Theorem ringdid 20403

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 20401	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
9	1, 2, 3, 4, 8	syl13anc 1399	1 ⊢ (𝜑 → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  +_gcplusg 17342  ._rcmulr 17343  Ringcrg 20372

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2732  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7416  df-ring 20374

This theorem is used by:  ringdi22  20405  rloccring  33711  vietalem  34089  zrhcntr  34489