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Theorem isdrngo1 32153
 Description: The predicate "is a division ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
Hypotheses
Ref Expression
isdivrng1.1
isdivrng1.2
isdivrng1.3 GId
isdivrng1.4
Assertion
Ref Expression
isdrngo1

Proof of Theorem isdrngo1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-drngo 26126 . . . 4 GId GId
21relopabi 4976 . . 3
3 1st2nd 6851 . . 3
42, 3mpan 675 . 2
5 relrngo 26097 . . . 4
6 1st2nd 6851 . . . 4
75, 6mpan 675 . . 3
9 isdivrng1.1 . . . . 5
10 isdivrng1.2 . . . . 5
119, 10opeq12i 4190 . . . 4
1211eqeq2i 2441 . . 3
13 fvex 5889 . . . . . . 7
1410, 13eqeltri 2507 . . . . . 6
15 isdivrngo 26151 . . . . . 6 GId GId
1614, 15ax-mp 5 . . . . 5 GId GId
17 isdivrng1.4 . . . . . . . . . 10
18 isdivrng1.3 . . . . . . . . . . 11 GId
1918sneqi 4008 . . . . . . . . . 10 GId
2017, 19difeq12i 3582 . . . . . . . . 9 GId
2120, 20xpeq12i 4873 . . . . . . . 8 GId GId
2221reseq2i 5119 . . . . . . 7 GId GId
2322eleq1i 2500 . . . . . 6 GId GId
2423anbi2i 699 . . . . 5 GId GId
2516, 24bitr4i 256 . . . 4
26 eleq1 2495 . . . . 5
27 eleq1 2495 . . . . . 6
2827anbi1d 710 . . . . 5
2926, 28bibi12d 323 . . . 4
3025, 29mpbiri 237 . . 3
3112, 30sylbir 217 . 2
324, 8, 31pm5.21nii 355 1
 Colors of variables: wff setvar class Syntax hints:   wb 188   wa 371   wceq 1438   wcel 1869  cvv 3082   cdif 3434  csn 3997  cop 4003   cxp 4849   crn 4852   cres 4853   wrel 4856  cfv 5599  c1st 6803  c2nd 6804  cgr 25906  GIdcgi 25907  crngo 26095  cdrng 26125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-fv 5607  df-ov 6306  df-1st 6805  df-2nd 6806  df-rngo 26096  df-drngo 26126 This theorem is referenced by:  divrngcl  32154  isdrngo2  32155  divrngpr  32244
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