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Mirrors > Home > MPE Home > Th. List > ringdi | Structured version Visualization version GIF version |
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 | ringi 18560 | . 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 475 | 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 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 .rcmulr 15942 Ringcrg 18547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-ring 18549 |
This theorem is referenced by: ringcom 18579 ringrz 18588 rngnegr 18595 ringsubdi 18599 ringlghm 18604 prdsringd 18612 imasring 18619 opprring 18631 issubrg2 18800 cntzsubr 18812 sralmod 19187 psrlmod 19401 psrdi 19406 mamudir 20210 mdetrlin 20408 mdetuni0 20427 ply1divex 23896 lfladdcl 34358 lflvsdi2 34366 dvhlveclem 36397 lidlrng 41927 |
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