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Theorem unitnegcl 13252
Description: The negative of a unit is a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
unitnegcl.1 𝑈 = (Unit‘𝑅)
unitnegcl.2 𝑁 = (invg𝑅)
Assertion
Ref Expression
unitnegcl ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋) ∈ 𝑈)

Proof of Theorem unitnegcl
StepHypRef Expression
1 simpl 109 . . 3 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → 𝑅 ∈ Ring)
2 ringgrp 13137 . . . . . 6 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
3 eqidd 2178 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (Base‘𝑅) = (Base‘𝑅))
4 unitnegcl.1 . . . . . . . 8 𝑈 = (Unit‘𝑅)
54a1i 9 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → 𝑈 = (Unit‘𝑅))
6 ringsrg 13177 . . . . . . . 8 (𝑅 ∈ Ring → 𝑅 ∈ SRing)
76adantr 276 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → 𝑅 ∈ SRing)
8 simpr 110 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → 𝑋𝑈)
93, 5, 7, 8unitcld 13230 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → 𝑋 ∈ (Base‘𝑅))
10 eqid 2177 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
11 unitnegcl.2 . . . . . . 7 𝑁 = (invg𝑅)
1210, 11grpinvcl 12875 . . . . . 6 ((𝑅 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑁𝑋) ∈ (Base‘𝑅))
132, 9, 12syl2an2r 595 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋) ∈ (Base‘𝑅))
14 eqid 2177 . . . . . 6 (∥r𝑅) = (∥r𝑅)
1510, 14, 11dvdsrneg 13225 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑁𝑋) ∈ (Base‘𝑅)) → (𝑁𝑋)(∥r𝑅)(𝑁‘(𝑁𝑋)))
1613, 15syldan 282 . . . 4 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋)(∥r𝑅)(𝑁‘(𝑁𝑋)))
1710, 11grpinvinv 12891 . . . . 5 ((𝑅 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑁‘(𝑁𝑋)) = 𝑋)
182, 9, 17syl2an2r 595 . . . 4 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁‘(𝑁𝑋)) = 𝑋)
1916, 18breqtrd 4029 . . 3 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋)(∥r𝑅)𝑋)
20 eqidd 2178 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (1r𝑅) = (1r𝑅))
21 eqidd 2178 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (∥r𝑅) = (∥r𝑅))
22 eqidd 2178 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (oppr𝑅) = (oppr𝑅))
23 eqidd 2178 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (∥r‘(oppr𝑅)) = (∥r‘(oppr𝑅)))
245, 20, 21, 22, 23, 7isunitd 13228 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑋𝑈 ↔ (𝑋(∥r𝑅)(1r𝑅) ∧ 𝑋(∥r‘(oppr𝑅))(1r𝑅))))
258, 24mpbid 147 . . . 4 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑋(∥r𝑅)(1r𝑅) ∧ 𝑋(∥r‘(oppr𝑅))(1r𝑅)))
2625simpld 112 . . 3 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → 𝑋(∥r𝑅)(1r𝑅))
2710, 14dvdsrtr 13223 . . 3 ((𝑅 ∈ Ring ∧ (𝑁𝑋)(∥r𝑅)𝑋𝑋(∥r𝑅)(1r𝑅)) → (𝑁𝑋)(∥r𝑅)(1r𝑅))
281, 19, 26, 27syl3anc 1238 . 2 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋)(∥r𝑅)(1r𝑅))
29 eqid 2177 . . . . 5 (oppr𝑅) = (oppr𝑅)
3029opprring 13202 . . . 4 (𝑅 ∈ Ring → (oppr𝑅) ∈ Ring)
