Theorem ringdi22 33198

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 20160	. . . 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 18860	. . 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 20181	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)))
12	1, 2, 3, 4, 6, 7, 9	ringdird 20182	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
13	1, 2, 3, 4, 6, 7, 10	ringdird 20182	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑇) = ((𝑋 · 𝑇) + (𝑌 · 𝑇)))
14	12, 13	oveq12d 7364	. 2 ⊢ (𝜑 → (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))
15	11, 14	eqtrd 2766	1 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  +_gcplusg 17161  ._rcmulr 17162  Ringcrg 20151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-mgm 18548  df-sgrp 18627  df-mnd 18643  df-grp 18849  df-ring 20153

This theorem is referenced by:  ssdifidlprm  33423