Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( 1st ‘ 𝑆 ) = ( 1st ‘ 𝑆 ) |
2 |
|
eqid |
⊢ ran ( 1st ‘ 𝑆 ) = ran ( 1st ‘ 𝑆 ) |
3 |
|
eqid |
⊢ ( 1st ‘ 𝑇 ) = ( 1st ‘ 𝑇 ) |
4 |
|
eqid |
⊢ ran ( 1st ‘ 𝑇 ) = ran ( 1st ‘ 𝑇 ) |
5 |
1 2 3 4
|
rngohomf |
⊢ ( ( 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) → 𝐺 : ran ( 1st ‘ 𝑆 ) ⟶ ran ( 1st ‘ 𝑇 ) ) |
6 |
5
|
3expa |
⊢ ( ( ( 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) → 𝐺 : ran ( 1st ‘ 𝑆 ) ⟶ ran ( 1st ‘ 𝑇 ) ) |
7 |
6
|
3adantl1 |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) → 𝐺 : ran ( 1st ‘ 𝑆 ) ⟶ ran ( 1st ‘ 𝑇 ) ) |
8 |
7
|
adantrl |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) → 𝐺 : ran ( 1st ‘ 𝑆 ) ⟶ ran ( 1st ‘ 𝑇 ) ) |
9 |
|
eqid |
⊢ ( 1st ‘ 𝑅 ) = ( 1st ‘ 𝑅 ) |
10 |
|
eqid |
⊢ ran ( 1st ‘ 𝑅 ) = ran ( 1st ‘ 𝑅 ) |
11 |
9 10 1 2
|
rngohomf |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → 𝐹 : ran ( 1st ‘ 𝑅 ) ⟶ ran ( 1st ‘ 𝑆 ) ) |
12 |
11
|
3expa |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ) ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → 𝐹 : ran ( 1st ‘ 𝑅 ) ⟶ ran ( 1st ‘ 𝑆 ) ) |
13 |
12
|
3adantl3 |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → 𝐹 : ran ( 1st ‘ 𝑅 ) ⟶ ran ( 1st ‘ 𝑆 ) ) |
14 |
13
|
adantrr |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) → 𝐹 : ran ( 1st ‘ 𝑅 ) ⟶ ran ( 1st ‘ 𝑆 ) ) |
15 |
|
fco |
⊢ ( ( 𝐺 : ran ( 1st ‘ 𝑆 ) ⟶ ran ( 1st ‘ 𝑇 ) ∧ 𝐹 : ran ( 1st ‘ 𝑅 ) ⟶ ran ( 1st ‘ 𝑆 ) ) → ( 𝐺 ∘ 𝐹 ) : ran ( 1st ‘ 𝑅 ) ⟶ ran ( 1st ‘ 𝑇 ) ) |
16 |
8 14 15
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) → ( 𝐺 ∘ 𝐹 ) : ran ( 1st ‘ 𝑅 ) ⟶ ran ( 1st ‘ 𝑇 ) ) |
17 |
|
eqid |
⊢ ( 2nd ‘ 𝑅 ) = ( 2nd ‘ 𝑅 ) |
18 |
|
eqid |
⊢ ( GId ‘ ( 2nd ‘ 𝑅 ) ) = ( GId ‘ ( 2nd ‘ 𝑅 ) ) |
19 |
10 17 18
|
rngo1cl |
⊢ ( 𝑅 ∈ RingOps → ( GId ‘ ( 2nd ‘ 𝑅 ) ) ∈ ran ( 1st ‘ 𝑅 ) ) |
20 |
19
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) → ( GId ‘ ( 2nd ‘ 𝑅 ) ) ∈ ran ( 1st ‘ 𝑅 ) ) |
21 |
20
|
adantr |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) → ( GId ‘ ( 2nd ‘ 𝑅 ) ) ∈ ran ( 1st ‘ 𝑅 ) ) |
22 |
|
fvco3 |
⊢ ( ( 𝐹 : ran ( 1st ‘ 𝑅 ) ⟶ ran ( 1st ‘ 𝑆 ) ∧ ( GId ‘ ( 2nd ‘ 𝑅 ) ) ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐺 ∘ 𝐹 ) ‘ ( GId ‘ ( 2nd ‘ 𝑅 ) ) ) = ( 𝐺 ‘ ( 𝐹 ‘ ( GId ‘ ( 2nd ‘ 𝑅 ) ) ) ) ) |
23 |
14 21 22
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) → ( ( 𝐺 ∘ 𝐹 ) ‘ ( GId ‘ ( 2nd ‘ 𝑅 ) ) ) = ( 𝐺 ‘ ( 𝐹 ‘ ( GId ‘ ( 2nd ‘ 𝑅 ) ) ) ) ) |
24 |
|
eqid |
⊢ ( 2nd ‘ 𝑆 ) = ( 2nd ‘ 𝑆 ) |
25 |
|
eqid |
⊢ ( GId ‘ ( 2nd ‘ 𝑆 ) ) = ( GId ‘ ( 2nd ‘ 𝑆 ) ) |
26 |
17 18 24 25
|
rngohom1 |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝐹 ‘ ( GId ‘ ( 2nd ‘ 𝑅 ) ) ) = ( GId ‘ ( 2nd ‘ 𝑆 ) ) ) |
27 |
26
|
3expa |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ) ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝐹 ‘ ( GId ‘ ( 2nd ‘ 𝑅 ) ) ) = ( GId ‘ ( 2nd ‘ 𝑆 ) ) ) |
28 |
27
|
3adantl3 |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝐹 ‘ ( GId ‘ ( 2nd ‘ 𝑅 ) ) ) = ( GId ‘ ( 2nd ‘ 𝑆 ) ) ) |
29 |
28
|
adantrr |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) → ( 𝐹 ‘ ( GId ‘ ( 2nd ‘ 𝑅 ) ) ) = ( GId ‘ ( 2nd ‘ 𝑆 ) ) ) |
30 |
29
|
fveq2d |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) → ( 𝐺 ‘ ( 𝐹 ‘ ( GId ‘ ( 2nd ‘ 𝑅 ) ) ) ) = ( 𝐺 ‘ ( GId ‘ ( 2nd ‘ 𝑆 ) ) ) ) |
31 |
|
eqid |
⊢ ( 2nd ‘ 𝑇 ) = ( 2nd ‘ 𝑇 ) |
32 |
|
eqid |
⊢ ( GId ‘ ( 2nd ‘ 𝑇 ) ) = ( GId ‘ ( 2nd ‘ 𝑇 ) ) |
33 |
24 25 31 32
|
rngohom1 |
⊢ ( ( 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) → ( 𝐺 ‘ ( GId ‘ ( 2nd ‘ 𝑆 ) ) ) = ( GId ‘ ( 2nd ‘ 𝑇 ) ) ) |
34 |
33
|
3expa |
⊢ ( ( ( 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) → ( 𝐺 ‘ ( GId ‘ ( 2nd ‘ 𝑆 ) ) ) = ( GId ‘ ( 2nd ‘ 𝑇 ) ) ) |
35 |
34
|
3adantl1 |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) → ( 𝐺 ‘ ( GId ‘ ( 2nd ‘ 𝑆 ) ) ) = ( GId ‘ ( 2nd ‘ 𝑇 ) ) ) |
36 |
35
|
adantrl |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) → ( 𝐺 ‘ ( GId ‘ ( 2nd ‘ 𝑆 ) ) ) = ( GId ‘ ( 2nd ‘ 𝑇 ) ) ) |
37 |
30 36
|
eqtrd |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) → ( 𝐺 ‘ ( 𝐹 ‘ ( GId ‘ ( 2nd ‘ 𝑅 ) ) ) ) = ( GId ‘ ( 2nd ‘ 𝑇 ) ) ) |
38 |
23 37
|
eqtrd |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) → ( ( 𝐺 ∘ 𝐹 ) ‘ ( GId ‘ ( 2nd ‘ 𝑅 ) ) ) = ( GId ‘ ( 2nd ‘ 𝑇 ) ) ) |
39 |
