Theorem ringdi22 33291

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 20223	. . . 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 18923	. . 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 20244	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)))
12	1, 2, 3, 4, 6, 7, 9	ringdird 20245	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
13	1, 2, 3, 4, 6, 7, 10	ringdird 20245	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑇) = ((𝑋 · 𝑇) + (𝑌 · 𝑇)))
14	12, 13	oveq12d 7385	. 2 ⊢ (𝜑 → (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))
15	11, 14	eqtrd 2771	1 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  ._rcmulr 17221  Ringcrg 20214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912  df-ring 20216

This theorem is referenced by:  ssdifidlprm  33518