Theorem ringdi22 33412

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 20294	. . . 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 18991	. . 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 20315	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)))
12	1, 2, 3, 4, 6, 7, 9	ringdird 20316	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
13	1, 2, 3, 4, 6, 7, 10	ringdird 20316	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑇) = ((𝑋 · 𝑇) + (𝑌 · 𝑇)))
14	12, 13	oveq12d 7416	. 2 ⊢ (𝜑 → (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))
15	11, 14	eqtrd 2799	1 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  +_gcplusg 17288  ._rcmulr 17289  Ringcrg 20285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-12 2214  ax-ext 2736  ax-nul 5258

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-iota 6479  df-fv 6531  df-ov 7401  df-mgm 18676  df-sgrp 18755  df-mnd 18771  df-grp 18980  df-ring 20287

This theorem is referenced by:  ssdifidlprm  33647