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Theorem unitgrp 15465
 Description: The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
unitmulcl.1 Unit
unitgrp.2 mulGrps
Assertion
Ref Expression
unitgrp

Proof of Theorem unitgrp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unitmulcl.1 . . . 4 Unit
2 unitgrp.2 . . . 4 mulGrps
31, 2unitgrpbas 15464 . . 3
43a1i 10 . 2
5 fvex 5555 . . . 4
63, 5eqeltri 2366 . . 3
7 eqid 2296 . . . . 5 mulGrp mulGrp
8 eqid 2296 . . . . 5
97, 8mgpplusg 15345 . . . 4 mulGrp
102, 9ressplusg 13266 . . 3
116, 10mp1i 11 . 2
121, 8unitmulcl 15462 . 2
13 eqid 2296 . . . . 5
1413, 1unitcl 15457 . . . 4
1513, 1unitcl 15457 . . . 4
1613, 1unitcl 15457 . . . 4
1714, 15, 163anim123i 1137 . . 3
1813, 8rngass 15373 . . 3
1917, 18sylan2 460 . 2
20 eqid 2296 . . 3
211, 201unit 15456 . 2
2213, 8, 20rnglidm 15380 . . 3
2314, 22sylan2 460 . 2
24 simpr 447 . . . 4
25 eqid 2296 . . . . 5 r r
26 eqid 2296 . . . . 5 oppr oppr
27 eqid 2296 . . . . 5 roppr roppr
281, 20, 25, 26, 27isunit 15455 . . . 4 r roppr
2924, 28sylib 188 . . 3 r roppr
3014adantl 452 . . . . . 6
3113, 25, 8dvdsr2 15445 . . . . . 6 r
3230, 31syl 15 . . . . 5 r
3326, 13opprbas 15427 . . . . . . 7 oppr
34 eqid 2296 . . . . . . 7 oppr oppr
3533, 27, 34dvdsr2 15445 . . . . . 6 roppr oppr
3630, 35syl 15 . . . . 5 roppr oppr
3732, 36anbi12d 691 . . . 4 r roppr oppr
38 reeanv 2720 . . . . 5 oppr oppr
39 simprl 732 . . . . . . . . . . . . 13 oppr
4030ad2antrr 706 . . . . . . . . . . . . 13 oppr
4113, 25, 8dvdsrmul 15446 . . . . . . . . . . . . 13 r
4239, 40, 41syl2anc 642 . . . . . . . . . . . 12 oppr r
43 simplll 734 . . . . . . . . . . . . . . 15 oppr
44 simplr 731 . . . . . . . . . . . . . . 15 oppr
4513, 8rngass 15373 . . . . . . . . . . . . . . 15
4643, 44, 40, 39, 45syl13anc 1184 . . . . . . . . . . . . . 14 oppr
47 simprrl 740 . . . . . . . . . . . . . . 15 oppr
4847oveq1d 5889 . . . . . . . . . . . . . 14 oppr
4913, 8, 26, 34opprmul 15424 . . . . . . . . . . . . . . . 16 oppr
50 simprrr 741 . . . . . . . . . . . . . . . 16 oppr oppr
5149, 50syl5eqr 2342 . . . . . . . . . . . . . . 15 oppr
5251oveq2d 5890 . . . . . . . . . . . . . 14 oppr
5346, 48, 523eqtr3d 2336 . . . . . . . . . . . . 13 oppr
5413, 8, 20rnglidm 15380 . . . . . . . . . . . . . 14
5543, 39, 54syl2anc 642 . . . . . . . . . . . . 13 oppr
5613, 8, 20rngridm 15381 . . . . . . . . . . . . . 14
5743, 44, 56syl2anc 642 . . . . . . . . . . . . 13 oppr
5853, 55, 573eqtr3d 2336 . . . . . . . . . . . 12 oppr
5942, 58, 513brtr3d 4068 . . . . . . . . . . 11 oppr r
6033, 27, 34dvdsrmul 15446 . . . . . . . . . . . . 13 ropproppr
6144, 40, 60syl2anc 642 . . . . . . . . . . . 12 oppr ropproppr
6213, 8, 26, 34opprmul 15424 . . . . . . . . . . . . 13 oppr
6362, 47syl5eq 2340 . . . . . . . . . . . 12 oppr oppr
6461, 63breqtrd 4063 . . . . . . . . . . 11 oppr roppr
651, 20, 25, 26, 27isunit 15455 . . . . . . . . . . 11 r roppr
6659, 64, 65sylanbrc 645 . . . . . . . . . 10 oppr
6766, 47jca 518 . . . . . . . . 9 oppr
6867expr 598 . . . . . . . 8 oppr
6968rexlimdva 2680 . . . . . . 7 oppr
7069expimpd 586 . . . . . 6 oppr
7170reximdv2 2665 . . . . 5 oppr
7238, 71syl5bir 209 . . . 4 oppr
7337, 72sylbid 206 . . 3 r roppr
7429, 73mpd 14 . 2
754, 11, 12, 19, 21, 23, 74isgrpde 14522 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   w3a 934   wceq 1632   wcel 1696  wrex 2557  cvv 2801   class class class wbr 4039  cfv 5271  (class class class)co 5874  cbs 13164   ↾s cress 13165   cplusg 13224  cmulr 13225  cgrp 14378  mulGrpcmgp 15341  crg 15353  cur 15355  opprcoppr 15420  rcdsr 15436  Unitcui 15437 This theorem is referenced by:  unitabl  15466  unitsubm  15468  unitinvcl  15472  unitinvinv  15473  unitlinv  15475  unitrinv  15476  isdrng2  15538  subrgugrp  15580  expghm  16466  nrginvrcn  18218  nrgtdrg  18219  dchrfi  20510  dchrghm  20511  dchrabs  20515  dchrptlem1  20519  dchrptlem2  20520  dchrptlem3  20521  dchrsum2  20523  idomodle  27615  proot1mul  27618  proot1hash  27622  proot1ex  27623 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-grp 14505  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440
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