Step |
Hyp |
Ref |
Expression |
1 |
|
rhmrcl1 |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝑅 ∈ 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 ∈ Ring ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( ( ( invr ‘ 𝑅 ) ‘ 𝐴 ) ( .r ‘ 𝑅 ) 𝐴 ) = ( 1r ‘ 𝑅 ) ) |
7 |
1 6
|
sylan |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( ( ( invr ‘ 𝑅 ) ‘ 𝐴 ) ( .r ‘ 𝑅 ) 𝐴 ) = ( 1r ‘ 𝑅 ) ) |
8 |
7
|
fveq2d |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ ( ( ( invr ‘ 𝑅 ) ‘ 𝐴 ) ( .r ‘ 𝑅 ) 𝐴 ) ) = ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ) |
9 |
|
simpl |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
11 |
10 2
|
unitss |
⊢ ( Unit ‘ 𝑅 ) ⊆ ( Base ‘ 𝑅 ) |
12 |
2 3
|
unitinvcl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( ( invr ‘ 𝑅 ) ‘ 𝐴 ) ∈ ( Unit ‘ 𝑅 ) ) |
13 |
1 12
|
sylan |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( ( invr ‘ 𝑅 ) ‘ 𝐴 ) ∈ ( Unit ‘ 𝑅 ) ) |
14 |
11 13
|
sselid |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( ( invr ‘ 𝑅 ) ‘ 𝐴 ) ∈ ( Base ‘ 𝑅 ) ) |
15 |
|
simpr |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝐴 ∈ ( Unit ‘ 𝑅 ) ) |
16 |
11 15
|
sselid |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝐴 ∈ ( Base ‘ 𝑅 ) ) |
17 |
|
eqid |
⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 ) |
18 |
10 4 17
|
rhmmul |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( ( invr ‘ 𝑅 ) ‘ 𝐴 ) ∈ ( Base ‘ 𝑅 ) ∧ 𝐴 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( ( ( invr ‘ 𝑅 ) ‘ 𝐴 ) ( .r ‘ 𝑅 ) 𝐴 ) ) = ( ( 𝐹 ‘ ( ( invr ‘ 𝑅 ) ‘ 𝐴 ) ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) ) |
19 |
9 14 16 18
|
syl3anc |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( 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 ∈ ( 𝑅 RingHom 𝑆 ) → ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑆 ) ) |
22 |
21
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑆 ) ) |
23 |
8 19 22
|
3eqtr3d |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( ( 𝐹 ‘ ( ( invr ‘ 𝑅 ) ‘ 𝐴 ) ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) = ( 1r ‘ 𝑆 ) ) |
24 |
|
rhmrcl2 |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝑆 ∈ Ring ) |
25 |
24
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝑆 ∈ Ring ) |
26 |
|
elrhmunit |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ ( Unit ‘ 𝑆 ) ) |
27 |
|
eqid |
⊢ ( Unit ‘ 𝑆 ) = ( Unit ‘ 𝑆 ) |
28 |
|
eqid |
⊢ ( invr ‘ 𝑆 ) = ( invr ‘ 𝑆 ) |
29 |
27 28 17 20
|
unitlinv |
⊢ ( ( 𝑆 ∈ Ring ∧ ( 𝐹 ‘ 𝐴 ) ∈ ( Unit ‘ 𝑆 ) ) → ( ( ( invr ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝐴 ) ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) = ( 1r ‘ 𝑆 ) ) |
30 |
25 26 29
|
syl2anc |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( ( ( invr ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝐴 ) ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝐴 ) ) = ( 1r ‘ 𝑆 ) ) |
31 |
23 30
|
eqtr4d |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( 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 ∈ Ring → ( ( mulGrp ‘ 𝑆 ) ↾s ( Unit ‘ 𝑆 ) ) ∈ Grp ) |
34 |
24 33
|
syl |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( ( mulGrp ‘ 𝑆 ) ↾s ( Unit ‘ 𝑆 ) ) ∈ Grp ) |
35 |
34
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( ( mulGrp ‘ 𝑆 ) ↾s ( Unit ‘ 𝑆 ) ) ∈ Grp ) |
36 |
|
elrhmunit |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( ( invr ‘ 𝑅 ) ‘ 𝐴 ) ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ ( ( invr ‘ 𝑅 ) ‘ 𝐴 ) ) ∈ ( Unit ‘ 𝑆 ) ) |
37 |
13 36
|
syldan |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ ( ( invr ‘ 𝑅 ) ‘ 𝐴 ) ) ∈ ( Unit ‘ 𝑆 ) ) |
38 |
27 28
|
unitinvcl |
⊢ ( ( 𝑆 ∈ Ring ∧ ( 𝐹 ‘ 𝐴 ) ∈ ( Unit ‘ 𝑆 ) ) → ( ( invr ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝐴 ) ) ∈ ( Unit ‘ 𝑆 ) ) |
39 |
25 26 38
|
syl2anc |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( ( invr ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝐴 ) ) ∈ ( Unit ‘ 𝑆 ) ) |
40 |
27 32
|
unitgrpbas |
⊢ ( Unit ‘ 𝑆 ) = ( Base ‘ ( ( mulGrp ‘ 𝑆 ) ↾s ( Unit ‘ 𝑆 ) ) ) |
41 |
|
fvex |
⊢ ( Unit ‘ 𝑆 ) ∈ V |
42 |
|
eqid |
⊢ ( mulGrp ‘ 𝑆 ) = ( mulGrp ‘ 𝑆 ) |
43 |
42 17
|
mgpplusg |
⊢ ( .r ‘ 𝑆 ) = ( +g ‘ ( mulGrp ‘ 𝑆 ) ) |
44 |
32 43
|
ressplusg |
⊢ ( ( Unit ‘ 𝑆 ) ∈ 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 ) ) ∈ Grp ∧ ( ( 𝐹 ‘ ( ( invr ‘ 𝑅 ) ‘ 𝐴 ) ) ∈ ( Unit ‘ 𝑆 ) ∧ ( ( invr ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝐴 ) ) ∈ ( Unit ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ ( 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 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( 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 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ ( ( invr ‘ 𝑅 ) ‘ 𝐴 ) ) = ( ( invr ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝐴 ) ) ) |