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Theorem isdrng 20733
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 6906 . . . 4 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
2 isdrng.u . . . 4 𝑈 = (Unit‘𝑅)
31, 2eqtr4di 2795 . . 3 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
4 fveq2 6906 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5 isdrng.b . . . . 5 𝐵 = (Base‘𝑅)
64, 5eqtr4di 2795 . . . 4 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7 fveq2 6906 . . . . . 6 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
8 isdrng.z . . . . . 6 0 = (0g𝑅)
97, 8eqtr4di 2795 . . . . 5 (𝑟 = 𝑅 → (0g𝑟) = 0 )
109sneqd 4638 . . . 4 (𝑟 = 𝑅 → {(0g𝑟)} = { 0 })
116, 10difeq12d 4127 . . 3 (𝑟 = 𝑅 → ((Base‘𝑟) ∖ {(0g𝑟)}) = (𝐵 ∖ { 0 }))
123, 11eqeq12d 2753 . 2 (𝑟 = 𝑅 → ((Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g𝑟)}) ↔ 𝑈 = (𝐵 ∖ { 0 })))
13 df-drng 20731 . 2 DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g𝑟)})}
1412, 13elrab2 3695 1 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  cdif 3948  {csn 4626  cfv 6561  Basecbs 17247  0gc0g 17484  Ringcrg 20230  Unitcui 20355  DivRingcdr 20729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-drng 20731
This theorem is referenced by:  drngunit  20734  drngui  20735  drngring  20736  isdrng2  20743  drngprop  20744  drngid  20746  drngdomn  20749  opprdrng  20764  drngpropd  20769  fidomndrng  20774  issubdrg  20781  imadrhmcl  20798  cntzsdrg  20803  zringndrg  21479  istdrg2  24186  cvsunit  25164  cphreccllem  25212  isdrng4  33298  sradrng  33633  assafld  33688  zrhunitpreima  33977  aks5lem7  42201
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