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Theorem isdrngo1 26572
 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 21994 . . . 4 GId GId
21relopabi 5000 . . 3
3 1st2nd 6393 . . 3
42, 3mpan 652 . 2
5 relrngo 21965 . . . 4
6 1st2nd 6393 . . . 4
75, 6mpan 652 . . 3
9 isdivrng1.1 . . . . 5
10 isdivrng1.2 . . . . 5
119, 10opeq12i 3989 . . . 4
1211eqeq2i 2446 . . 3
13 fvex 5742 . . . . . . 7
1410, 13eqeltri 2506 . . . . . 6
15 isdivrngo 22019 . . . . . 6 GId GId
1614, 15ax-mp 8 . . . . 5 GId GId
17 isdivrng1.4 . . . . . . . . . 10
18 isdivrng1.3 . . . . . . . . . . 11 GId
1918sneqi 3826 . . . . . . . . . 10 GId
2017, 19difeq12i 3463 . . . . . . . . 9 GId
2120, 20xpeq12i 4900 . . . . . . . 8 GId GId
2221reseq2i 5143 . . . . . . 7 GId GId
2322eleq1i 2499 . . . . . 6 GId GId
2423anbi2i 676 . . . . 5 GId GId
2516, 24bitr4i 244 . . . 4
26 eleq1 2496 . . . . 5
27 eleq1 2496 . . . . . 6
2827anbi1d 686 . . . . 5
2926, 28bibi12d 313 . . . 4
3025, 29mpbiri 225 . . 3
3112, 30sylbir 205 . 2
324, 8, 31pm5.21nii 343 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725  cvv 2956   cdif 3317  csn 3814  cop 3817   cxp 4876   crn 4879   cres 4880   wrel 4883  cfv 5454  c1st 6347  c2nd 6348  cgr 21774  GIdcgi 21775  crngo 21963  cdrng 21993 This theorem is referenced by:  divrngcl  26573  isdrngo2  26574  divrngpr  26663 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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-rngo 21964  df-drngo 21994
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