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Theorem ringdid 33219
Description: Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Thierry Arnoux, 4-May-2025.)
Hypotheses
Ref Expression
ringdid.b 𝐵 = (Base‘𝑅)
ringdid.p + = (+g𝑅)
ringdid.m · = (.r𝑅)
ringdid.r (𝜑𝑅 ∈ Ring)
ringdid.x (𝜑𝑋𝐵)
ringdid.y (𝜑𝑌𝐵)
ringdid.z (𝜑𝑍𝐵)
Assertion
Ref Expression
ringdid (𝜑 → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))

Proof of Theorem ringdid
StepHypRef Expression
1 ringdid.r . 2 (𝜑𝑅 ∈ Ring)
2 ringdid.x . 2 (𝜑𝑋𝐵)
3 ringdid.y . 2 (𝜑𝑌𝐵)
4 ringdid.z . 2 (𝜑𝑍𝐵)
5 ringdid.b . . 3 𝐵 = (Base‘𝑅)
6 ringdid.p . . 3 + = (+g𝑅)
7 ringdid.m . . 3 · = (.r𝑅)
85, 6, 7ringdi 20278 . 2 ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
91, 2, 3, 4, 8syl13anc 1371 1 (𝜑 → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = 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:  ringdi22  33221  rloccring  33257  zrhcntr  33942
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