Theorem flddivrng 38510

Description: A field is a division ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
flddivrng	⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)

Proof of Theorem flddivrng

Step	Hyp	Ref	Expression
1		df-fld 38503	. . 3 ⊢ Fld = (DivRingOps ∩ Com2)
2		inss1 4191	. . 3 ⊢ (DivRingOps ∩ Com2) ⊆ DivRingOps
3	1, 2	eqsstri 3985	. 2 ⊢ Fld ⊆ DivRingOps
4	3	sseli 3935	1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)

Syntax hints:   → wi 4   ∈ wcel 2145   ∩ cin 3906  DivRingOpscdrng 38459  Com2ccm2 38500  Fldcfld 38502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-in 3914  df-ss 3924  df-fld 38503

This theorem is referenced by:  isfld2  38516  isfldidl  38579