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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 20733 | . . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 }))) |
| 5 | 4 | simprbi 496 | . . 3 ⊢ (𝑅 ∈ DivRing → 𝑈 = (𝐵 ∖ { 0 })) |
| 6 | 5 | eleq2d 2827 | . 2 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (𝐵 ∖ { 0 }))) |
| 7 | eldifsn 4786 | . 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 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 {csn 4626 ‘cfv 6561 Basecbs 17247 0gc0g 17484 Ringcrg 20230 Unitcui 20355 DivRingcdr 20729 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-drng 20731 |
| This theorem is referenced by: drngunz 20747 drnginvrcl 20753 drnginvrn0 20754 drnginvrl 20756 drnginvrr 20757 issubdrg 20781 sdrgunit 20797 abvdiv 20830 qsssubdrg 21444 redvr 21635 drnguc1p 26213 lgseisenlem3 27421 sdrgdvcl 33301 sdrginvcl 33302 ornglmullt 33337 orngrmullt 33338 isarchiofld 33347 1arithufd 33576 ply1asclunit 33599 ply1dg1rt 33604 qqhval2lem 33982 qqhf 33987 matunitlindf 37625 fldhmf1 42091 lincreslvec3 48399 isldepslvec2 48402 |
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