Theorem ringdid 20179

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	⊢ + = (+g‘𝑅)
ringdid.m	⊢ · = (.r‘𝑅)
ringdid.r	⊢ (𝜑 → 𝑅 ∈ Ring)
ringdid.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ringdid.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
ringdid.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
ringdid	⊢ (𝜑 → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))

Proof of Theorem ringdid

Step	Hyp	Ref	Expression
1		ringdid.r	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
2		ringdid.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		ringdid.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4		ringdid.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
5		ringdid.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
6		ringdid.p	. . 3 ⊢ + = (+g‘𝑅)
7		ringdid.m	. . 3 ⊢ · = (.r‘𝑅)
8	5, 6, 7	ringdi 20177	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
9	1, 2, 3, 4, 8	syl13anc 1374	1 ⊢ (𝜑 → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  +_gcplusg 17158  ._rcmulr 17159  Ringcrg 20149

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703  ax-nul 5244

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-iota 6437  df-fv 6489  df-ov 7349  df-ring 20151

This theorem is referenced by:  ringdi22  33193  rloccring  33232  zrhcntr  33987