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Metamath Proof Explorer

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 )