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Mirrors > Home > MPE Home > Th. List > drngui | Structured version Visualization version GIF version |
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 2823 | . . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
4 | drngui.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
5 | 2, 3, 4 | isdrng 19508 | . . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 }))) |
6 | 1, 5 | mpbi 232 | . . 3 ⊢ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })) |
7 | 6 | simpri 488 | . 2 ⊢ (Unit‘𝑅) = (𝐵 ∖ { 0 }) |
8 | 7 | eqcomi 2832 | 1 ⊢ (𝐵 ∖ { 0 }) = (Unit‘𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 {csn 4569 ‘cfv 6357 Basecbs 16485 0gc0g 16715 Ringcrg 19299 Unitcui 19391 DivRingcdr 19504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-drng 19506 |
This theorem is referenced by: cnflddiv 20577 cnfldinv 20578 cnsubdrglem 20598 cnmgpabl 20608 cnmsubglem 20610 gzrngunit 20613 zringunit 20637 expghm 20645 psgninv 20728 zrhpsgnmhm 20730 amgmlem 25569 dchrghm 25834 dchrabs 25838 sum2dchr 25852 lgseisenlem4 25956 qrngdiv 26202 proot1ex 39808 amgmwlem 44910 amgmlemALT 44911 |
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