Theorem isdrng 20650

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 6828	. . . 4 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
2		isdrng.u	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
3	1, 2	eqtr4di 2786	. . 3 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
4		fveq2 6828	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5		isdrng.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
6	4, 5	eqtr4di 2786	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7		fveq2 6828	. . . . . 6 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
8		isdrng.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
9	7, 8	eqtr4di 2786	. . . . 5 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
10	9	sneqd 4587	. . . 4 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = { 0 })
11	6, 10	difeq12d 4076	. . 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 20648	. 2 ⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})}
14	12, 13	elrab2 3646	1 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∖ cdif 3895  {csn 4575  ‘cfv 6486  Basecbs 17122  0_gc0g 17345  Ringcrg 20153  Unitcui 20275  DivRingcdr 20646

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 2705

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 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6442  df-fv 6494  df-drng 20648

This theorem is referenced by:  drngunit  20651  drngui  20652  drngring  20653  isdrng2  20660  drngprop  20661  drngid  20663  drngdomn  20666  opprdrng  20681  drngpropd  20686  fidomndrng  20690  issubdrg  20697  imadrhmcl  20714  cntzsdrg  20719  zringndrg  21407  istdrg2  24094  cvsunit  25059  cphreccllem  25106  isdrng4  33268  sradrng  33615  assafld  33671  zrhunitpreima  34010  aks5lem7  42313