Theorem srgdi 20162

Description: Distributive law for the multiplication operation of a semiring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Thierry Arnoux, 1-Apr-2018.)

Hypotheses

Ref	Expression
srgdi.b	⊢ 𝐵 = (Base‘𝑅)
srgdi.p	⊢ + = (+g‘𝑅)
srgdi.t	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
srgdi	⊢ ((𝑅 ∈ SRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))

Proof of Theorem srgdi

Step	Hyp	Ref	Expression
1		srgdi.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
2		srgdi.p	. . 3 ⊢ + = (+g‘𝑅)
3		srgdi.t	. . 3 ⊢ · = (.r‘𝑅)
4	1, 2, 3	srgdilem 20157	. 2 ⊢ ((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 494	1 ⊢ ((𝑅 ∈ SRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  +_gcplusg 17276  ._rcmulr 17277  SRingcsrg 20151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2708  ax-nul 5281

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-ov 7413  df-srg 20152

This theorem is referenced by:  srgcom4lem  20178  srglmhm  20186  srgbinomlem  20195