Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 19437 | . . . 4 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 }))) |
5 | 4 | simprbi 497 | . . 3 ⊢ (𝑅 ∈ DivRing → 𝑈 = (𝐵 ∖ { 0 })) |
6 | 5 | eleq2d 2898 | . 2 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∈ (𝐵 ∖ { 0 }))) |
7 | eldifsn 4713 | . 2 ⊢ (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) | |
8 | 6, 7 | syl6bb 288 | 1 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3016 ∖ cdif 3932 {csn 4559 ‘cfv 6349 Basecbs 16473 0gc0g 16703 Ringcrg 19228 Unitcui 19320 DivRingcdr 19433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-iota 6308 df-fv 6357 df-drng 19435 |
This theorem is referenced by: drngunz 19448 drnginvrcl 19450 drnginvrn0 19451 drnginvrl 19452 drnginvrr 19453 issubdrg 19491 abvdiv 19539 qsssubdrg 20534 redvr 20691 drnguc1p 24693 lgseisenlem3 25881 ornglmullt 30808 orngrmullt 30809 isarchiofld 30818 qqhval2lem 31122 qqhf 31127 matunitlindf 34772 lincreslvec3 44435 isldepslvec2 44438 |
Copyright terms: Public domain | W3C validator |