Theorem isdrng 20361

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 6892	. . . 4 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
2		isdrng.u	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
3	1, 2	eqtr4di 2791	. . 3 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
4		fveq2 6892	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5		isdrng.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
6	4, 5	eqtr4di 2791	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7		fveq2 6892	. . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
8		isdrng.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
9	7, 8	eqtr4di 2791	. . . . 5 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
10	9	sneqd 4641	. . . 4 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = { 0 })
11	6, 10	difeq12d 4124	. . 3 ⊢ (𝑟 = 𝑅 → ((Base‘𝑟) ∖ {(0g‘𝑟)}) = (𝐵 ∖ { 0 }))
12	3, 11	eqeq12d 2749	. 2 ⊢ (𝑟 = 𝑅 → ((Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)}) ↔ 𝑈 = (𝐵 ∖ { 0 })))
13		df-drng 20359	. 2 ⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})}
14	12, 13	elrab2 3687	1 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ∖ cdif 3946  {csn 4629  ‘cfv 6544  Basecbs 17144  0_gc0g 17385  Ringcrg 20056  Unitcui 20169  DivRingcdr 20357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-drng 20359

This theorem is referenced by:  drngunit  20362  drngui  20363  drngring  20364  isdrng2  20371  drngprop  20372  drngid  20375  opprdrng  20389  drngpropd  20394  issubdrg  20401  imadrhmcl  20413  cntzsdrg  20418  drngdomn  20921  fidomndrng  20926  zringndrg  21038  istdrg2  23682  cvsunit  24647  cphreccllem  24695  isdrng4  32395  sradrng  32673  zrhunitpreima  32958