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Theorem issdrg 27473
 Description: Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.)
Assertion
Ref Expression
issdrg SubDRing SubRing s

Proof of Theorem issdrg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sdrg 27472 . . . . 5 SubDRing SubRing s
21dmmptss 5358 . . . 4 SubDRing
3 elfvdm 5749 . . . 4 SubDRing SubDRing
42, 3sseldi 3338 . . 3 SubDRing
5 fveq2 5720 . . . . . . 7 SubRing SubRing
6 oveq1 6080 . . . . . . . 8 s s
76eleq1d 2501 . . . . . . 7 s s
85, 7rabeqbidv 2943 . . . . . 6 SubRing s SubRing s
9 fvex 5734 . . . . . . 7 SubRing
109rabex 4346 . . . . . 6 SubRing s
118, 1, 10fvmpt 5798 . . . . 5 SubDRing SubRing s
1211eleq2d 2502 . . . 4 SubDRing SubRing s
13 oveq2 6081 . . . . . 6 s s
1413eleq1d 2501 . . . . 5 s s
1514elrab 3084 . . . 4 SubRing s SubRing s
1612, 15syl6bb 253 . . 3 SubDRing SubRing s
174, 16biadan2 624 . 2 SubDRing SubRing s
18 3anass 940 . 2 SubRing s SubRing s
1917, 18bitr4i 244 1 SubDRing SubRing s
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  crab 2701   cdm 4870  cfv 5446  (class class class)co 6073   ↾s cress 13462  cdr 15827  SubRingcsubrg 15856  SubDRingcsdrg 27471 This theorem is referenced by:  issdrg2  27474 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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-sdrg 27472
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