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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 482 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑅 ∈ Ring) | |
2 | ringgrp 20256 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
3 | eqid 2735 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | unitnegcl.1 | . . . . . . 7 ⊢ 𝑈 = (Unit‘𝑅) | |
5 | 3, 4 | unitcl 20392 | . . . . . 6 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ (Base‘𝑅)) |
6 | unitnegcl.2 | . . . . . . 7 ⊢ 𝑁 = (invg‘𝑅) | |
7 | 3, 6 | grpinvcl 19018 | . . . . . 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 2735 | . . . . . 6 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
10 | 3, 9, 6 | dvdsrneg 20387 | . . . . 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 19036 | . . . . 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 5174 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘𝑅)𝑋) |
15 | simpr 484 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑈) | |
16 | eqid 2735 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
17 | eqid 2735 | . . . . . 6 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
18 | eqid 2735 | . . . . . 6 ⊢ (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)) | |
19 | 4, 16, 9, 17, 18 | isunit 20390 | . . . . 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 20385 | . . 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 1370 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘𝑅)(1r‘𝑅)) |
24 | 17 | opprring 20364 | . . . 4 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring) |
25 | 24 | adantr 480 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (oppr‘𝑅) ∈ Ring) |
26 | 17, 3 | opprbas 20358 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘(oppr‘𝑅)) |
27 | 17, 6 | opprneg 20368 | . . . . . 6 ⊢ 𝑁 = (invg‘(oppr‘𝑅)) |
28 | 26, 18, 27 | dvdsrneg 20387 | . . . . 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 5174 | . . 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 20385 | . . 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 1370 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋)(∥r‘(oppr‘𝑅))(1r‘𝑅)) |
34 | 4, 16, 9, 17, 18 | isunit 20390 | . 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 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 Basecbs 17245 Grpcgrp 18964 invgcminusg 18965 1rcur 20199 Ringcrg 20251 opprcoppr 20350 ∥rcdsr 20371 Unitcui 20372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 |
This theorem is referenced by: irredneg 20447 deg1invg 26160 nzrneg1ne0 48074 invginvrid 48212 lincresunit3lem3 48320 lincresunitlem1 48321 |
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