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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 2724 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | eqid 2724 | . . . 4 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
4 | eqid 2724 | . . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
5 | eqid 2724 | . . . 4 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)) | |
6 | 1, 2, 3, 4, 5 | isunit 20267 | . . 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 20259 | . 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 1533 ∈ wcel 2098 class class class wbr 5139 ‘cfv 6534 Basecbs 17145 1rcur 20078 opprcoppr 20227 ∥rcdsr 20248 Unitcui 20249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fv 6542 df-ov 7405 df-dvdsr 20251 df-unit 20252 |
This theorem is referenced by: unitss 20270 unitmulcl 20274 unitgrp 20277 ringinvcl 20286 unitnegcl 20291 ringunitnzdiv 20292 unitdvcl 20299 dvrid 20300 dvrcan1 20303 dvrcan3 20304 dvreq1 20305 irredrmul 20321 subrguss 20481 subrginv 20482 subrgunit 20484 isdrng2 20593 unitrrg 21195 gzrngunitlem 21296 gzrngunit 21297 zringunit 21323 matinv 22503 cramerimp 22512 unitnmn0 24509 nminvr 24510 nrginvrcnlem 24532 ig1peu 26031 dchrelbas3 27090 dchrmulcl 27101 isdrng4 32864 kerunit 32906 fldhmf1 41452 invginvrid 47257 lincresunit3lem3 47368 lincresunit3lem1 47373 |
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