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Theorem unitpropd 15807
 Description: The set of units depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
Hypotheses
Ref Expression
rngidpropd.1
rngidpropd.2
rngidpropd.3
Assertion
Ref Expression
unitpropd Unit Unit
Distinct variable groups:   ,,   ,,   ,,   ,,

Proof of Theorem unitpropd
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rngidpropd.1 . . . . . . 7
2 rngidpropd.2 . . . . . . 7
3 rngidpropd.3 . . . . . . 7
41, 2, 3rngidpropd 15805 . . . . . 6
54breq2d 4227 . . . . 5 r r
64breq2d 4227 . . . . 5 roppr roppr
75, 6anbi12d 693 . . . 4 r roppr r roppr
81, 2, 3dvdsrpropd 15806 . . . . . 6 r r
98breqd 4226 . . . . 5 r r
10 eqid 2438 . . . . . . . . 9 oppr oppr
11 eqid 2438 . . . . . . . . 9
1210, 11opprbas 15739 . . . . . . . 8 oppr
131, 12syl6eq 2486 . . . . . . 7 oppr
14 eqid 2438 . . . . . . . . 9 oppr oppr
15 eqid 2438 . . . . . . . . 9
1614, 15opprbas 15739 . . . . . . . 8 oppr
172, 16syl6eq 2486 . . . . . . 7 oppr
183ancom2s 779 . . . . . . . 8
19 eqid 2438 . . . . . . . . 9
20 eqid 2438 . . . . . . . . 9 oppr oppr
2111, 19, 10, 20opprmul 15736 . . . . . . . 8 oppr
22 eqid 2438 . . . . . . . . 9
23 eqid 2438 . . . . . . . . 9 oppr oppr
2415, 22, 14, 23opprmul 15736 . . . . . . . 8 oppr
2518, 21, 243eqtr4g 2495 . . . . . . 7 oppr oppr
2613, 17, 25dvdsrpropd 15806 . . . . . 6 roppr roppr
2726breqd 4226 . . . . 5 roppr roppr
289, 27anbi12d 693 . . . 4 r roppr r roppr
297, 28bitrd 246 . . 3 r roppr r roppr
30 eqid 2438 . . . 4 Unit Unit
31 eqid 2438 . . . 4
32 eqid 2438 . . . 4 r r
33 eqid 2438 . . . 4 roppr roppr
3430, 31, 32, 10, 33isunit 15767 . . 3 Unit r roppr
35 eqid 2438 . . . 4 Unit Unit
36 eqid 2438 . . . 4
37 eqid 2438 . . . 4 r r
38 eqid 2438 . . . 4 roppr roppr
3935, 36, 37, 14, 38isunit 15767 . . 3 Unit r roppr
4029, 34, 393bitr4g 281 . 2 Unit Unit
4140eqrdv 2436 1 Unit Unit
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726   class class class wbr 4215  cfv 5457  (class class class)co 6084  cbs 13474  cmulr 13535  cur 15667  opprcoppr 15732  rcdsr 15748  Unitcui 15749 This theorem is referenced by:  invrpropd  15808  drngprop  15851  drngpropd  15867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-plusg 13547  df-mulr 13548  df-0g 13732  df-mgp 15654  df-ur 15670  df-oppr 15733  df-dvdsr 15751  df-unit 15752
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