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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 2771 | . . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
4 | drngui.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
5 | 2, 3, 4 | isdrng 18957 | . . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 }))) |
6 | 1, 5 | mpbi 220 | . . 3 ⊢ (𝑅 ∈ Ring ∧ (Unit‘𝑅) = (𝐵 ∖ { 0 })) |
7 | 6 | simpri 473 | . 2 ⊢ (Unit‘𝑅) = (𝐵 ∖ { 0 }) |
8 | 7 | eqcomi 2780 | 1 ⊢ (𝐵 ∖ { 0 }) = (Unit‘𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∖ cdif 3720 {csn 4316 ‘cfv 6029 Basecbs 16060 0gc0g 16304 Ringcrg 18751 Unitcui 18843 DivRingcdr 18953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-iota 5992 df-fv 6037 df-drng 18955 |
This theorem is referenced by: cnflddiv 19987 cnfldinv 19988 cnsubdrglem 20008 cnmgpabl 20018 cnmsubglem 20020 gzrngunit 20023 zringunit 20047 expghm 20055 psgninv 20139 zrhpsgnmhm 20141 amgmlem 24933 dchrghm 25198 dchrabs 25202 sum2dchr 25216 lgseisenlem4 25320 qrngdiv 25530 proot1ex 38302 amgmwlem 43076 amgmlemALT 43077 |
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