Step |
Hyp |
Ref |
Expression |
1 |
|
drngpropd.1 |
|- ( ph -> B = ( Base ` K ) ) |
2 |
|
drngpropd.2 |
|- ( ph -> B = ( Base ` L ) ) |
3 |
|
drngpropd.3 |
|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) ) |
4 |
|
drngpropd.4 |
|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) ) |
5 |
1 2 4
|
unitpropd |
|- ( ph -> ( Unit ` K ) = ( Unit ` L ) ) |
6 |
5
|
adantr |
|- ( ( ph /\ K e. Ring ) -> ( Unit ` K ) = ( Unit ` L ) ) |
7 |
1 2
|
eqtr3d |
|- ( ph -> ( Base ` K ) = ( Base ` L ) ) |
8 |
7
|
adantr |
|- ( ( ph /\ K e. Ring ) -> ( Base ` K ) = ( Base ` L ) ) |
9 |
1
|
adantr |
|- ( ( ph /\ K e. Ring ) -> B = ( Base ` K ) ) |
10 |
2
|
adantr |
|- ( ( ph /\ K e. Ring ) -> B = ( Base ` L ) ) |
11 |
3
|
adantlr |
|- ( ( ( ph /\ K e. Ring ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) ) |
12 |
9 10 11
|
grpidpropd |
|- ( ( ph /\ K e. Ring ) -> ( 0g ` K ) = ( 0g ` L ) ) |
13 |
12
|
sneqd |
|- ( ( ph /\ K e. Ring ) -> { ( 0g ` K ) } = { ( 0g ` L ) } ) |
14 |
8 13
|
difeq12d |
|- ( ( ph /\ K e. Ring ) -> ( ( Base ` K ) \ { ( 0g ` K ) } ) = ( ( Base ` L ) \ { ( 0g ` L ) } ) ) |
15 |
6 14
|
eqeq12d |
|- ( ( ph /\ K e. Ring ) -> ( ( Unit ` K ) = ( ( Base ` K ) \ { ( 0g ` K ) } ) <-> ( Unit ` L ) = ( ( Base ` L ) \ { ( 0g ` L ) } ) ) ) |
16 |
15
|
pm5.32da |
|- ( ph -> ( ( K e. Ring /\ ( Unit ` K ) = ( ( Base ` K ) \ { ( 0g ` K ) } ) ) <-> ( K e. Ring /\ ( Unit ` L ) = ( ( Base ` L ) \ { ( 0g ` L ) } ) ) ) ) |
17 |
1 2 3 4
|
ringpropd |
|- ( ph -> ( K e. Ring <-> L e. Ring ) ) |
18 |
17
|
anbi1d |
|- ( ph -> ( ( K e. Ring /\ ( Unit ` L ) = ( ( Base ` L ) \ { ( 0g ` L ) } ) ) <-> ( L e. Ring /\ ( Unit ` L ) = ( ( Base ` L ) \ { ( 0g ` L ) } ) ) ) ) |
19 |
16 18
|
bitrd |
|- ( ph -> ( ( K e. Ring /\ ( Unit ` K ) = ( ( Base ` K ) \ { ( 0g ` K ) } ) ) <-> ( L e. Ring /\ ( Unit ` L ) = ( ( Base ` L ) \ { ( 0g ` L ) } ) ) ) ) |
20 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
21 |
|
eqid |
|- ( Unit ` K ) = ( Unit ` K ) |
22 |
|
eqid |
|- ( 0g ` K ) = ( 0g ` K ) |
23 |
20 21 22
|
isdrng |
|- ( K e. DivRing <-> ( K e. Ring /\ ( Unit ` K ) = ( ( Base ` K ) \ { ( 0g ` K ) } ) ) ) |
24 |
|
eqid |
|- ( Base ` L ) = ( Base ` L ) |
25 |
|
eqid |
|- ( Unit ` L ) = ( Unit ` L ) |
26 |
|
eqid |
|- ( 0g ` L ) = ( 0g ` L ) |
27 |
24 25 26
|
isdrng |
|- ( L e. DivRing <-> ( L e. Ring /\ ( Unit ` L ) = ( ( Base ` L ) \ { ( 0g ` L ) } ) ) ) |
28 |
19 23 27
|
3bitr4g |
|- ( ph -> ( K e. DivRing <-> L e. DivRing ) ) |