Theorem ringdi22 33203

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 20263	. . . 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 18981	. . 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 33201	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)))
12	1, 2, 3, 4, 6, 7, 9	ringdird 33202	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
13	1, 2, 3, 4, 6, 7, 10	ringdird 33202	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑇) = ((𝑋 · 𝑇) + (𝑌 · 𝑇)))
14	12, 13	oveq12d 7461	. 2 ⊢ (𝜑 → (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))
15	11, 14	eqtrd 2780	1 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ‘cfv 6568  (class class class)co 7443  Basecbs 17252  +_gcplusg 17305  ._rcmulr 17306  Ringcrg 20254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6520  df-fv 6576  df-ov 7446  df-mgm 18672  df-sgrp 18751  df-mnd 18767  df-grp 18970  df-ring 20256

This theorem is referenced by:  ssdifidlprm  33443