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Theorem isdrng 18874
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 6304 . . . 4 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
2 isdrng.u . . . 4 𝑈 = (Unit‘𝑅)
31, 2syl6eqr 2776 . . 3 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
4 fveq2 6304 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5 isdrng.b . . . . 5 𝐵 = (Base‘𝑅)
64, 5syl6eqr 2776 . . . 4 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7 fveq2 6304 . . . . . 6 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
8 isdrng.z . . . . . 6 0 = (0g𝑅)
97, 8syl6eqr 2776 . . . . 5 (𝑟 = 𝑅 → (0g𝑟) = 0 )
109sneqd 4297 . . . 4 (𝑟 = 𝑅 → {(0g𝑟)} = { 0 })
116, 10difeq12d 3837 . . 3 (𝑟 = 𝑅 → ((Base‘𝑟) ∖ {(0g𝑟)}) = (𝐵 ∖ { 0 }))
123, 11eqeq12d 2739 . 2 (𝑟 = 𝑅 → ((Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g𝑟)}) ↔ 𝑈 = (𝐵 ∖ { 0 })))
13 df-drng 18872 . 2 DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g𝑟)})}
1412, 13elrab2 3472 1 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1596  wcel 2103  cdif 3677  {csn 4285  cfv 6001  Basecbs 15980  0gc0g 16223  Ringcrg 18668  Unitcui 18760  DivRingcdr 18870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-iota 5964  df-fv 6009  df-drng 18872
This theorem is referenced by:  drngunit  18875  drngui  18876  drngring  18877  isdrng2  18880  drngprop  18881  drngid  18884  opprdrng  18894  drngpropd  18897  issubdrg  18928  drngdomn  19426  fidomndrng  19430  zringndrg  19961  istdrg2  22103  cvsunit  23052  cphreccllem  23099  zrhunitpreima  30252  cntzsdrg  38191
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