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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 485 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ Ring) | |
2 | ringgrp 19305 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
3 | eqid 2824 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | unitnegcl.1 | . . . . . . 7 ⊢ 𝑈 = (Unit‘𝑅) | |
5 | 3, 4 | unitcl 19412 | . . . . . 6 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ (Base‘𝑅)) |
6 | unitnegcl.2 | . . . . . . 7 ⊢ 𝑁 = (invg‘𝑅) | |
7 | 3, 6 | grpinvcl 18154 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑁‘𝑋) ∈ (Base‘𝑅)) |
8 | 2, 5, 7 | syl2an 597 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) ∈ (Base‘𝑅)) |
9 | eqid 2824 | . . . . . 6 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
10 | 3, 9, 6 | dvdsrneg 19407 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘𝑋) ∈ (Base‘𝑅)) → (𝑁‘𝑋)(∥r‘𝑅)(𝑁‘(𝑁‘𝑋))) |
11 | 8, 10 | syldan 593 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘𝑅)(𝑁‘(𝑁‘𝑋))) |
12 | 3, 6 | grpinvinv 18169 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
13 | 2, 5, 12 | syl2an 597 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘(𝑁‘𝑋)) = 𝑋) |
14 | 11, 13 | breqtrd 5095 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘𝑅)𝑋) |
15 | simpr 487 | . . . . 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 19410 | . . . . 5 ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋(∥r‘𝑅)(1r‘𝑅) ∧ 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅))) |
20 | 15, 19 | sylib 220 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋(∥r‘𝑅)(1r‘𝑅) ∧ 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅))) |
21 | 20 | simpld 497 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑋(∥r‘𝑅)(1r‘𝑅)) |
22 | 3, 9 | dvdsrtr 19405 | . . 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 1367 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘𝑅)(1r‘𝑅)) |
24 | 17 | opprring 19384 | . . . 4 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring) |
25 | 24 | adantr 483 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (oppr‘𝑅) ∈ Ring) |
26 | 17, 3 | opprbas 19382 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅)) |
27 | 17, 6 | opprneg 19388 | . . . . . 6 ⊢ 𝑁 = (invg‘(oppr‘𝑅)) |
28 | 26, 18, 27 | dvdsrneg 19407 | . . . . 5 ⊢ (((oppr‘𝑅) ∈ Ring ∧ (𝑁‘𝑋) ∈ (Base‘𝑅)) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(𝑁‘(𝑁‘𝑋))) |
29 | 25, 8, 28 | syl2anc 586 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(𝑁‘(𝑁‘𝑋))) |
30 | 29, 13 | breqtrd 5095 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))𝑋) |
31 | 20 | simprd 498 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑋(∥r‘(oppr‘𝑅))(1r‘𝑅)) |
32 | 26, 18 | dvdsrtr 19405 | . . 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 1367 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(1r‘𝑅)) |
34 | 4, 16, 9, 17, 18 | isunit 19410 | . 2 ⊢ ((𝑁‘𝑋) ∈ 𝑈 ↔ ((𝑁‘𝑋)(∥r‘𝑅)(1r‘𝑅) ∧ (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(1r‘𝑅))) |
35 | 23, 33, 34 | sylanbrc 585 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 Basecbs 16486 Grpcgrp 18106 invgcminusg 18107 1rcur 19254 Ringcrg 19300 opprcoppr 19375 ∥rcdsr 19391 Unitcui 19392 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-plusg 16581 df-mulr 16582 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-dvdsr 19394 df-unit 19395 |
This theorem is referenced by: irredneg 19463 deg1invg 24703 nzrneg1ne0 44147 invginvrid 44422 lincresunit3lem3 44536 lincresunitlem1 44537 |
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