ringdi22 - Metamath Proof Explorer

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Theorem ringdi22 20493

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 20469	. . . 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 19158	. . 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 20491	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)))
12	1, 2, 3, 4, 6, 7, 9	ringdird 20492	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
13	1, 2, 3, 4, 6, 7, 10	ringdird 20492	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑇) = ((𝑋 · 𝑇) + (𝑌 · 𝑇)))
14	12, 13	oveq12d 7438	. 2 ⊢ (𝜑 → (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))
15	11, 14	eqtrd 2795	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 1569 ∈ wcel 2144 ‘cfv 6538 (class class class)co 7420 Basecbs 17387 +_gcplusg 17428 ._rcmulr 17429 Ringcrg 20459

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-12 2212 ax-ext 2732 ax-nul 5259

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-sb 2099 df-clab 2739 df-cleq 2752 df-clel 2835 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-ss 3896 df-dif 4047 df-un 4151 df-nul 4279 df-if 4482 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-br 5103 df-iota 6494 df-fv 6546 df-ov 7423 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-grp 19147 df-ring 20461

This theorem is used by: ssdifidlprm 21642

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