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Metamath Proof Explorer

Theorem lidl1el

Description: An ideal contains 1 iff it is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015) (Revised by Wolf Lammen, 6-Sep-2020)

Ref Expression
Hypotheses lidlcl.u 
|- U = ( LIdeal ` R )
lidlcl.b 
|- B = ( Base ` R )
lidl1el.o 
|- .1. = ( 1r ` R )
Assertion lidl1el 
|- ( ( R e. Ring /\ I e. U ) -> ( .1. e. I <-> I = B ) )

Proof

Step Hyp Ref Expression
1 lidlcl.u 
 |-  U = ( LIdeal ` R )
2 lidlcl.b 
 |-  B = ( Base ` R )
3 lidl1el.o 
 |-  .1. = ( 1r ` R )
4 2 1 lidlss 
 |-  ( I e. U -> I C_ B )
5 4 ad2antlr 
 |-  ( ( ( R e. Ring /\ I e. U ) /\ .1. e. I ) -> I C_ B )
6 eqid 
 |-  ( .r ` R ) = ( .r ` R )
7 2 6 3 ringridm 
 |-  ( ( R e. Ring /\ a e. B ) -> ( a ( .r ` R ) .1. ) = a )
8 7 ad2ant2rl 
 |-  ( ( ( R e. Ring /\ I e. U ) /\ ( .1. e. I /\ a e. B ) ) -> ( a ( .r ` R ) .1. ) = a )
9 1 2 6 lidlmcl 
 |-  ( ( ( R e. Ring /\ I e. U ) /\ ( a e. B /\ .1. e. I ) ) -> ( a ( .r ` R ) .1. ) e. I )
10 9 ancom2s 
 |-  ( ( ( R e. Ring /\ I e. U ) /\ ( .1. e. I /\ a e. B ) ) -> ( a ( .r ` R ) .1. ) e. I )
11 8 10 eqeltrrd 
 |-  ( ( ( R e. Ring /\ I e. U ) /\ ( .1. e. I /\ a e. B ) ) -> a e. I )
12 11 expr 
 |-  ( ( ( R e. Ring /\ I e. U ) /\ .1. e. I ) -> ( a e. B -> a e. I ) )
13 12 ssrdv 
 |-  ( ( ( R e. Ring /\ I e. U ) /\ .1. e. I ) -> B C_ I )
14 5 13 eqssd 
 |-  ( ( ( R e. Ring /\ I e. U ) /\ .1. e. I ) -> I = B )
15 14 ex 
 |-  ( ( R e. Ring /\ I e. U ) -> ( .1. e. I -> I = B ) )
16 2 3 ringidcl 
 |-  ( R e. Ring -> .1. e. B )
17 16 adantr 
 |-  ( ( R e. Ring /\ I e. U ) -> .1. e. B )
18 eleq2 
 |-  ( I = B -> ( .1. e. I <-> .1. e. B ) )
19 17 18 syl5ibrcom 
 |-  ( ( R e. Ring /\ I e. U ) -> ( I = B -> .1. e. I ) )
20 15 19 impbid 
 |-  ( ( R e. Ring /\ I e. U ) -> ( .1. e. I <-> I = B ) )