref
Expression
Hypotheses
srgdi.b
⊢ 𝐵 = ( Base ‘ 𝑅 )
srgdi.p
⊢ + = ( +g ‘ 𝑅 )
srgdi.t
⊢ · = ( .r ‘ 𝑅 )
Assertion
srgdi
⊢ ( ( 𝑅 ∈ SRing ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 · ( 𝑌 + 𝑍 ) ) = ( ( 𝑋 · 𝑌 ) + ( 𝑋 · 𝑍 ) ) )
Proof
Step
Hyp
Ref
Expression
1
srgdi.b
⊢ 𝐵 = ( Base ‘ 𝑅 )
2
srgdi.p
⊢ + = ( +g ‘ 𝑅 )
3
srgdi.t
⊢ · = ( .r ‘ 𝑅 )
4
1 2 3
srgi
⊢ ( ( 𝑅 ∈ SRing ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 · ( 𝑌 + 𝑍 ) ) = ( ( 𝑋 · 𝑌 ) + ( 𝑋 · 𝑍 ) ) ∧ ( ( 𝑋 + 𝑌 ) · 𝑍 ) = ( ( 𝑋 · 𝑍 ) + ( 𝑌 · 𝑍 ) ) ) )
5
4
simpld
⊢ ( ( 𝑅 ∈ SRing ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 · ( 𝑌 + 𝑍 ) ) = ( ( 𝑋 · 𝑌 ) + ( 𝑋 · 𝑍 ) ) )