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Theorem mulgass3 15732
 Description: An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
mulgass3.b
mulgass3.m .g
mulgass3.t
Assertion
Ref Expression
mulgass3

Proof of Theorem mulgass3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . . . . 6 oppr oppr
21opprrng 15726 . . . . 5 oppr
32adantr 452 . . . 4 oppr
4 simpr1 963 . . . 4
5 simpr3 965 . . . 4
6 simpr2 964 . . . 4
7 mulgass3.b . . . . . 6
81, 7opprbas 15724 . . . . 5 oppr
9 eqid 2435 . . . . 5 .goppr .goppr
10 eqid 2435 . . . . 5 oppr oppr
118, 9, 10mulgass2 15700 . . . 4 oppr .gopproppr .gopproppr
123, 4, 5, 6, 11syl13anc 1186 . . 3 .gopproppr .gopproppr
13 mulgass3.t . . . 4
147, 13, 1, 10opprmul 15721 . . 3 .gopproppr .goppr
157, 13, 1, 10opprmul 15721 . . . 4 oppr
1615oveq2i 6084 . . 3 .gopproppr .goppr
1712, 14, 163eqtr3g 2490 . 2 .goppr .goppr
18 mulgass3.m . . . . 5 .g
197a1i 11 . . . . 5
208a1i 11 . . . . 5 oppr
21 ssv 3360 . . . . . 6
2221a1i 11 . . . . 5
23 ovex 6098 . . . . . 6
2423a1i 11 . . . . 5
25 eqid 2435 . . . . . . . 8
261, 25oppradd 15725 . . . . . . 7 oppr
2726oveqi 6086 . . . . . 6 oppr
2827a1i 11 . . . . 5 oppr
2918, 9, 19, 20, 22, 24, 28mulgpropd 14913 . . . 4 .goppr
3029oveqd 6090 . . 3 .goppr
3130oveq2d 6089 . 2 .goppr
3229oveqd 6090 . 2 .goppr
3317, 31, 323eqtr4d 2477 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  cvv 2948   wss 3312  cfv 5446  (class class class)co 6073  cz 10272  cbs 13459   cplusg 13519  cmulr 13520  .gcmg 14679  crg 15650  opprcoppr 15717 This theorem is referenced by:  zlmassa  16795 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-seq 11314  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-plusg 13532  df-mulr 13533  df-0g 13717  df-mnd 14680  df-grp 14802  df-minusg 14803  df-mulg 14805  df-mgp 15639  df-rng 15653  df-ur 15655  df-oppr 15718
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