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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringdid | Structured version Visualization version GIF version |
Description: Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Thierry Arnoux, 4-May-2025.) |
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) |
Ref | Expression |
---|---|
ringdid | ⊢ (𝜑 → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringdid.r | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | ringdid.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | ringdid.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | ringdid.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
5 | ringdid.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
6 | ringdid.p | . . 3 ⊢ + = (+g‘𝑅) | |
7 | ringdid.m | . . 3 ⊢ · = (.r‘𝑅) | |
8 | 5, 6, 7 | ringdi 20278 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍))) |
9 | 1, 2, 3, 4, 8 | syl13anc 1371 | 1 ⊢ (𝜑 → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 Ringcrg 20251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-ring 20253 |
This theorem is referenced by: ringdi22 33221 rloccring 33257 zrhcntr 33942 |
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