3130adantr 276 . . 3 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (oppr𝑅) ∈ Ring)
3229, 10opprbasg 13200 . . . . . . . . 9 (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(oppr𝑅)))
3332eleq2d 2247 . . . . . . . 8 (𝑅 ∈ Ring → ((𝑁𝑋) ∈ (Base‘𝑅) ↔ (𝑁𝑋) ∈ (Base‘(oppr𝑅))))
3433adantr 276 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → ((𝑁𝑋) ∈ (Base‘𝑅) ↔ (𝑁𝑋) ∈ (Base‘(oppr𝑅))))
3513, 34mpbid 147 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋) ∈ (Base‘(oppr𝑅)))
36 eqid 2177 . . . . . . 7 (Base‘(oppr𝑅)) = (Base‘(oppr𝑅))
37 eqid 2177 . . . . . . 7 (∥r‘(oppr𝑅)) = (∥r‘(oppr𝑅))
38 eqid 2177 . . . . . . 7 (invg‘(oppr𝑅)) = (invg‘(oppr𝑅))
3936, 37, 38dvdsrneg 13225 . . . . . 6 (((oppr𝑅) ∈ Ring ∧ (𝑁𝑋) ∈ (Base‘(oppr𝑅))) → (𝑁𝑋)(∥r‘(oppr𝑅))((invg‘(oppr𝑅))‘(𝑁𝑋)))
4030, 35, 39syl2an2r 595 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋)(∥r‘(oppr𝑅))((invg‘(oppr𝑅))‘(𝑁𝑋)))
4129, 11opprnegg 13206 . . . . . . . 8 (𝑅 ∈ Ring → 𝑁 = (invg‘(oppr𝑅)))
4241fveq1d 5517 . . . . . . 7 (𝑅 ∈ Ring → (𝑁‘(𝑁𝑋)) = ((invg‘(oppr𝑅))‘(𝑁𝑋)))
4342breq2d 4015 . . . . . 6 (𝑅 ∈ Ring → ((𝑁𝑋)(∥r‘(oppr𝑅))(𝑁‘(𝑁𝑋)) ↔ (𝑁𝑋)(∥r‘(oppr𝑅))((invg‘(oppr𝑅))‘(𝑁𝑋))))
4443adantr 276 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → ((𝑁𝑋)(∥r‘(oppr𝑅))(𝑁‘(𝑁𝑋)) ↔ (𝑁𝑋)(∥r‘(oppr𝑅))((invg‘(oppr𝑅))‘(𝑁𝑋))))
4540, 44mpbird 167 . . . 4 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋)(∥r‘(oppr𝑅))(𝑁‘(𝑁𝑋)))
4645, 18breqtrd 4029 . . 3 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋)(∥r‘(oppr𝑅))𝑋)
4725simprd 114 . . 3 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → 𝑋(∥r‘(oppr𝑅))(1r𝑅))
4836, 37dvdsrtr 13223 . . 3 (((oppr𝑅) ∈ Ring ∧ (𝑁𝑋)(∥r‘(oppr𝑅))𝑋𝑋(∥r‘(oppr𝑅))(1r𝑅)) → (𝑁𝑋)(∥r‘(oppr𝑅))(1r𝑅))
4931, 46, 47, 48syl3anc 1238 . 2 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋)(∥r‘(oppr𝑅))(1r𝑅))
505, 20, 21, 22, 23, 7isunitd 13228 . 2 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → ((𝑁𝑋) ∈ 𝑈 ↔ ((𝑁𝑋)(∥r𝑅)(1r𝑅) ∧ (𝑁𝑋)(∥r‘(oppr𝑅))(1r𝑅))))
5128, 49, 50mpbir2and 944 1 ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋) ∈ 𝑈)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148   class class class wbr 4003  cfv 5216  Basecbs 12456  Grpcgrp 12831  invgcminusg 12832  1rcur 13095  SRingcsrg 13099  Ringcrg 13132  opprcoppr 13192  rcdsr 13208  Unitcui 13209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-lttrn 7924  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-tpos 6245  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-plusg 12543  df-mulr 12544  df-0g 12697  df-mgm 12729  df-sgrp 12762  df-mnd 12772  df-grp 12834  df-minusg 12835  df-cmn 13043  df-abl 13044  df-mgp 13084  df-ur 13096  df-srg 13100  df-ring 13134  df-oppr 13193  df-dvdsr 13211  df-unit 13212
This theorem is referenced by:  aprsym  13295
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