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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 2736 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 3 | eqid 2736 | . . . 4 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
| 4 | eqid 2736 | . . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
| 5 | eqid 2736 | . . . 4 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)) | |
| 6 | 1, 2, 3, 4, 5 | isunit 20374 | . . 3 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋(∥r‘𝑅)(1r‘𝑅) ∧ 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅))) | 
| 7 | 6 | simplbi 497 | . 2 ⊢ (𝑋 ∈ 𝑈 → 𝑋(∥r‘𝑅)(1r‘𝑅)) | 
| 8 | unitcl.1 | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
| 9 | 8, 3 | dvdsrcl 20366 | . 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 1539 ∈ wcel 2107 class class class wbr 5142 ‘cfv 6560 Basecbs 17248 1rcur 20179 opprcoppr 20334 ∥rcdsr 20355 Unitcui 20356 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-dvdsr 20358 df-unit 20359 | 
| This theorem is referenced by: unitss 20377 unitmulcl 20381 unitgrp 20384 ringinvcl 20393 unitnegcl 20398 ringunitnzdiv 20399 unitdvcl 20406 dvrid 20407 dvrcan1 20410 dvrcan3 20411 dvreq1 20412 irredrmul 20428 subrguss 20588 subrginv 20589 subrgunit 20591 unitrrg 20704 isdrng2 20744 gzrngunitlem 21451 gzrngunit 21452 zringunit 21478 matinv 22684 cramerimp 22693 unitnmn0 24690 nminvr 24691 nrginvrcnlem 24713 ig1peu 26215 dchrelbas3 27283 dchrmulcl 27294 isdrng4 33299 kerunit 33350 dvdsruasso2 33415 unitmulrprm 33557 1arithidomlem1 33564 1arithidomlem2 33565 1arithidom 33566 ply1unit 33601 m1pmeq 33609 fldhmf1 42092 invginvrid 48288 lincresunit3lem3 48396 lincresunit3lem1 48401 | 
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