Theorem idomringd 13775

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 13774	. 2 |- ( ph -> R e. CRing )
3	2	crngringd 13505	1 |- ( ph -> R e. Ring )

Syntax hints:   -> wi 4   e. wcel 2164   Ringcrg 13492  IDomncidom 13753

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-in 3159  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-cring 13495  df-idom 13756

This theorem is referenced by:  lgseisenlem3  15188