Theorem rrextdrg 33603

Description: An extension of ℝ is a division ring. (Contributed by Thierry Arnoux, 2-May-2018.)

Assertion

Ref	Expression
rrextdrg	⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ DivRing)

Proof of Theorem rrextdrg

Step	Hyp	Ref	Expression
1		eqid 2728	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2728	. . . 4 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))
3		eqid 2728	. . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅)
4	1, 2, 3	isrrext 33601	. . 3 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))))))
5	4	simp1bi 1143	. 2 ⊢ (𝑅 ∈ ℝExt → (𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing))
6	5	simprd 495	1 ⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ DivRing)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099   × cxp 5676   ↾ cres 5680  ‘cfv 6548  0cc0 11138  Basecbs 17179  distcds 17241  DivRingcdr 20623  metUnifcmetu 21269  ℤModczlm 21425  chrcchr 21426  UnifStcuss 24157  CUnifSpccusp 24201  NrmRingcnrg 24487  NrmModcnlm 24488   ℝExt crrext 33595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5684  df-res 5690  df-iota 6500  df-fv 6556  df-rrext 33600

This theorem is referenced by:  rrhfe  33613  rrhcne  33614  rrhqima  33615  rrh0  33616  sitgclg  33962