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Mirrors > Home > MPE Home > Th. List > unitnegcl | Structured version Visualization version GIF version |
Description: The negative of a unit is a unit. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
unitnegcl.1 | ⊢ 𝑈 = (Unit‘𝑅) |
unitnegcl.2 | ⊢ 𝑁 = (invg‘𝑅) |
Ref | Expression |
---|---|
unitnegcl | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 472 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ Ring) | |
2 | ringgrp 18375 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
3 | eqid 2610 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | unitnegcl.1 | . . . . . . 7 ⊢ 𝑈 = (Unit‘𝑅) | |
5 | 3, 4 | unitcl 18482 | . . . . . 6 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ (Base‘𝑅)) |
6 | unitnegcl.2 | . . . . . . 7 ⊢ 𝑁 = (invg‘𝑅) | |
7 | 3, 6 | grpinvcl 17290 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑁‘𝑋) ∈ (Base‘𝑅)) |
8 | 2, 5, 7 | syl2an 493 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) ∈ (Base‘𝑅)) |
9 | eqid 2610 | . . . . . 6 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
10 | 3, 9, 6 | dvdsrneg 18477 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘𝑋) ∈ (Base‘𝑅)) → (𝑁‘𝑋)(∥r‘𝑅)(𝑁‘(𝑁‘𝑋))) |
11 | 8, 10 | syldan 486 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘𝑅)(𝑁‘(𝑁‘𝑋))) |
12 | 3, 6 | grpinvinv 17305 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
13 | 2, 5, 12 | syl2an 493 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
14 | 11, 13 | breqtrd 4609 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘𝑅)𝑋) |
15 | simpr 476 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑈) | |
16 | eqid 2610 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
17 | eqid 2610 | . . . . . 6 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
18 | eqid 2610 | . . . . . 6 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)) | |
19 | 4, 16, 9, 17, 18 | isunit 18480 | . . . . 5 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋(∥r‘𝑅)(1r‘𝑅) ∧ 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅))) |
20 | 15, 19 | sylib 207 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋(∥r‘𝑅)(1r‘𝑅) ∧ 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅))) |
21 | 20 | simpld 474 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑋(∥r‘𝑅)(1r‘𝑅)) |
22 | 3, 9 | dvdsrtr 18475 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘𝑋)(∥r‘𝑅)𝑋 ∧ 𝑋(∥r‘𝑅)(1r‘𝑅)) → (𝑁‘𝑋)(∥r‘𝑅)(1r‘𝑅)) |
23 | 1, 14, 21, 22 | syl3anc 1318 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘𝑅)(1r‘𝑅)) |
24 | 17 | opprring 18454 | . . . 4 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring) |
25 | 24 | adantr 480 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (oppr‘𝑅) ∈ Ring) |
26 | 17, 3 | opprbas 18452 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅)) |
27 | 17, 6 | opprneg 18458 | . . . . . 6 ⊢ 𝑁 = (invg‘(oppr‘𝑅)) |
28 | 26, 18, 27 | dvdsrneg 18477 | . . . . 5 ⊢ (((oppr‘𝑅) ∈ Ring ∧ (𝑁‘𝑋) ∈ (Base‘𝑅)) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(𝑁‘(𝑁‘𝑋))) |
29 | 25, 8, 28 | syl2anc 691 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(𝑁‘(𝑁‘𝑋))) |
30 | 29, 13 | breqtrd 4609 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))𝑋) |
31 | 20 | simprd 478 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅)) |
32 | 26, 18 | dvdsrtr 18475 | . . 3 ⊢ (((oppr‘𝑅) ∈ Ring ∧ (𝑁‘𝑋)(∥r‘(oppr‘𝑅))𝑋 ∧ 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅)) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(1r‘𝑅)) |
33 | 25, 30, 31, 32 | syl3anc 1318 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(1r‘𝑅)) |
34 | 4, 16, 9, 17, 18 | isunit 18480 | . 2 ⊢ ((𝑁‘𝑋) ∈ 𝑈 ↔ ((𝑁‘𝑋)(∥r‘𝑅)(1r‘𝑅) ∧ (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(1r‘𝑅))) |
35 | 23, 33, 34 | sylanbrc 695 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 Grpcgrp 17245 invgcminusg 17246 1rcur 18324 Ringcrg 18370 opprcoppr 18445 ∥rcdsr 18461 Unitcui 18462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-mulr 15782 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 |
This theorem is referenced by: irredneg 18533 deg1invg 23670 nzrneg1ne0 41659 invginvrid 41942 lincresunit3lem3 42057 lincresunitlem1 42058 |
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