Theorem ringdi22 33235

Description: Expand the product of two sums in a ring. (Contributed by Thierry Arnoux, 3-Jun-2025.)

Hypotheses

Ref	Expression
ringdi22.1	⊢ 𝐵 = (Base‘𝑅)
ringdi22.2	⊢ + = (+g‘𝑅)
ringdi22.3	⊢ · = (.r‘𝑅)
ringdi22.4	⊢ (𝜑 → 𝑅 ∈ Ring)
ringdi22.5	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ringdi22.6	⊢ (𝜑 → 𝑌 ∈ 𝐵)
ringdi22.7	⊢ (𝜑 → 𝑍 ∈ 𝐵)
ringdi22.8	⊢ (𝜑 → 𝑇 ∈ 𝐵)

Assertion

Ref	Expression
ringdi22	⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))

Proof of Theorem ringdi22

Step	Hyp	Ref	Expression
1		ringdi22.1	. . 3 ⊢ 𝐵 = (Base‘𝑅)
2		ringdi22.2	. . 3 ⊢ + = (+g‘𝑅)
3		ringdi22.3	. . 3 ⊢ · = (.r‘𝑅)
4		ringdi22.4	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
5	4	ringgrpd 20239	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
6		ringdi22.5	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		ringdi22.6	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8	1, 2, 5, 6, 7	grpcld 18965	. . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
9		ringdi22.7	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
10		ringdi22.8	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝐵)
11	1, 2, 3, 4, 8, 9, 10	ringdid 20260	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)))
12	1, 2, 3, 4, 6, 7, 9	ringdird 20261	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
13	1, 2, 3, 4, 6, 7, 10	ringdird 20261	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑇) = ((𝑋 · 𝑇) + (𝑌 · 𝑇)))
14	12, 13	oveq12d 7449	. 2 ⊢ (𝜑 → (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))
15	11, 14	eqtrd 2777	1 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  +_gcplusg 17297  ._rcmulr 17298  Ringcrg 20230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-ring 20232

This theorem is referenced by:  ssdifidlprm  33486