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| Description: The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.) | 
| Ref | Expression | 
|---|---|
| drngui.b | ⊢ 𝐵 = (Base‘𝑅) | 
| drngui.z | ⊢ 0 = (0g‘𝑅) | 
| drngui.r | ⊢ 𝑅 ∈ DivRing | 
| Ref | Expression | 
|---|---|
| drngui | ⊢ (𝐵 ∖ { 0 }) = (Unit‘𝑅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | drngui.r | . . . 4 ⊢ 𝑅 ∈ DivRing | |
| 2 | drngui.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
| 3 | eqid 2737 | . . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
| 4 | drngui.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
| 5 | 2, 3, 4 | isdrng 20733 | . . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 }))) | 
| 6 | 1, 5 | mpbi 230 | . . 3 ⊢ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })) | 
| 7 | 6 | simpri 485 | . 2 ⊢ (Unit‘𝑅) = (𝐵 ∖ { 0 }) | 
| 8 | 7 | eqcomi 2746 | 1 ⊢ (𝐵 ∖ { 0 }) = (Unit‘𝑅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∖ 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-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: cnflddiv 21413 cnflddivOLD 21414 cnfldinv 21415 cnsubdrglem 21436 cnmgpabl 21446 cnmsubglem 21448 gzrngunit 21451 zringunit 21477 expghm 21486 psgninv 21600 zrhpsgnmhm 21602 amgmlem 27033 dchrghm 27300 dchrabs 27304 sum2dchr 27318 lgseisenlem4 27422 qrngdiv 27668 proot1ex 43208 amgmwlem 49321 amgmlemALT 49322 | 
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