Theorem isdrng 19771

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 6717	. . . 4 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
2		isdrng.u	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
3	1, 2	eqtr4di 2796	. . 3 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
4		fveq2 6717	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5		isdrng.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
6	4, 5	eqtr4di 2796	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7		fveq2 6717	. . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
8		isdrng.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
9	7, 8	eqtr4di 2796	. . . . 5 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
10	9	sneqd 4553	. . . 4 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = { 0 })
11	6, 10	difeq12d 4038	. . 3 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) ∖ {(0g‘𝑟)}) = (𝐵 ∖ { 0 }))
12	3, 11	eqeq12d 2753	. 2 ⊢ (𝑟 = 𝑅 → ((Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)}) ↔ 𝑈 = (𝐵 ∖ { 0 })))
13		df-drng 19769	. 2 ⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})}
14	12, 13	elrab2 3605	1 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ∖ cdif 3863  {csn 4541  ‘cfv 6380  Basecbs 16760  0_gc0g 16944  Ringcrg 19562  Unitcui 19657  DivRingcdr 19767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-drng 19769

This theorem is referenced by:  drngunit  19772  drngui  19773  drngring  19774  isdrng2  19777  drngprop  19778  drngid  19781  opprdrng  19791  drngpropd  19794  issubdrg  19825  cntzsdrg  19846  drngdomn  20341  fidomndrng  20345  zringndrg  20455  istdrg2  23075  cvsunit  24028  cphreccllem  24075  sradrng  31387  zrhunitpreima  31640