Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) |
2 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
3 |
|
eqid |
⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 ) |
4 |
2 3
|
unitss |
⊢ ( Unit ‘ 𝑅 ) ⊆ ( Base ‘ 𝑅 ) |
5 |
|
simpr |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝐴 ∈ ( Unit ‘ 𝑅 ) ) |
6 |
4 5
|
sselid |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝐴 ∈ ( Base ‘ 𝑅 ) ) |
7 |
|
rhmrcl1 |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝑅 ∈ Ring ) |
8 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
9 |
2 8
|
ringidcl |
⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
10 |
1 7 9
|
3syl |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
11 |
|
eqid |
⊢ ( ∥r ‘ 𝑅 ) = ( ∥r ‘ 𝑅 ) |
12 |
|
eqid |
⊢ ( oppr ‘ 𝑅 ) = ( oppr ‘ 𝑅 ) |
13 |
|
eqid |
⊢ ( ∥r ‘ ( oppr ‘ 𝑅 ) ) = ( ∥r ‘ ( oppr ‘ 𝑅 ) ) |
14 |
3 8 11 12 13
|
isunit |
⊢ ( 𝐴 ∈ ( Unit ‘ 𝑅 ) ↔ ( 𝐴 ( ∥r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ∧ 𝐴 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( 1r ‘ 𝑅 ) ) ) |
15 |
5 14
|
sylib |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐴 ( ∥r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ∧ 𝐴 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( 1r ‘ 𝑅 ) ) ) |
16 |
15
|
simpld |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝐴 ( ∥r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) |
17 |
|
eqid |
⊢ ( ∥r ‘ 𝑆 ) = ( ∥r ‘ 𝑆 ) |
18 |
2 11 17
|
rhmdvdsr |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Base ‘ 𝑅 ) ∧ ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) ∧ 𝐴 ( ∥r ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ) |
19 |
1 6 10 16 18
|
syl31anc |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ) |
20 |
|
eqid |
⊢ ( 1r ‘ 𝑆 ) = ( 1r ‘ 𝑆 ) |
21 |
8 20
|
rhm1 |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑆 ) ) |
22 |
21
|
breq2d |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ↔ ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 1r ‘ 𝑆 ) ) ) |
23 |
22
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ↔ ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 1r ‘ 𝑆 ) ) ) |
24 |
19 23
|
mpbid |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 1r ‘ 𝑆 ) ) |
25 |
|
rhmopp |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 ∈ ( ( oppr ‘ 𝑅 ) RingHom ( oppr ‘ 𝑆 ) ) ) |
26 |
25
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝐹 ∈ ( ( oppr ‘ 𝑅 ) RingHom ( oppr ‘ 𝑆 ) ) ) |
27 |
15
|
simprd |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → 𝐴 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( 1r ‘ 𝑅 ) ) |
28 |
12 2
|
opprbas |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ ( oppr ‘ 𝑅 ) ) |
29 |
|
eqid |
⊢ ( ∥r ‘ ( oppr ‘ 𝑆 ) ) = ( ∥r ‘ ( oppr ‘ 𝑆 ) ) |
30 |
28 13 29
|
rhmdvdsr |
⊢ ( ( ( 𝐹 ∈ ( ( oppr ‘ 𝑅 ) RingHom ( oppr ‘ 𝑆 ) ) ∧ 𝐴 ∈ ( Base ‘ 𝑅 ) ∧ ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) ∧ 𝐴 ( ∥r ‘ ( oppr ‘ 𝑅 ) ) ( 1r ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ) |
31 |
26 6 10 27 30
|
syl31anc |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ) |
32 |
21
|
breq2d |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ↔ ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 1r ‘ 𝑆 ) ) ) |
33 |
32
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 𝐹 ‘ ( 1r ‘ 𝑅 ) ) ↔ ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 1r ‘ 𝑆 ) ) ) |
34 |
31 33
|
mpbid |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 1r ‘ 𝑆 ) ) |
35 |
|
eqid |
⊢ ( Unit ‘ 𝑆 ) = ( Unit ‘ 𝑆 ) |
36 |
|
eqid |
⊢ ( oppr ‘ 𝑆 ) = ( oppr ‘ 𝑆 ) |
37 |
35 20 17 36 29
|
isunit |
⊢ ( ( 𝐹 ‘ 𝐴 ) ∈ ( Unit ‘ 𝑆 ) ↔ ( ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ 𝑆 ) ( 1r ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝐴 ) ( ∥r ‘ ( oppr ‘ 𝑆 ) ) ( 1r ‘ 𝑆 ) ) ) |
38 |
24 34 37
|
sylanbrc |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐴 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ ( Unit ‘ 𝑆 ) ) |