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Theorem ringdi 20278
Description: Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
Hypotheses
Ref Expression
ringdi.b 𝐵 = (Base‘𝑅)
ringdi.p + = (+g𝑅)
ringdi.t · = (.r𝑅)
Assertion
Ref Expression
ringdi ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))

Proof of Theorem ringdi
StepHypRef Expression
1 ringdi.b . . 3 𝐵 = (Base‘𝑅)
2 ringdi.p . . 3 + = (+g𝑅)
3 ringdi.t . . 3 · = (.r𝑅)
41, 2, 3ringdilem 20267 . 2 ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))
54simpld 494 1 ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  Ringcrg 20251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-ring 20253
This theorem is referenced by:  ringcomlem  20293  ringnegr  20317  ringlghm  20326  prdsringd  20335  imasring  20344  issubrg2  20609  cntzsubr  20623  sralmod  21212  psrlmod  21998  psrdi  22003  mamudir  22424  mdetrlin  22624  mdetuni0  22643  ply1divex  26191  ringdid  33219  erler  33252  rlocaddval  33255  lfladdcl  39053  lflvsdi2  39061  dvhlveclem  41091
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