Theorem isdrng 20666

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 6834	. . . 4 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
2		isdrng.u	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
3	1, 2	eqtr4di 2789	. . 3 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
4		fveq2 6834	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5		isdrng.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
6	4, 5	eqtr4di 2789	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7		fveq2 6834	. . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
8		isdrng.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
9	7, 8	eqtr4di 2789	. . . . 5 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
10	9	sneqd 4592	. . . 4 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = { 0 })
11	6, 10	difeq12d 4079	. . 3 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) ∖ {(0g‘𝑟)}) = (𝐵 ∖ { 0 }))
12	3, 11	eqeq12d 2752	. 2 ⊢ (𝑟 = 𝑅 → ((Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)}) ↔ 𝑈 = (𝐵 ∖ { 0 })))
13		df-drng 20664	. 2 ⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})}
14	12, 13	elrab2 3649	1 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∖ cdif 3898  {csn 4580  ‘cfv 6492  Basecbs 17136  0_gc0g 17359  Ringcrg 20168  Unitcui 20291  DivRingcdr 20662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-drng 20664

This theorem is referenced by:  drngunit  20667  drngui  20668  drngring  20669  isdrng2  20676  drngprop  20677  drngid  20679  drngdomn  20682  opprdrng  20697  drngpropd  20702  fidomndrng  20706  issubdrg  20713  imadrhmcl  20730  cntzsdrg  20735  zringndrg  21423  istdrg2  24122  cvsunit  25087  cphreccllem  25134  isdrng4  33377  sradrng  33738  assafld  33794  zrhunitpreima  34133  aks5lem7  42450