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Mirrors > Home > MPE Home > Th. List > unitcl | Structured version Visualization version GIF version |
Description: A unit is an element of the base set. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
unitcl.1 | ⊢ 𝐵 = (Base‘𝑅) |
unitcl.2 | ⊢ 𝑈 = (Unit‘𝑅) |
Ref | Expression |
---|---|
unitcl | ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unitcl.2 | . . . 4 ⊢ 𝑈 = (Unit‘𝑅) | |
2 | eqid 2818 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | eqid 2818 | . . . 4 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
4 | eqid 2818 | . . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
5 | eqid 2818 | . . . 4 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)) | |
6 | 1, 2, 3, 4, 5 | isunit 19336 | . . 3 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋(∥r‘𝑅)(1r‘𝑅) ∧ 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅))) |
7 | 6 | simplbi 498 | . 2 ⊢ (𝑋 ∈ 𝑈 → 𝑋(∥r‘𝑅)(1r‘𝑅)) |
8 | unitcl.1 | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
9 | 8, 3 | dvdsrcl 19328 | . 2 ⊢ (𝑋(∥r‘𝑅)(1r‘𝑅) → 𝑋 ∈ 𝐵) |
10 | 7, 9 | syl 17 | 1 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 Basecbs 16471 1rcur 19180 opprcoppr 19301 ∥rcdsr 19317 Unitcui 19318 |
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 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
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-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-dvdsr 19320 df-unit 19321 |
This theorem is referenced by: unitss 19339 unitmulcl 19343 unitgrp 19346 ringinvcl 19355 unitnegcl 19360 unitdvcl 19366 dvrid 19367 dvrcan1 19370 dvrcan3 19371 dvreq1 19372 irredrmul 19386 isdrng2 19441 subrguss 19479 subrginv 19480 subrgunit 19482 unitrrg 19994 gzrngunitlem 20538 gzrngunit 20539 zringunit 20563 matinv 21214 cramerimp 21223 unitnmn0 23204 nminvr 23205 nrginvrcnlem 23227 ig1peu 24692 dchrelbas3 25741 dchrmulcl 25752 kerunit 30823 invginvrid 44343 lincresunit3lem3 44457 lincresunit3lem1 44462 |
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