|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 482 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ Ring) | |
| 2 | ringgrp 20235 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
| 3 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 4 | unitnegcl.1 | . . . . . . 7 ⊢ 𝑈 = (Unit‘𝑅) | |
| 5 | 3, 4 | unitcl 20375 | . . . . . 6 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ (Base‘𝑅)) | 
| 6 | unitnegcl.2 | . . . . . . 7 ⊢ 𝑁 = (invg‘𝑅) | |
| 7 | 3, 6 | grpinvcl 19005 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑁‘𝑋) ∈ (Base‘𝑅)) | 
| 8 | 2, 5, 7 | syl2an 596 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) ∈ (Base‘𝑅)) | 
| 9 | eqid 2737 | . . . . . 6 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
| 10 | 3, 9, 6 | dvdsrneg 20370 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘𝑋) ∈ (Base‘𝑅)) → (𝑁‘𝑋)(∥r‘𝑅)(𝑁‘(𝑁‘𝑋))) | 
| 11 | 8, 10 | syldan 591 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘𝑅)(𝑁‘(𝑁‘𝑋))) | 
| 12 | 3, 6 | grpinvinv 19023 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑁‘(𝑁‘𝑋)) = 𝑋) | 
| 13 | 2, 5, 12 | syl2an 596 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘(𝑁‘𝑋)) = 𝑋) | 
| 14 | 11, 13 | breqtrd 5169 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘𝑅)𝑋) | 
| 15 | simpr 484 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑈) | |
| 16 | eqid 2737 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 17 | eqid 2737 | . . . . . 6 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
| 18 | eqid 2737 | . . . . . 6 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)) | |
| 19 | 4, 16, 9, 17, 18 | isunit 20373 | . . . . 5 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋(∥r‘𝑅)(1r‘𝑅) ∧ 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅))) | 
| 20 | 15, 19 | sylib 218 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋(∥r‘𝑅)(1r‘𝑅) ∧ 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅))) | 
| 21 | 20 | simpld 494 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑋(∥r‘𝑅)(1r‘𝑅)) | 
| 22 | 3, 9 | dvdsrtr 20368 | . . 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 1373 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘𝑅)(1r‘𝑅)) | 
| 24 | 17 | opprring 20347 | . . . 4 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring) | 
| 25 | 24 | adantr 480 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (oppr‘𝑅) ∈ Ring) | 
| 26 | 17, 3 | opprbas 20341 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅)) | 
| 27 | 17, 6 | opprneg 20351 | . . . . . 6 ⊢ 𝑁 = (invg‘(oppr‘𝑅)) | 
| 28 | 26, 18, 27 | dvdsrneg 20370 | . . . . 5 ⊢ (((oppr‘𝑅) ∈ Ring ∧ (𝑁‘𝑋) ∈ (Base‘𝑅)) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(𝑁‘(𝑁‘𝑋))) | 
| 29 | 25, 8, 28 | syl2anc 584 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(𝑁‘(𝑁‘𝑋))) | 
| 30 | 29, 13 | breqtrd 5169 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))𝑋) | 
| 31 | 20 | simprd 495 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅)) | 
| 32 | 26, 18 | dvdsrtr 20368 | . . 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 1373 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(1r‘𝑅)) | 
| 34 | 4, 16, 9, 17, 18 | isunit 20373 | . 2 ⊢ ((𝑁‘𝑋) ∈ 𝑈 ↔ ((𝑁‘𝑋)(∥r‘𝑅)(1r‘𝑅) ∧ (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(1r‘𝑅))) | 
| 35 | 23, 33, 34 | sylanbrc 583 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) ∈ 𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 Grpcgrp 18951 invgcminusg 18952 1rcur 20178 Ringcrg 20230 opprcoppr 20333 ∥rcdsr 20354 Unitcui 20355 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 | 
| This theorem is referenced by: irredneg 20430 deg1invg 26145 nzrneg1ne0 48146 invginvrid 48283 lincresunit3lem3 48391 lincresunitlem1 48392 | 
| Copyright terms: Public domain | W3C validator |