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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 2793 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | eqid 2793 | . . . 4 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
4 | eqid 2793 | . . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
5 | eqid 2793 | . . . 4 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)) | |
6 | 1, 2, 3, 4, 5 | isunit 19085 | . . 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 19077 | . 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 1520 ∈ wcel 2079 class class class wbr 4956 ‘cfv 6217 Basecbs 16300 1rcur 18929 opprcoppr 19050 ∥rcdsr 19066 Unitcui 19067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-ov 7010 df-dvdsr 19069 df-unit 19070 |
This theorem is referenced by: unitss 19088 unitmulcl 19092 unitgrp 19095 ringinvcl 19104 unitnegcl 19109 unitdvcl 19115 dvrid 19116 dvrcan1 19119 dvrcan3 19120 dvreq1 19121 irredrmul 19135 isdrng2 19190 subrguss 19228 subrginv 19229 subrgunit 19231 unitrrg 19743 gzrngunitlem 20280 gzrngunit 20281 zringunit 20305 matinv 20958 cramerimp 20967 unitnmn0 22948 nminvr 22949 nrginvrcnlem 22971 ig1peu 24436 dchrelbas3 25484 dchrmulcl 25495 kerunit 30505 invginvrid 43849 lincresunit3lem3 43963 lincresunit3lem1 43968 |
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