Step |
Hyp |
Ref |
Expression |
1 |
|
ringlz.1 |
|- Z = ( GId ` G ) |
2 |
|
ringlz.2 |
|- X = ran G |
3 |
|
ringlz.3 |
|- G = ( 1st ` R ) |
4 |
|
ringlz.4 |
|- H = ( 2nd ` R ) |
5 |
3
|
rngogrpo |
|- ( R e. RingOps -> G e. GrpOp ) |
6 |
2 1
|
grpoidcl |
|- ( G e. GrpOp -> Z e. X ) |
7 |
2 1
|
grpolid |
|- ( ( G e. GrpOp /\ Z e. X ) -> ( Z G Z ) = Z ) |
8 |
5 6 7
|
syl2anc2 |
|- ( R e. RingOps -> ( Z G Z ) = Z ) |
9 |
8
|
adantr |
|- ( ( R e. RingOps /\ A e. X ) -> ( Z G Z ) = Z ) |
10 |
9
|
oveq1d |
|- ( ( R e. RingOps /\ A e. X ) -> ( ( Z G Z ) H A ) = ( Z H A ) ) |
11 |
3 2 1
|
rngo0cl |
|- ( R e. RingOps -> Z e. X ) |
12 |
11
|
adantr |
|- ( ( R e. RingOps /\ A e. X ) -> Z e. X ) |
13 |
|
simpr |
|- ( ( R e. RingOps /\ A e. X ) -> A e. X ) |
14 |
12 12 13
|
3jca |
|- ( ( R e. RingOps /\ A e. X ) -> ( Z e. X /\ Z e. X /\ A e. X ) ) |
15 |
3 4 2
|
rngodir |
|- ( ( R e. RingOps /\ ( Z e. X /\ Z e. X /\ A e. X ) ) -> ( ( Z G Z ) H A ) = ( ( Z H A ) G ( Z H A ) ) ) |
16 |
14 15
|
syldan |
|- ( ( R e. RingOps /\ A e. X ) -> ( ( Z G Z ) H A ) = ( ( Z H A ) G ( Z H A ) ) ) |
17 |
5
|
adantr |
|- ( ( R e. RingOps /\ A e. X ) -> G e. GrpOp ) |
18 |
|
simpl |
|- ( ( R e. RingOps /\ A e. X ) -> R e. RingOps ) |
19 |
3 4 2
|
rngocl |
|- ( ( R e. RingOps /\ Z e. X /\ A e. X ) -> ( Z H A ) e. X ) |
20 |
18 12 13 19
|
syl3anc |
|- ( ( R e. RingOps /\ A e. X ) -> ( Z H A ) e. X ) |
21 |
2 1
|
grporid |
|- ( ( G e. GrpOp /\ ( Z H A ) e. X ) -> ( ( Z H A ) G Z ) = ( Z H A ) ) |
22 |
21
|
eqcomd |
|- ( ( G e. GrpOp /\ ( Z H A ) e. X ) -> ( Z H A ) = ( ( Z H A ) G Z ) ) |
23 |
17 20 22
|
syl2anc |
|- ( ( R e. RingOps /\ A e. X ) -> ( Z H A ) = ( ( Z H A ) G Z ) ) |
24 |
10 16 23
|
3eqtr3d |
|- ( ( R e. RingOps /\ A e. X ) -> ( ( Z H A ) G ( Z H A ) ) = ( ( Z H A ) G Z ) ) |
25 |
2
|
grpolcan |
|- ( ( G e. GrpOp /\ ( ( Z H A ) e. X /\ Z e. X /\ ( Z H A ) e. X ) ) -> ( ( ( Z H A ) G ( Z H A ) ) = ( ( Z H A ) G Z ) <-> ( Z H A ) = Z ) ) |
26 |
17 20 12 20 25
|
syl13anc |
|- ( ( R e. RingOps /\ A e. X ) -> ( ( ( Z H A ) G ( Z H A ) ) = ( ( Z H A ) G Z ) <-> ( Z H A ) = Z ) ) |
27 |
24 26
|
mpbid |
|- ( ( R e. RingOps /\ A e. X ) -> ( Z H A ) = Z ) |