Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( Base ‘ ( oppr ‘ 𝑅 ) ) = ( Base ‘ ( oppr ‘ 𝑅 ) ) |
2 |
|
eqid |
⊢ ( 1r ‘ ( oppr ‘ 𝑅 ) ) = ( 1r ‘ ( oppr ‘ 𝑅 ) ) |
3 |
|
eqid |
⊢ ( 1r ‘ ( oppr ‘ 𝑆 ) ) = ( 1r ‘ ( oppr ‘ 𝑆 ) ) |
4 |
|
eqid |
⊢ ( .r ‘ ( oppr ‘ 𝑅 ) ) = ( .r ‘ ( oppr ‘ 𝑅 ) ) |
5 |
|
eqid |
⊢ ( .r ‘ ( oppr ‘ 𝑆 ) ) = ( .r ‘ ( oppr ‘ 𝑆 ) ) |
6 |
|
rhmrcl1 |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝑅 ∈ Ring ) |
7 |
|
eqid |
⊢ ( oppr ‘ 𝑅 ) = ( oppr ‘ 𝑅 ) |
8 |
7
|
opprringb |
⊢ ( 𝑅 ∈ Ring ↔ ( oppr ‘ 𝑅 ) ∈ Ring ) |
9 |
6 8
|
sylib |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( oppr ‘ 𝑅 ) ∈ Ring ) |
10 |
|
rhmrcl2 |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝑆 ∈ Ring ) |
11 |
|
eqid |
⊢ ( oppr ‘ 𝑆 ) = ( oppr ‘ 𝑆 ) |
12 |
11
|
opprringb |
⊢ ( 𝑆 ∈ Ring ↔ ( oppr ‘ 𝑆 ) ∈ Ring ) |
13 |
10 12
|
sylib |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( oppr ‘ 𝑆 ) ∈ Ring ) |
14 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
15 |
7 14
|
oppr1 |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ ( oppr ‘ 𝑅 ) ) |
16 |
15
|
eqcomi |
⊢ ( 1r ‘ ( oppr ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) |
17 |
|
eqid |
⊢ ( 1r ‘ 𝑆 ) = ( 1r ‘ 𝑆 ) |
18 |
11 17
|
oppr1 |
⊢ ( 1r ‘ 𝑆 ) = ( 1r ‘ ( oppr ‘ 𝑆 ) ) |
19 |
18
|
eqcomi |
⊢ ( 1r ‘ ( oppr ‘ 𝑆 ) ) = ( 1r ‘ 𝑆 ) |
20 |
16 19
|
rhm1 |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( 𝐹 ‘ ( 1r ‘ ( oppr ‘ 𝑅 ) ) ) = ( 1r ‘ ( oppr ‘ 𝑆 ) ) ) |
21 |
|
simpl |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ∧ 𝑦 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ) ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) |
22 |
|
simprr |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ∧ 𝑦 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ) ) → 𝑦 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ) |
23 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
24 |
7 23
|
opprbas |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ ( oppr ‘ 𝑅 ) ) |
25 |
22 24
|
eleqtrrdi |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ∧ 𝑦 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ) ) → 𝑦 ∈ ( Base ‘ 𝑅 ) ) |
26 |
|
simprl |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ∧ 𝑦 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ) ) → 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ) |
27 |
26 24
|
eleqtrrdi |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ∧ 𝑦 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ) ) → 𝑥 ∈ ( Base ‘ 𝑅 ) ) |
28 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
29 |
|
eqid |
⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 ) |
30 |
23 28 29
|
rhmmul |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) ) = ( ( 𝐹 ‘ 𝑦 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑥 ) ) ) |
31 |
21 25 27 30
|
syl3anc |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ∧ 𝑦 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) ) = ( ( 𝐹 ‘ 𝑦 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑥 ) ) ) |
32 |
23 28 7 4
|
opprmul |
⊢ ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) = ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) |
33 |
32
|
fveq2i |
⊢ ( 𝐹 ‘ ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ) = ( 𝐹 ‘ ( 𝑦 ( .r ‘ 𝑅 ) 𝑥 ) ) |
34 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
35 |
34 29 11 5
|
opprmul |
⊢ ( ( 𝐹 ‘ 𝑥 ) ( .r ‘ ( oppr ‘ 𝑆 ) ) ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑦 ) ( .r ‘ 𝑆 ) ( 𝐹 ‘ 𝑥 ) ) |
36 |
31 33 35
|
3eqtr4g |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ( 𝑥 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ∧ 𝑦 ∈ ( Base ‘ ( oppr ‘ 𝑅 ) ) ) ) → ( 𝐹 ‘ ( 𝑥 ( .r ‘ ( oppr ‘ 𝑅 ) ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( .r ‘ ( oppr ‘ 𝑆 ) ) ( 𝐹 ‘ 𝑦 ) ) ) |
37 |
|
ringgrp |
⊢ ( ( oppr ‘ 𝑅 ) ∈ Ring → ( oppr ‘ 𝑅 ) ∈ Grp ) |
38 |
9 37
|
syl |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( oppr ‘ 𝑅 ) ∈ Grp ) |
39 |
|
ringgrp |
⊢ ( ( oppr ‘ 𝑆 ) ∈ Ring → ( oppr ‘ 𝑆 ) ∈ Grp ) |
40 |
13 39
|
syl |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( oppr ‘ 𝑆 ) ∈ Grp ) |
41 |
23 34
|
rhmf |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ) |
42 |
|
rhmghm |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ) |
43 |
42
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) → 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ) |
44 |
|
simplr |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) → 𝑥 ∈ ( Base ‘ 𝑅 ) ) |
45 |
|
simpr |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) → 𝑦 ∈ ( Base ‘ 𝑅 ) ) |
46 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
47 |
|
eqid |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) |
48 |
23 46 47
|
ghmlin |
⊢ ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) |
49 |
43 44 45 48
|
syl3anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) |
50 |
49
|
ralrimiva |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝑥 ∈ ( Base ‘ 𝑅 ) ) → ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) |
51 |
50
|
ralrimiva |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) |
52 |
41 51
|
jca |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
53 |
38 40 52
|
jca31 |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( ( ( oppr ‘ 𝑅 ) ∈ Grp ∧ ( oppr ‘ 𝑆 ) ∈ Grp ) ∧ ( 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
54 |
11 34
|
opprbas |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ ( oppr ‘ 𝑆 ) ) |
55 |
7 46
|
oppradd |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ ( oppr ‘ 𝑅 ) ) |
56 |
11 47
|
oppradd |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ ( oppr ‘ 𝑆 ) ) |
57 |
24 54 55 56
|
isghm |
⊢ ( 𝐹 ∈ ( ( oppr ‘ 𝑅 ) GrpHom ( oppr ‘ 𝑆 ) ) ↔ ( ( ( oppr ‘ 𝑅 ) ∈ Grp ∧ ( oppr ‘ 𝑆 ) ∈ Grp ) ∧ ( 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ∧ ∀ 𝑥 ∈ ( Base ‘ 𝑅 ) ∀ 𝑦 ∈ ( Base ‘ 𝑅 ) ( 𝐹 ‘ ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
58 |
53 57
|
sylibr |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 ∈ ( ( oppr ‘ 𝑅 ) GrpHom ( oppr ‘ 𝑆 ) ) ) |
59 |
1 2 3 4 5 9 13 20 36 58
|
isrhm2d |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 ∈ ( ( oppr ‘ 𝑅 ) RingHom ( oppr ‘ 𝑆 ) ) ) |