Theorem fldcrngd 40450

Description: A field is a commutative ring. EDITORIAL: Shortens recrng 20875. Also recrng 20875 should be named resrng. Also fldcrng 36210 is misnamed. (Contributed by SN, 23-Nov-2024.)

Hypothesis

Ref	Expression
fldcrngd.1	⊢ (𝜑 → 𝑅 ∈ Field)

Assertion

Ref	Expression
fldcrngd	⊢ (𝜑 → 𝑅 ∈ CRing)

Proof of Theorem fldcrngd

Step	Hyp	Ref	Expression
1		fldcrngd.1	. 2 ⊢ (𝜑 → 𝑅 ∈ Field)
2		isfld 20049	. . 3 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
3	2	simprbi 498	. 2 ⊢ (𝑅 ∈ Field → 𝑅 ∈ CRing)
4	1, 3	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ CRing)

Syntax hints:   → wi 4   ∈ wcel 2104  CRingccrg 19833  DivRingcdr 20040  Fieldcfield 20041

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3439  df-in 3899  df-field 20043

This theorem is referenced by:  prjcrv0  40665