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Theorem isdrng 20750
Description: The predicate "is a division ring". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
isdrng.b 𝐵 = (Base‘𝑅)
isdrng.u 𝑈 = (Unit‘𝑅)
isdrng.z 0 = (0g𝑅)
Assertion
Ref Expression
isdrng (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))

Proof of Theorem isdrng
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . . 4 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
2 isdrng.u . . . 4 𝑈 = (Unit‘𝑅)
31, 2eqtr4di 2793 . . 3 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
4 fveq2 6907 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5 isdrng.b . . . . 5 𝐵 = (Base‘𝑅)
64, 5eqtr4di 2793 . . . 4 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7 fveq2 6907 . . . . . 6 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
8 isdrng.z . . . . . 6 0 = (0g𝑅)
97, 8eqtr4di 2793 . . . . 5 (𝑟 = 𝑅 → (0g𝑟) = 0 )
109sneqd 4643 . . . 4 (𝑟 = 𝑅 → {(0g𝑟)} = { 0 })
116, 10difeq12d 4137 . . 3 (𝑟 = 𝑅 → ((Base‘𝑟) ∖ {(0g𝑟)}) = (𝐵 ∖ { 0 }))
123, 11eqeq12d 2751 . 2 (𝑟 = 𝑅 → ((Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g𝑟)}) ↔ 𝑈 = (𝐵 ∖ { 0 })))
13 df-drng 20748 . 2 DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g𝑟)})}
1412, 13elrab2 3698 1 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  cdif 3960  {csn 4631  cfv 6563  Basecbs 17245  0gc0g 17486  Ringcrg 20251  Unitcui 20372  DivRingcdr 20746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-drng 20748
This theorem is referenced by:  drngunit  20751  drngui  20752  drngring  20753  isdrng2  20760  drngprop  20761  drngid  20763  drngdomn  20766  opprdrng  20781  drngpropd  20786  fidomndrng  20791  issubdrg  20798  imadrhmcl  20815  cntzsdrg  20820  zringndrg  21497  istdrg2  24202  cvsunit  25178  cphreccllem  25226  isdrng4  33279  sradrng  33613  assafld  33665  zrhunitpreima  33939  aks5lem7  42182
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