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Theorem istdrg 18196
 Description: Express the predicate " is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
istrg.1 mulGrp
istdrg.1 Unit
Assertion
Ref Expression
istdrg TopDRing s

Proof of Theorem istdrg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3531 . . 3
21anbi1i 678 . 2 s s
3 fveq2 5729 . . . . . 6 mulGrp mulGrp
4 istrg.1 . . . . . 6 mulGrp
53, 4syl6eqr 2487 . . . . 5 mulGrp
6 fveq2 5729 . . . . . 6 Unit Unit
7 istdrg.1 . . . . . 6 Unit
86, 7syl6eqr 2487 . . . . 5 Unit
95, 8oveq12d 6100 . . . 4 mulGrps Unit s
109eleq1d 2503 . . 3 mulGrps Unit s
11 df-tdrg 18191 . . 3 TopDRing mulGrps Unit
1210, 11elrab2 3095 . 2 TopDRing s
13 df-3an 939 . 2 s s
142, 12, 133bitr4i 270 1 TopDRing s
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   w3a 937   wceq 1653   wcel 1726   cin 3320  cfv 5455  (class class class)co 6082   ↾s cress 13471  mulGrpcmgp 15649  Unitcui 15745  cdr 15836  ctgp 18102  ctrg 18186  TopDRingctdrg 18187 This theorem is referenced by:  tdrgunit  18197  tdrgtrg  18203  tdrgdrng  18204  istdrg2  18208  nrgtdrg  18729 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-iota 5419  df-fv 5463  df-ov 6085  df-tdrg 18191
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