ringdi22 - Metamath Proof Explorer

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Theorem ringdi22 20520

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

**Hypotheses**
Ref	Expression
ringdid.b	⊢ 𝐵 = (Base‘𝑅)
ringdid.p	⊢ + = (+_g‘𝑅)
ringdid.m	⊢ · = (._r‘𝑅)
ringdid.r	⊢ (𝜑 → 𝑅 ∈ Ring)
ringdid.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ringdid.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
ringdid.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
ringdi22.t	⊢ (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		ringdid.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
2		ringdid.p	. . 3 ⊢ + = (+_g‘𝑅)
3		ringdid.m	. . 3 ⊢ · = (._r‘𝑅)
4		ringdid.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
5	4	ringgrpd 20496	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
6		ringdid.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		ringdid.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8	1, 2, 5, 6, 7	grpcld 19184	. . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
9		ringdid.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
10		ringdi22.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝐵)
11	1, 2, 3, 4, 8, 9, 10	ringdid 20518	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)))
12	1, 2, 3, 4, 6, 7, 9	ringdird 20519	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
13	1, 2, 3, 4, 6, 7, 10	ringdird 20519	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑇) = ((𝑋 · 𝑇) + (𝑌 · 𝑇)))
14	12, 13	oveq12d 7458	. 2 ⊢ (𝜑 → (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))
15	11, 14	eqtrd 2798	1 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 ‘cfv 6555 (class class class)co 7440 Basecbs 17413 +_gcplusg 17454 ._rcmulr 17455 Ringcrg 20486

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-12 2216 ax-ext 2735 ax-nul 5263

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2102 df-clab 2742 df-cleq 2755 df-clel 2839 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-ss 3901 df-dif 4050 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-iota 6509 df-fv 6563 df-ov 7443 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-grp 19173 df-ring 20488

This theorem is used by: ssdifidlprm 21669

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