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Theorem ringdir 20194

Description: Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)

**Hypotheses**
Ref	Expression
ringdi.b	⊢ 𝐵 = (Base‘𝑅)
ringdi.p	⊢ + = (+_g‘𝑅)
ringdi.t	⊢ · = (._r‘𝑅)

**Assertion**
Ref	Expression
ringdir	⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))

**Proof of Theorem ringdir**
Step	Hyp	Ref	Expression
1		ringdi.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
2		ringdi.p	. . 3 ⊢ + = (+_g‘𝑅)
3		ringdi.t	. . 3 ⊢ · = (._r‘𝑅)
4	1, 2, 3	ringdilem 20182	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))
5	4	simprd 495	1 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 +_gcplusg 17226 ._rcmulr 17227 Ringcrg 20166

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-12 2167 ax-ext 2699 ax-nul 5300

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-ral 3058 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-iota 6494 df-fv 6550 df-ov 7417 df-ring 20168

This theorem is referenced by: ringo2times 20204 ringcomlem 20208 ringnegl 20231 mulgass2 20238 ringrghm 20242 prdsringd 20250 imasring 20259 dvrdir 20344 issubrg2 20524 cntzsubr 20538 sralmod 21073 frlmphl 21708 psrlmod 21896 psrdir 21902 evlslem1 22021 mamudi 22296 mdetrlin 22497 ringdird 32932 rlocaddval 32976 mxidlprm 33177 q1pdir 33263 r1pcyc 33267 lflvscl 38543 lflvsdi1 38544 dvhlveclem 40575

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