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Mirrors > Home > MPE Home > Th. List > drngunit | Structured version Visualization version GIF version |
Description: Elementhood in the set of units when 𝑅 is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
isdrng.b | ⊢ 𝐵 = (Base‘𝑅) |
isdrng.u | ⊢ 𝑈 = (Unit‘𝑅) |
isdrng.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
drngunit | ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isdrng.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
2 | isdrng.u | . . . . 5 ⊢ 𝑈 = (Unit‘𝑅) | |
3 | isdrng.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
4 | 1, 2, 3 | isdrng 20749 | . . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 }))) |
5 | 4 | simprbi 496 | . . 3 ⊢ (𝑅 ∈ DivRing → 𝑈 = (𝐵 ∖ { 0 })) |
6 | 5 | eleq2d 2824 | . 2 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (𝐵 ∖ { 0 }))) |
7 | eldifsn 4790 | . 2 ⊢ (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) | |
8 | 6, 7 | bitrdi 287 | 1 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∖ cdif 3959 {csn 4630 ‘cfv 6562 Basecbs 17244 0gc0g 17485 Ringcrg 20250 Unitcui 20371 DivRingcdr 20745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-iota 6515 df-fv 6570 df-drng 20747 |
This theorem is referenced by: drngunz 20763 drnginvrcl 20769 drnginvrn0 20770 drnginvrl 20772 drnginvrr 20773 issubdrg 20797 sdrgunit 20813 abvdiv 20846 qsssubdrg 21461 redvr 21652 drnguc1p 26227 lgseisenlem3 27435 sdrgdvcl 33280 sdrginvcl 33281 ornglmullt 33316 orngrmullt 33317 isarchiofld 33326 1arithufd 33555 ply1asclunit 33578 ply1dg1rt 33583 qqhval2lem 33943 qqhf 33948 matunitlindf 37604 fldhmf1 42071 lincreslvec3 48327 isldepslvec2 48330 |
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