Theorem flddivrng 38446

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 38439	. . 3 ⊢ Fld = (DivRingOps ∩ Com2)
2		inss1 4183	. . 3 ⊢ (DivRingOps ∩ Com2) ⊆ DivRingOps
3	1, 2	eqsstri 3977	. 2 ⊢ Fld ⊆ DivRingOps
4	3	sseli 3927	1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)

Syntax hints:   → wi 4   ∈ wcel 2136   ∩ cin 3898  DivRingOpscdrng 38395  Com2ccm2 38436  Fldcfld 38438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1557  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-v 3450  df-in 3906  df-ss 3916  df-fld 38439

This theorem is referenced by:  isfld2  38452  isfldidl  38515