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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 | β’ π΅ = (Baseβπ ) |
unitcl.2 | β’ π = (Unitβπ ) |
Ref | Expression |
---|---|
unitcl | β’ (π β π β π β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unitcl.2 | . . . 4 β’ π = (Unitβπ ) | |
2 | eqid 2736 | . . . 4 β’ (1rβπ ) = (1rβπ ) | |
3 | eqid 2736 | . . . 4 β’ (β₯rβπ ) = (β₯rβπ ) | |
4 | eqid 2736 | . . . 4 β’ (opprβπ ) = (opprβπ ) | |
5 | eqid 2736 | . . . 4 β’ (β₯rβ(opprβπ )) = (β₯rβ(opprβπ )) | |
6 | 1, 2, 3, 4, 5 | isunit 20086 | . . 3 β’ (π β π β (π(β₯rβπ )(1rβπ ) β§ π(β₯rβ(opprβπ ))(1rβπ ))) |
7 | 6 | simplbi 498 | . 2 β’ (π β π β π(β₯rβπ )(1rβπ )) |
8 | unitcl.1 | . . 3 β’ π΅ = (Baseβπ ) | |
9 | 8, 3 | dvdsrcl 20078 | . 2 β’ (π(β₯rβπ )(1rβπ ) β π β π΅) |
10 | 7, 9 | syl 17 | 1 β’ (π β π β π β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 class class class wbr 5105 βcfv 6496 Basecbs 17083 1rcur 19913 opprcoppr 20048 β₯rcdsr 20067 Unitcui 20068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fv 6504 df-ov 7360 df-dvdsr 20070 df-unit 20071 |
This theorem is referenced by: unitss 20089 unitmulcl 20093 unitgrp 20096 ringinvcl 20105 unitnegcl 20110 unitdvcl 20116 dvrid 20117 dvrcan1 20120 dvrcan3 20121 dvreq1 20122 irredrmul 20136 isdrng2 20198 subrguss 20237 subrginv 20238 subrgunit 20240 unitrrg 20763 gzrngunitlem 20862 gzrngunit 20863 zringunit 20887 matinv 22026 cramerimp 22035 unitnmn0 24032 nminvr 24033 nrginvrcnlem 24055 ig1peu 25536 dchrelbas3 26586 dchrmulcl 26597 kerunit 32114 fldhmf1 40547 invginvrid 46433 lincresunit3lem3 46545 lincresunit3lem1 46550 |
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