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Theorem istdrg2 18207
 Description: A topological-ring division ring is a topological division ring iff the group of nonzero elements is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
istdrg2.m mulGrp
istdrg2.b
istdrg2.z
Assertion
Ref Expression
istdrg2 TopDRing s

Proof of Theorem istdrg2
StepHypRef Expression
1 istdrg2.m . . 3 mulGrp
2 eqid 2436 . . 3 Unit Unit
31, 2istdrg 18195 . 2 TopDRing s Unit
4 istdrg2.b . . . . . . . . 9
5 istdrg2.z . . . . . . . . 9
64, 2, 5isdrng 15839 . . . . . . . 8 Unit
76simprbi 451 . . . . . . 7 Unit
87adantl 453 . . . . . 6 Unit
98oveq2d 6097 . . . . 5 s Unit s
109eleq1d 2502 . . . 4 s Unit s
1110pm5.32i 619 . . 3 s Unit s
12 df-3an 938 . . 3 s Unit s Unit
13 df-3an 938 . . 3 s s
1411, 12, 133bitr4i 269 . 2 s Unit s
153, 14bitri 241 1 TopDRing s
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725   cdif 3317  csn 3814  cfv 5454  (class class class)co 6081  cbs 13469   ↾s cress 13470  c0g 13723  mulGrpcmgp 15648  crg 15660  Unitcui 15744  cdr 15835  ctgp 18101  ctrg 18185  TopDRingctdrg 18186 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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 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-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  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-iota 5418  df-fv 5462  df-ov 6084  df-drng 15837  df-tdrg 18190
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