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| Mirrors > Home > MPE Home > Th. List > ringgrp | Structured version Visualization version GIF version | ||
| Description: A ring is a group. (Contributed by NM, 15-Sep-2011.) |
| Ref | Expression |
|---|---|
| ringgrp | ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 2 | eqid 2737 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
| 3 | eqid 2737 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
| 4 | eqid 2737 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 5 | 1, 2, 3, 4 | isring 20234 | . 2 ⊢ (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ (mulGrp‘𝑅) ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)(𝑦(+g‘𝑅)𝑧)) = ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)(𝑥(.r‘𝑅)𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧))))) |
| 6 | 5 | simp1bi 1146 | 1 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) |
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