Theorem flddivrng 37986

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 37979	. . 3 ⊢ Fld = (DivRingOps ∩ Com2)
2		inss1 4245	. . 3 ⊢ (DivRingOps ∩ Com2) ⊆ DivRingOps
3	1, 2	eqsstri 4030	. 2 ⊢ Fld ⊆ DivRingOps
4	3	sseli 3991	1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)

Syntax hints:   → wi 4   ∈ wcel 2106   ∩ cin 3962  DivRingOpscdrng 37935  Com2ccm2 37976  Fldcfld 37978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-in 3970  df-ss 3980  df-fld 37979

This theorem is referenced by:  isfld2  37992  isfldidl  38055