ringdi22 - Metamath Proof Explorer

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Theorem ringdi22 20404

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 20380	. . . 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 19070	. . 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 20402	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)))
12	1, 2, 3, 4, 6, 7, 9	ringdird 20403	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
13	1, 2, 3, 4, 6, 7, 10	ringdird 20403	. . 3 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝑇) = ((𝑋 · 𝑇) + (𝑌 · 𝑇)))
14	12, 13	oveq12d 7434	. 2 ⊢ (𝜑 → (((𝑋 + 𝑌) · 𝑍) + ((𝑋 + 𝑌) · 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))
15	11, 14	eqtrd 2797	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 2145 ‘cfv 6537 (class class class)co 7416 Basecbs 17303 +_gcplusg 17344 ._rcmulr 17345 Ringcrg 20371

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-12 2215 ax-ext 2734 ax-nul 5267

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 2100 df-clab 2741 df-cleq 2754 df-clel 2837 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-sbc 3743 df-dif 3905 df-un 3907 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-iota 6493 df-fv 6545 df-ov 7419 df-mgm 18732 df-sgrp 18821 df-mnd 18837 df-grp 19059 df-ring 20373

This theorem is used by: ssdifidlprm 21548

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