Step |
Hyp |
Ref |
Expression |
1 |
|
rrgnz.t |
|- E = ( RLReg ` R ) |
2 |
|
rrgnz.z |
|- .0. = ( 0g ` R ) |
3 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
4 |
3 2
|
nzrnz |
|- ( R e. NzRing -> ( 1r ` R ) =/= .0. ) |
5 |
4
|
neneqd |
|- ( R e. NzRing -> -. ( 1r ` R ) = .0. ) |
6 |
|
nzrring |
|- ( R e. NzRing -> R e. Ring ) |
7 |
6
|
adantr |
|- ( ( R e. NzRing /\ .0. e. E ) -> R e. Ring ) |
8 |
|
simpr |
|- ( ( R e. NzRing /\ .0. e. E ) -> .0. e. E ) |
9 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
10 |
9 3
|
ringidcl |
|- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) ) |
11 |
7 10
|
syl |
|- ( ( R e. NzRing /\ .0. e. E ) -> ( 1r ` R ) e. ( Base ` R ) ) |
12 |
|
eqid |
|- ( .r ` R ) = ( .r ` R ) |
13 |
9 12 2 7 11
|
ringlzd |
|- ( ( R e. NzRing /\ .0. e. E ) -> ( .0. ( .r ` R ) ( 1r ` R ) ) = .0. ) |
14 |
1 9 12 2
|
rrgeq0 |
|- ( ( R e. Ring /\ .0. e. E /\ ( 1r ` R ) e. ( Base ` R ) ) -> ( ( .0. ( .r ` R ) ( 1r ` R ) ) = .0. <-> ( 1r ` R ) = .0. ) ) |
15 |
14
|
biimpa |
|- ( ( ( R e. Ring /\ .0. e. E /\ ( 1r ` R ) e. ( Base ` R ) ) /\ ( .0. ( .r ` R ) ( 1r ` R ) ) = .0. ) -> ( 1r ` R ) = .0. ) |
16 |
7 8 11 13 15
|
syl31anc |
|- ( ( R e. NzRing /\ .0. e. E ) -> ( 1r ` R ) = .0. ) |
17 |
5 16
|
mtand |
|- ( R e. NzRing -> -. .0. e. E ) |