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| Description: Distributive law for the multiplication operation of a semiring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Thierry Arnoux, 1-Apr-2018.) | 
| Ref | Expression | 
|---|---|
| srgdi.b | ⊢ 𝐵 = (Base‘𝑅) | 
| srgdi.p | ⊢ + = (+g‘𝑅) | 
| srgdi.t | ⊢ · = (.r‘𝑅) | 
| Ref | Expression | 
|---|---|
| srgdir | ⊢ ((𝑅 ∈ SRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | srgdi.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
| 2 | srgdi.p | . . 3 ⊢ + = (+g‘𝑅) | |
| 3 | srgdi.t | . . 3 ⊢ · = (.r‘𝑅) | |
| 4 | 1, 2, 3 | srgdilem 20190 | . 2 ⊢ ((𝑅 ∈ SRing ∧ (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 ∈ SRing ∧ (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 1086 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 .rcmulr 17299 SRingcsrg 20184 | 
| 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-srg 20185 | 
| This theorem is referenced by: srgo2times 20210 srgcom4lem 20211 srgmulgass 20215 srgrmhm 20220 | 
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