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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 476 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ Ring) | |
2 | ringgrp 18905 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
3 | eqid 2824 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | unitnegcl.1 | . . . . . . 7 ⊢ 𝑈 = (Unit‘𝑅) | |
5 | 3, 4 | unitcl 19012 | . . . . . 6 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ (Base‘𝑅)) |
6 | unitnegcl.2 | . . . . . . 7 ⊢ 𝑁 = (invg‘𝑅) | |
7 | 3, 6 | grpinvcl 17820 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑁‘𝑋) ∈ (Base‘𝑅)) |
8 | 2, 5, 7 | syl2an 591 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) ∈ (Base‘𝑅)) |
9 | eqid 2824 | . . . . . 6 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
10 | 3, 9, 6 | dvdsrneg 19007 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘𝑋) ∈ (Base‘𝑅)) → (𝑁‘𝑋)(∥r‘𝑅)(𝑁‘(𝑁‘𝑋))) |
11 | 8, 10 | syldan 587 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘𝑅)(𝑁‘(𝑁‘𝑋))) |
12 | 3, 6 | grpinvinv 17835 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
13 | 2, 5, 12 | syl2an 591 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
14 | 11, 13 | breqtrd 4898 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘𝑅)𝑋) |
15 | simpr 479 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑈) | |
16 | eqid 2824 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
17 | eqid 2824 | . . . . . 6 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
18 | eqid 2824 | . . . . . 6 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)) | |
19 | 4, 16, 9, 17, 18 | isunit 19010 | . . . . 5 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋(∥r‘𝑅)(1r‘𝑅) ∧ 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅))) |
20 | 15, 19 | sylib 210 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋(∥r‘𝑅)(1r‘𝑅) ∧ 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅))) |
21 | 20 | simpld 490 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑋(∥r‘𝑅)(1r‘𝑅)) |
22 | 3, 9 | dvdsrtr 19005 | . . 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 1496 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘𝑅)(1r‘𝑅)) |
24 | 17 | opprring 18984 | . . . 4 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring) |
25 | 24 | adantr 474 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (oppr‘𝑅) ∈ Ring) |
26 | 17, 3 | opprbas 18982 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅)) |
27 | 17, 6 | opprneg 18988 | . . . . . 6 ⊢ 𝑁 = (invg‘(oppr‘𝑅)) |
28 | 26, 18, 27 | dvdsrneg 19007 | . . . . 5 ⊢ (((oppr‘𝑅) ∈ Ring ∧ (𝑁‘𝑋) ∈ (Base‘𝑅)) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(𝑁‘(𝑁‘𝑋))) |
29 | 25, 8, 28 | syl2anc 581 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(𝑁‘(𝑁‘𝑋))) |
30 | 29, 13 | breqtrd 4898 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))𝑋) |
31 | 20 | simprd 491 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅)) |
32 | 26, 18 | dvdsrtr 19005 | . . 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 1496 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(1r‘𝑅)) |
34 | 4, 16, 9, 17, 18 | isunit 19010 | . 2 ⊢ ((𝑁‘𝑋) ∈ 𝑈 ↔ ((𝑁‘𝑋)(∥r‘𝑅)(1r‘𝑅) ∧ (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(1r‘𝑅))) |
35 | 23, 33, 34 | sylanbrc 580 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 class class class wbr 4872 ‘cfv 6122 Basecbs 16221 Grpcgrp 17775 invgcminusg 17776 1rcur 18854 Ringcrg 18900 opprcoppr 18975 ∥rcdsr 18991 Unitcui 18992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-tpos 7616 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-3 11414 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-plusg 16317 df-mulr 16318 df-0g 16454 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-grp 17778 df-minusg 17779 df-mgp 18843 df-ur 18855 df-ring 18902 df-oppr 18976 df-dvdsr 18994 df-unit 18995 |
This theorem is referenced by: irredneg 19063 deg1invg 24264 nzrneg1ne0 42715 invginvrid 42994 lincresunit3lem3 43109 lincresunitlem1 43110 |
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