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Theorem unitpropd 17159
 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 17157 . . . . . 6
54breq2d 4459 . . . . 5 r r
64breq2d 4459 . . . . 5 roppr roppr
75, 6anbi12d 710 . . . 4 r roppr r roppr
81, 2, 3dvdsrpropd 17158 . . . . . 6 r r
98breqd 4458 . . . . 5 r r
10 eqid 2467 . . . . . . . . 9 oppr oppr
11 eqid 2467 . . . . . . . . 9
1210, 11opprbas 17091 . . . . . . . 8 oppr
131, 12syl6eq 2524 . . . . . . 7 oppr
14 eqid 2467 . . . . . . . . 9 oppr oppr
15 eqid 2467 . . . . . . . . 9
1614, 15opprbas 17091 . . . . . . . 8 oppr
172, 16syl6eq 2524 . . . . . . 7 oppr
183ancom2s 800 . . . . . . . 8
19 eqid 2467 . . . . . . . . 9
20 eqid 2467 . . . . . . . . 9 oppr oppr
2111, 19, 10, 20opprmul 17088 . . . . . . . 8 oppr
22 eqid 2467 . . . . . . . . 9
23 eqid 2467 . . . . . . . . 9 oppr oppr
2415, 22, 14, 23opprmul 17088 . . . . . . . 8 oppr
2518, 21, 243eqtr4g 2533 . . . . . . 7 oppr oppr
2613, 17, 25dvdsrpropd 17158 . . . . . 6 roppr roppr
2726breqd 4458 . . . . 5 roppr roppr
289, 27anbi12d 710 . . . 4 r roppr r roppr
297, 28bitrd 253 . . 3 r roppr r roppr
30 eqid 2467 . . . 4 Unit Unit
31 eqid 2467 . . . 4
32 eqid 2467 . . . 4 r r
33 eqid 2467 . . . 4 roppr roppr
3430, 31, 32, 10, 33isunit 17119 . . 3 Unit r roppr
35 eqid 2467 . . . 4 Unit Unit
36 eqid 2467 . . . 4
37 eqid 2467 . . . 4 r r
38 eqid 2467 . . . 4 roppr roppr
3935, 36, 37, 14, 38isunit 17119 . . 3 Unit r roppr
4029, 34, 393bitr4g 288 . 2 Unit Unit
4140eqrdv 2464 1 Unit Unit
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1379   wcel 1767   class class class wbr 4447  cfv 5588  (class class class)co 6285  cbs 14493  cmulr 14559  cur 16967  opprcoppr 17084  rcdsr 17100  Unitcui 17101 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-tpos 6956  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-plusg 14571  df-mulr 14572  df-0g 14700  df-mgp 16956  df-ur 16968  df-oppr 17085  df-dvdsr 17103  df-unit 17104 This theorem is referenced by:  invrpropd  17160  drngprop  17219  drngpropd  17235
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