Theorem isdrng 20818

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 = (0g‘𝑅)

Assertion

Ref	Expression
isdrng	⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))

Proof of Theorem isdrng

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6883	. . . 4 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
2		isdrng.u	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
3	1, 2	eqtr4di 2816	. . 3 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
4		fveq2 6883	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5		isdrng.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
6	4, 5	eqtr4di 2816	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7		fveq2 6883	. . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
8		isdrng.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
9	7, 8	eqtr4di 2816	. . . . 5 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
10	9	sneqd 4602	. . . 4 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = { 0 })
11	6, 10	difeq12d 4083	. . 3 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) ∖ {(0g‘𝑟)}) = (𝐵 ∖ { 0 }))
12	3, 11	eqeq12d 2779	. 2 ⊢ (𝑟 = 𝑅 → ((Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)}) ↔ 𝑈 = (𝐵 ∖ { 0 })))
13		df-drng 20816	. 2 ⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})}
14	12, 13	elrab2 3655	1 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ∖ cdif 3903  {csn 4590  ‘cfv 6538  Basecbs 17270  0_gc0g 17493  Ringcrg 20316  Unitcui 20438  DivRingcdr 20814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-drng 20816

This theorem is referenced by:  drngunit  20819  drngui  20820  drngring  20821  isdrng4  20826  isdrng2  20830  drngprop  20831  drngid  20833  drngdomn  20836  opprdrng  20849  drngpropd  20854  fidomndrng  20858  issubdrg  20864  imadrhmcl  20881  cntzsdrg  20886  zringndrg  21599  istdrg2  24316  cvsunit  25271  cphreccllem  25318  sradrng  33950  assafld  34005  zrhunitpreima  34344  aks5lem7  42945