Theorem fldcrngd 14616

Description: A field is a commutative ring. (Contributed by SN, 23-Nov-2024.)

Hypothesis

Ref	Expression
fldcrngd.1	|- ( ph -> R e. Field )

Assertion

Ref	Expression
fldcrngd	|- ( ph -> R e. CRing )

Proof of Theorem fldcrngd

Step	Hyp	Ref	Expression
1		fldcrngd.1	. 2 |- ( ph -> R e. Field )
2		isfld 14614	. . 3 |- ( R e. Field <-> ( R e. DivRing /\ R e. CRing ) )
3	2	simprbi 275	. 2 |- ( R e. Field -> R e. CRing )
4	1, 3	syl 14	1 |- ( ph -> R e. CRing )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   CRingccrg 14301   DivRingcdr 14602  Fieldcfield 14603

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-field 14605

This theorem is used by: (None)