Theorem idomringd 13811

Description: An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)

Hypothesis

Ref	Expression
idomringd.1	|- ( ph -> R e. IDomn )

Assertion

Ref	Expression
idomringd	|- ( ph -> R e. Ring )

Proof of Theorem idomringd

Step	Hyp	Ref	Expression
1		idomringd.1	. . 3 |- ( ph -> R e. IDomn )
2	1	idomcringd 13810	. 2 |- ( ph -> R e. CRing )
3	2	crngringd 13541	1 |- ( ph -> R e. Ring )

Syntax hints:   -> wi 4   e. wcel 2167   Ringcrg 13528  IDomncidom 13789

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-in 3163  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-cring 13531  df-idom 13792

This theorem is referenced by:  lgseisenlem3  15280