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Theorem ringdi 20159
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 20150 . 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 1540  wcel 2105  cfv 6543  (class class class)co 7412  Basecbs 17151  +gcplusg 17204  .rcmulr 17205  Ringcrg 20134
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-ring 20136
This theorem is referenced by:  ringcomlem  20174  ringnegr  20198  ringlghm  20207  prdsringd  20216  imasring  20225  issubrg2  20490  cntzsubr  20504  sralmod  21039  psrlmod  21833  psrdi  21838  mamudir  22225  mdetrlin  22425  mdetuni0  22444  ply1divex  25993  lfladdcl  38408  lflvsdi2  38416  dvhlveclem  40446
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