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Theorem isdrngrd 17936
 Description: Properties that determine a division ring. (reciprocal) is normally dependent on i.e. read it as ." This version of isdrngd 17935 requires a right reciprocal instead of left. (Contributed by NM, 10-Aug-2013.)
Hypotheses
Ref Expression
isdrngd.b
isdrngd.t
isdrngd.z
isdrngd.u
isdrngd.r
isdrngd.n
isdrngd.o
isdrngd.i
isdrngd.j
isdrngrd.k
Assertion
Ref Expression
isdrngrd
Distinct variable groups:   ,,   , ,   ,,   ,   ,,   ,,   , ,
Allowed substitution hint:   ()

Proof of Theorem isdrngrd
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 isdrngd.b . . . 4
2 eqid 2429 . . . . 5 oppr oppr
3 eqid 2429 . . . . 5
42, 3opprbas 17792 . . . 4 oppr
51, 4syl6eq 2486 . . 3 oppr
6 eqidd 2430 . . 3 oppr oppr
7 isdrngd.z . . . 4
8 eqid 2429 . . . . 5
92, 8oppr0 17796 . . . 4 oppr
107, 9syl6eq 2486 . . 3 oppr
11 isdrngd.u . . . 4
12 eqid 2429 . . . . 5
132, 12oppr1 17797 . . . 4 oppr
1411, 13syl6eq 2486 . . 3 oppr
15 isdrngd.r . . . 4
162opprring 17794 . . . 4 oppr
1715, 16syl 17 . . 3 oppr
18 eleq1 2501 . . . . . . 7
19 neeq1 2712 . . . . . . 7
2018, 19anbi12d 715 . . . . . 6
21203anbi2d 1340 . . . . 5
22 oveq1 6312 . . . . . 6 oppr oppr
2322neeq1d 2708 . . . . 5 oppr oppr
2421, 23imbi12d 321 . . . 4 oppr oppr
25 eleq1 2501 . . . . . . . 8
26 neeq1 2712 . . . . . . . 8
2725, 26anbi12d 715 . . . . . . 7
28273anbi3d 1341 . . . . . 6
29 oveq2 6313 . . . . . . 7 oppr oppr
3029neeq1d 2708 . . . . . 6 oppr oppr
3128, 30imbi12d 321 . . . . 5 oppr oppr
32 isdrngd.t . . . . . . . . . 10
33323ad2ant1 1026 . . . . . . . . 9
3433oveqd 6322 . . . . . . . 8
35 eqid 2429 . . . . . . . . 9
36 eqid 2429 . . . . . . . . 9 oppr oppr
373, 35, 2, 36opprmul 17789 . . . . . . . 8 oppr
3834, 37syl6eqr 2488 . . . . . . 7 oppr
39 isdrngd.n . . . . . . 7
4038, 39eqnetrrd 2725 . . . . . 6 oppr
41403com23 1211 . . . . 5 oppr
4231, 41chvarv 2070 . . . 4 oppr
4324, 42chvarv 2070 . . 3 oppr
44 isdrngd.o . . 3
45 isdrngd.i . . 3
46 isdrngd.j . . 3
473, 35, 2, 36opprmul 17789 . . . 4 oppr
4832adantr 466 . . . . . 6
4948oveqd 6322 . . . . 5
50 isdrngrd.k . . . . 5
5149, 50eqtr3d 2472 . . . 4
5247, 51syl5eq 2482 . . 3 oppr
535, 6, 10, 14, 17, 43, 44, 45, 46, 52isdrngd 17935 . 2 oppr
542opprdrng 17934 . 2 oppr
5553, 54sylibr 215 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   w3a 982   wceq 1437   wcel 1870   wne 2625  cfv 5601  (class class class)co 6305  cbs 15084  cmulr 15153  c0g 15297  cur 17670  crg 17715  opprcoppr 17785  cdr 17910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-dvr 17846  df-drng 17912 This theorem is referenced by:  erngdvlem4-rN  34275
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