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| Description: Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) | 
| Ref | Expression | 
|---|---|
| ringdi.b | ⊢ 𝐵 = (Base‘𝑅) | 
| ringdi.p | ⊢ + = (+g‘𝑅) | 
| ringdi.t | ⊢ · = (.r‘𝑅) | 
| Ref | Expression | 
|---|---|
| ringdi | ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍))) | 
| 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 20247 | . 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 | simpld 494 | 1 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 .rcmulr 17299 Ringcrg 20231 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-12 2176 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-ring 20233 | 
| This theorem is referenced by: ringdid 20261 ringcomlem 20277 ringnegr 20301 ringlghm 20310 prdsringd 20319 imasring 20328 issubrg2 20593 cntzsubr 20607 sralmod 21195 psrlmod 21981 psrdi 21986 mamudir 22409 mdetrlin 22609 mdetuni0 22628 ply1divex 26177 erler 33270 rlocaddval 33273 lfladdcl 39073 lflvsdi2 39081 dvhlveclem 41111 | 
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