Step |
Hyp |
Ref |
Expression |
1 |
|
rhmrcl1 |
|- ( F e. ( R RingHom S ) -> R e. Ring ) |
2 |
|
eqid |
|- ( Unit ` R ) = ( Unit ` R ) |
3 |
|
eqid |
|- ( invr ` R ) = ( invr ` R ) |
4 |
|
eqid |
|- ( .r ` R ) = ( .r ` R ) |
5 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
6 |
2 3 4 5
|
unitlinv |
|- ( ( R e. Ring /\ A e. ( Unit ` R ) ) -> ( ( ( invr ` R ) ` A ) ( .r ` R ) A ) = ( 1r ` R ) ) |
7 |
1 6
|
sylan |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( ( ( invr ` R ) ` A ) ( .r ` R ) A ) = ( 1r ` R ) ) |
8 |
7
|
fveq2d |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( F ` ( ( ( invr ` R ) ` A ) ( .r ` R ) A ) ) = ( F ` ( 1r ` R ) ) ) |
9 |
|
simpl |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> F e. ( R RingHom S ) ) |
10 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
11 |
10 2
|
unitss |
|- ( Unit ` R ) C_ ( Base ` R ) |
12 |
2 3
|
unitinvcl |
|- ( ( R e. Ring /\ A e. ( Unit ` R ) ) -> ( ( invr ` R ) ` A ) e. ( Unit ` R ) ) |
13 |
1 12
|
sylan |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( ( invr ` R ) ` A ) e. ( Unit ` R ) ) |
14 |
11 13
|
sselid |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( ( invr ` R ) ` A ) e. ( Base ` R ) ) |
15 |
|
simpr |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> A e. ( Unit ` R ) ) |
16 |
11 15
|
sselid |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> A e. ( Base ` R ) ) |
17 |
|
eqid |
|- ( .r ` S ) = ( .r ` S ) |
18 |
10 4 17
|
rhmmul |
|- ( ( F e. ( R RingHom S ) /\ ( ( invr ` R ) ` A ) e. ( Base ` R ) /\ A e. ( Base ` R ) ) -> ( F ` ( ( ( invr ` R ) ` A ) ( .r ` R ) A ) ) = ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) ) |
19 |
9 14 16 18
|
syl3anc |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( F ` ( ( ( invr ` R ) ` A ) ( .r ` R ) A ) ) = ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) ) |
20 |
|
eqid |
|- ( 1r ` S ) = ( 1r ` S ) |
21 |
5 20
|
rhm1 |
|- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) ) |
22 |
21
|
adantr |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) ) |
23 |
8 19 22
|
3eqtr3d |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( 1r ` S ) ) |
24 |
|
rhmrcl2 |
|- ( F e. ( R RingHom S ) -> S e. Ring ) |
25 |
24
|
adantr |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> S e. Ring ) |
26 |
|
elrhmunit |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( F ` A ) e. ( Unit ` S ) ) |
27 |
|
eqid |
|- ( Unit ` S ) = ( Unit ` S ) |
28 |
|
eqid |
|- ( invr ` S ) = ( invr ` S ) |
29 |
27 28 17 20
|
unitlinv |
|- ( ( S e. Ring /\ ( F ` A ) e. ( Unit ` S ) ) -> ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) = ( 1r ` S ) ) |
30 |
25 26 29
|
syl2anc |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) = ( 1r ` S ) ) |
31 |
23 30
|
eqtr4d |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) ) |
32 |
|
eqid |
|- ( ( mulGrp ` S ) |`s ( Unit ` S ) ) = ( ( mulGrp ` S ) |`s ( Unit ` S ) ) |
33 |
27 32
|
unitgrp |
|- ( S e. Ring -> ( ( mulGrp ` S ) |`s ( Unit ` S ) ) e. Grp ) |
34 |
24 33
|
syl |
|- ( F e. ( R RingHom S ) -> ( ( mulGrp ` S ) |`s ( Unit ` S ) ) e. Grp ) |
35 |
34
|
adantr |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( ( mulGrp ` S ) |`s ( Unit ` S ) ) e. Grp ) |
36 |
|
elrhmunit |
|- ( ( F e. ( R RingHom S ) /\ ( ( invr ` R ) ` A ) e. ( Unit ` R ) ) -> ( F ` ( ( invr ` R ) ` A ) ) e. ( Unit ` S ) ) |
37 |
13 36
|
syldan |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( F ` ( ( invr ` R ) ` A ) ) e. ( Unit ` S ) ) |
38 |
27 28
|
unitinvcl |
|- ( ( S e. Ring /\ ( F ` A ) e. ( Unit ` S ) ) -> ( ( invr ` S ) ` ( F ` A ) ) e. ( Unit ` S ) ) |
39 |
25 26 38
|
syl2anc |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( ( invr ` S ) ` ( F ` A ) ) e. ( Unit ` S ) ) |
40 |
27 32
|
unitgrpbas |
|- ( Unit ` S ) = ( Base ` ( ( mulGrp ` S ) |`s ( Unit ` S ) ) ) |
41 |
|
fvex |
|- ( Unit ` S ) e. _V |
42 |
|
eqid |
|- ( mulGrp ` S ) = ( mulGrp ` S ) |
43 |
42 17
|
mgpplusg |
|- ( .r ` S ) = ( +g ` ( mulGrp ` S ) ) |
44 |
32 43
|
ressplusg |
|- ( ( Unit ` S ) e. _V -> ( .r ` S ) = ( +g ` ( ( mulGrp ` S ) |`s ( Unit ` S ) ) ) ) |
45 |
41 44
|
ax-mp |
|- ( .r ` S ) = ( +g ` ( ( mulGrp ` S ) |`s ( Unit ` S ) ) ) |
46 |
40 45
|
grprcan |
|- ( ( ( ( mulGrp ` S ) |`s ( Unit ` S ) ) e. Grp /\ ( ( F ` ( ( invr ` R ) ` A ) ) e. ( Unit ` S ) /\ ( ( invr ` S ) ` ( F ` A ) ) e. ( Unit ` S ) /\ ( F ` A ) e. ( Unit ` S ) ) ) -> ( ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) <-> ( F ` ( ( invr ` R ) ` A ) ) = ( ( invr ` S ) ` ( F ` A ) ) ) ) |
47 |
35 37 39 26 46
|
syl13anc |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( ( ( F ` ( ( invr ` R ) ` A ) ) ( .r ` S ) ( F ` A ) ) = ( ( ( invr ` S ) ` ( F ` A ) ) ( .r ` S ) ( F ` A ) ) <-> ( F ` ( ( invr ` R ) ` A ) ) = ( ( invr ` S ) ` ( F ` A ) ) ) ) |
48 |
31 47
|
mpbid |
|- ( ( F e. ( R RingHom S ) /\ A e. ( Unit ` R ) ) -> ( F ` ( ( invr ` R ) ` A ) ) = ( ( invr ` S ) ` ( F ` A ) ) ) |