Theorem fldcrngd 14476

Description: A field is a commutative ring. (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 14474	. . 3 ⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
3	2	simprbi 275	. 2 ⊢ (𝑅 ∈ Field → 𝑅 ∈ CRing)
4	1, 3	syl 14	1 ⊢ (𝜑 → 𝑅 ∈ CRing)

Syntax hints:   → wi 4   ∈ wcel 2205  CRingccrg 14162  DivRingcdr 14462  Fieldcfield 14463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3219  df-field 14465

This theorem is referenced by: (None)