Theorem flddivrng 35279

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 35272	. . 3 ⊢ Fld = (DivRingOps ∩ Com2)
2		inss1 4207	. . 3 ⊢ (DivRingOps ∩ Com2) ⊆ DivRingOps
3	1, 2	eqsstri 4003	. 2 ⊢ Fld ⊆ DivRingOps
4	3	sseli 3965	1 ⊢ (𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)

Syntax hints:   → wi 4   ∈ wcel 2114   ∩ cin 3937  DivRingOpscdrng 35228  Com2ccm2 35269  Fldcfld 35271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-in 3945  df-ss 3954  df-fld 35272

This theorem is referenced by:  isfld2  35285  isfldidl  35348