MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isdrng Structured version   Visualization version   GIF version

Theorem isdrng 18522
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 6087 . . . 4 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
2 isdrng.u . . . 4 𝑈 = (Unit‘𝑅)
31, 2syl6eqr 2661 . . 3 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
4 fveq2 6087 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5 isdrng.b . . . . 5 𝐵 = (Base‘𝑅)
64, 5syl6eqr 2661 . . . 4 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7 fveq2 6087 . . . . . 6 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
8 isdrng.z . . . . . 6 0 = (0g𝑅)
97, 8syl6eqr 2661 . . . . 5 (𝑟 = 𝑅 → (0g𝑟) = 0 )
109sneqd 4136 . . . 4 (𝑟 = 𝑅 → {(0g𝑟)} = { 0 })
116, 10difeq12d 3690 . . 3 (𝑟 = 𝑅 → ((Base‘𝑟) ∖ {(0g𝑟)}) = (𝐵 ∖ { 0 }))
123, 11eqeq12d 2624 . 2 (𝑟 = 𝑅 → ((Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g𝑟)}) ↔ 𝑈 = (𝐵 ∖ { 0 })))
13 df-drng 18520 . 2 DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g𝑟)})}
1412, 13elrab2 3332 1 (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wcel 1976  cdif 3536  {csn 4124  cfv 5789  Basecbs 15643  0gc0g 15871  Ringcrg 18318  Unitcui 18410  DivRingcdr 18518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5753  df-fv 5797  df-drng 18520
This theorem is referenced by:  drngunit  18523  drngui  18524  drngring  18525  isdrng2  18528  drngprop  18529  drngid  18532  opprdrng  18542  drngpropd  18545  issubdrg  18576  drngdomn  19072  fidomndrng  19076  istdrg2  21738  cvsunit  22686  cphreccllem  22730  zrhunitpreima  29143  cntzsdrg  36574
  Copyright terms: Public domain W3C validator