Theorem lidlunitel

Description: If an ideal I contains a unit J , then it is the whole ring. (Contributed by Thierry Arnoux, 19-Mar-2025)

	Ref	Expression
Hypotheses	lidlunitel.1	|- B = ( Base ` R )
	lidlunitel.2	|- U = ( Unit ` R )
	lidlunitel.3	|- ( ph -> J e. U )
	lidlunitel.4	|- ( ph -> J e. I )
	lidlunitel.5	|- ( ph -> R e. Ring )
	lidlunitel.6	|- ( ph -> I e. ( LIdeal ` R ) )
Assertion	lidlunitel	|- ( ph -> I = B )

Proof

Step	Hyp	Ref	Expression
1		lidlunitel.1	|- B = ( Base ` R )
2		lidlunitel.2	|- U = ( Unit ` R )
3		lidlunitel.3	|- ( ph -> J e. U )
4		lidlunitel.4	|- ( ph -> J e. I )
5		lidlunitel.5	|- ( ph -> R e. Ring )
6		lidlunitel.6	|- ( ph -> I e. ( LIdeal ` R ) )
7		eqid	|- ( invr ` R ) = ( invr ` R )
8		eqid	|- ( .r ` R ) = ( .r ` R )
9		eqid	|- ( 1r ` R ) = ( 1r ` R )
10	2 7 8 9	unitlinv	|- ( ( R e. Ring /\ J e. U ) -> ( ( ( invr ` R ) ` J ) ( .r ` R ) J ) = ( 1r ` R ) )
11	5 3 10	syl2anc	|- ( ph -> ( ( ( invr ` R ) ` J ) ( .r ` R ) J ) = ( 1r ` R ) )
12	1 2	unitss	|- U C_ B
13	2 7	unitinvcl	|- ( ( R e. Ring /\ J e. U ) -> ( ( invr ` R ) ` J ) e. U )
14	5 3 13	syl2anc	|- ( ph -> ( ( invr ` R ) ` J ) e. U )
15	12 14	sselid	|- ( ph -> ( ( invr ` R ) ` J ) e. B )
16		eqid	|- ( LIdeal ` R ) = ( LIdeal ` R )
17	16 1 8	lidlmcl	|- ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) ) /\ ( ( ( invr ` R ) ` J ) e. B /\ J e. I ) ) -> ( ( ( invr ` R ) ` J ) ( .r ` R ) J ) e. I )
18	5 6 15 4 17	syl22anc	|- ( ph -> ( ( ( invr ` R ) ` J ) ( .r ` R ) J ) e. I )
19	11 18	eqeltrrd	|- ( ph -> ( 1r ` R ) e. I )
20	16 1 9	lidl1el	|- ( ( R e. Ring /\ I e. ( LIdeal ` R ) ) -> ( ( 1r ` R ) e. I <-> I = B ) )
21	20	biimpa	|- ( ( ( R e. Ring /\ I e. ( LIdeal ` R ) ) /\ ( 1r ` R ) e. I ) -> I = B )
22	5 6 19 21	syl21anc	|- ( ph -> I = B )