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Theorem srgdir 20099
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 + = (+g𝑅)
srgdi.t · = (.r𝑅)
Assertion
Ref Expression
srgdir ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))

Proof of Theorem srgdir
StepHypRef Expression
1 srgdi.b . . 3 𝐵 = (Base‘𝑅)
2 srgdi.p . . 3 + = (+g𝑅)
3 srgdi.t . . 3 · = (.r𝑅)
41, 2, 3srgdilem 20093 . 2 ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))
54simprd 495 1 ((𝑅 ∈ SRing ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wcel 2105  cfv 6543  (class class class)co 7412  Basecbs 17151  +gcplusg 17204  .rcmulr 17205  SRingcsrg 20087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-srg 20088
This theorem is referenced by:  srgo2times  20113  srgcom4lem  20114  srgmulgass  20118  srgrmhm  20123
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