Step |
Hyp |
Ref |
Expression |
1 |
|
brric |
|- ( R ~=r S <-> ( R RingIso S ) =/= (/) ) |
2 |
|
n0 |
|- ( ( R RingIso S ) =/= (/) <-> E. f f e. ( R RingIso S ) ) |
3 |
1 2
|
bitri |
|- ( R ~=r S <-> E. f f e. ( R RingIso S ) ) |
4 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
5 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
6 |
4 5
|
rimf1o |
|- ( f e. ( R RingIso S ) -> f : ( Base ` R ) -1-1-onto-> ( Base ` S ) ) |
7 |
|
f1ofo |
|- ( f : ( Base ` R ) -1-1-onto-> ( Base ` S ) -> f : ( Base ` R ) -onto-> ( Base ` S ) ) |
8 |
|
foima |
|- ( f : ( Base ` R ) -onto-> ( Base ` S ) -> ( f " ( Base ` R ) ) = ( Base ` S ) ) |
9 |
6 7 8
|
3syl |
|- ( f e. ( R RingIso S ) -> ( f " ( Base ` R ) ) = ( Base ` S ) ) |
10 |
9
|
oveq2d |
|- ( f e. ( R RingIso S ) -> ( S |`s ( f " ( Base ` R ) ) ) = ( S |`s ( Base ` S ) ) ) |
11 |
|
rimrcl2 |
|- ( f e. ( R RingIso S ) -> S e. Ring ) |
12 |
5
|
ressid |
|- ( S e. Ring -> ( S |`s ( Base ` S ) ) = S ) |
13 |
11 12
|
syl |
|- ( f e. ( R RingIso S ) -> ( S |`s ( Base ` S ) ) = S ) |
14 |
10 13
|
eqtr2d |
|- ( f e. ( R RingIso S ) -> S = ( S |`s ( f " ( Base ` R ) ) ) ) |
15 |
14
|
adantr |
|- ( ( f e. ( R RingIso S ) /\ R e. CRing ) -> S = ( S |`s ( f " ( Base ` R ) ) ) ) |
16 |
|
eqid |
|- ( S |`s ( f " ( Base ` R ) ) ) = ( S |`s ( f " ( Base ` R ) ) ) |
17 |
|
rimrhm |
|- ( f e. ( R RingIso S ) -> f e. ( R RingHom S ) ) |
18 |
17
|
adantr |
|- ( ( f e. ( R RingIso S ) /\ R e. CRing ) -> f e. ( R RingHom S ) ) |
19 |
|
simpr |
|- ( ( f e. ( R RingIso S ) /\ R e. CRing ) -> R e. CRing ) |
20 |
19
|
crngringd |
|- ( ( f e. ( R RingIso S ) /\ R e. CRing ) -> R e. Ring ) |
21 |
4
|
subrgid |
|- ( R e. Ring -> ( Base ` R ) e. ( SubRing ` R ) ) |
22 |
20 21
|
syl |
|- ( ( f e. ( R RingIso S ) /\ R e. CRing ) -> ( Base ` R ) e. ( SubRing ` R ) ) |
23 |
16 18 19 22
|
imacrhmcl |
|- ( ( f e. ( R RingIso S ) /\ R e. CRing ) -> ( S |`s ( f " ( Base ` R ) ) ) e. CRing ) |
24 |
15 23
|
eqeltrd |
|- ( ( f e. ( R RingIso S ) /\ R e. CRing ) -> S e. CRing ) |
25 |
24
|
ex |
|- ( f e. ( R RingIso S ) -> ( R e. CRing -> S e. CRing ) ) |
26 |
25
|
exlimiv |
|- ( E. f f e. ( R RingIso S ) -> ( R e. CRing -> S e. CRing ) ) |
27 |
26
|
imp |
|- ( ( E. f f e. ( R RingIso S ) /\ R e. CRing ) -> S e. CRing ) |
28 |
3 27
|
sylanb |
|- ( ( R ~=r S /\ R e. CRing ) -> S e. CRing ) |