9 10 1
|
rngohomadd |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( 1st ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) |
40 |
39
|
ex |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( 1st ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
41 |
40
|
3expa |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ) ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( 1st ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
42 |
41
|
3adantl3 |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( 1st ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
43 |
42
|
imp |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( 1st ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) |
44 |
43
|
adantlrr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( 1st ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) |
45 |
44
|
fveq2d |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝐺 ‘ ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) ) = ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 1st ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
46 |
9 10 1 2
|
rngohomcl |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ) |
47 |
9 10 1 2
|
rngohomcl |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) |
48 |
46 47
|
anim12dan |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) ) |
49 |
48
|
ex |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) ) ) |
50 |
49
|
3expa |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ) ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) ) ) |
51 |
50
|
3adantl3 |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) ) ) |
52 |
51
|
imp |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) ) |
53 |
52
|
adantlrr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) ) |
54 |
1 2 3
|
rngohomadd |
⊢ ( ( ( 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ∧ ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 1st ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 1st ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
55 |
54
|
ex |
⊢ ( ( 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) → ( ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 1st ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 1st ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
56 |
55
|
3expa |
⊢ ( ( ( 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) → ( ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 1st ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 1st ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
57 |
56
|
3adantl1 |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) → ( ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 1st ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 1st ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
58 |
57
|
imp |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ∧ ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 1st ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 1st ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
59 |
58
|
adantlrl |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 1st ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 1st ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
60 |
53 59
|
syldan |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 1st ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 1st ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
61 |
45 60
|
eqtrd |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝐺 ‘ ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 1st ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
62 |
9 10
|
rngogcl |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
63 |
62
|
3expb |
⊢ ( ( 𝑅 ∈ RingOps ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
64 |
63
|
3ad2antl1 |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
65 |
64
|
adantlr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
66 |
|
fvco3 |
⊢ ( ( 𝐹 : ran ( 1st ‘ 𝑅 ) ⟶ ran ( 1st ‘ 𝑆 ) ∧ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( 𝐺 ‘ ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) ) ) |
67 |
14 66
|
sylan |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( 𝐺 ‘ ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) ) ) |
