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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 2651 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | eqid 2651 | . . . 4 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
4 | eqid 2651 | . . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
5 | eqid 2651 | . . . 4 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)) | |
6 | 1, 2, 3, 4, 5 | isunit 18703 | . . 3 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋(∥r‘𝑅)(1r‘𝑅) ∧ 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅))) |
7 | 6 | simplbi 475 | . 2 ⊢ (𝑋 ∈ 𝑈 → 𝑋(∥r‘𝑅)(1r‘𝑅)) |
8 | unitcl.1 | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
9 | 8, 3 | dvdsrcl 18695 | . 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 1523 ∈ wcel 2030 class class class wbr 4685 ‘cfv 5926 Basecbs 15904 1rcur 18547 opprcoppr 18668 ∥rcdsr 18684 Unitcui 18685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-dvdsr 18687 df-unit 18688 |
This theorem is referenced by: unitss 18706 unitmulcl 18710 unitgrp 18713 ringinvcl 18722 unitnegcl 18727 unitdvcl 18733 dvrid 18734 dvrcan1 18737 dvrcan3 18738 dvreq1 18739 irredrmul 18753 isdrng2 18805 subrguss 18843 subrginv 18844 subrgunit 18846 unitrrg 19341 gzrngunitlem 19859 gzrngunit 19860 zringunit 19884 matinv 20531 cramerimp 20540 unitnmn0 22519 nminvr 22520 nrginvrcnlem 22542 ig1peu 23976 dchrelbas3 25008 dchrmulcl 25019 kerunit 29951 invginvrid 42473 lincresunit3lem3 42588 lincresunit3lem1 42593 |
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