Theorem flddivrng 38006

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 37999	. . 3 ⊢ Fld = (DivRingOps ∩ Com2)
2		inss1 4237	. . 3 ⊢ (DivRingOps ∩ Com2) ⊆ DivRingOps
3	1, 2	eqsstri 4030	. 2 ⊢ Fld ⊆ DivRingOps
4	3	sseli 3979	1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)

Syntax hints:   → wi 4   ∈ wcel 2108   ∩ cin 3950  DivRingOpscdrng 37955  Com2ccm2 37996  Fldcfld 37998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-in 3958  df-ss 3968  df-fld 37999

This theorem is referenced by:  isfld2  38012  isfldidl  38075