68 |
65 67
|
syldan |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( 𝐺 ‘ ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) ) ) |
69 |
|
fvco3 |
⊢ ( ( 𝐹 : ran ( 1st ‘ 𝑅 ) ⟶ ran ( 1st ‘ 𝑆 ) ∧ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) = ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ) |
70 |
14 69
|
sylan |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) = ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ) |
71 |
|
fvco3 |
⊢ ( ( 𝐹 : ran ( 1st ‘ 𝑅 ) ⟶ ran ( 1st ‘ 𝑆 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) |
72 |
14 71
|
sylan |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) |
73 |
70 72
|
anim12dan |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) = ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
74 |
|
oveq12 |
⊢ ( ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) = ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) → ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 1st ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 1st ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
75 |
73 74
|
syl |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 1st ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 1st ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
76 |
61 68 75
|
3eqtr4d |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 1st ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) ) |
77 |
9 10 17 24
|
rngohommul |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) |
78 |
77
|
ex |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
79 |
78
|
3expa |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ) ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
80 |
79
|
3adantl3 |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
81 |
80
|
imp |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) |
82 |
81
|
adantlrr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) |
83 |
82
|
fveq2d |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝐺 ‘ ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) ) = ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) ) |
84 |
1 2 24 31
|
rngohommul |
⊢ ( ( ( 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ∧ ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 2nd ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
85 |
84
|
ex |
⊢ ( ( 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) → ( ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 2nd ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
86 |
85
|
3expa |
⊢ ( ( ( 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) → ( ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 2nd ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
87 |
86
|
3adantl1 |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) → ( ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 2nd ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
88 |
87
|
imp |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ∧ ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 2nd ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
89 |
88
|
adantlrl |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( ( 𝐹 ‘ 𝑥 ) ∈ ran ( 1st ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ran ( 1st ‘ 𝑆 ) ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 2nd ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
90 |
53 89
|
syldan |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝐺 ‘ ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 2nd ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
91 |
83 90
|
eqtrd |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝐺 ‘ ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 2nd ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
92 |
9 17 10
|
rngocl |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
93 |
92
|
3expb |
⊢ ( ( 𝑅 ∈ RingOps ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
94 |
93
|
3ad2antl1 |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
95 |
94
|
adantlr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
96 |
|
fvco3 |
⊢ ( ( 𝐹 : ran ( 1st ‘ 𝑅 ) ⟶ ran ( 1st ‘ 𝑆 ) ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) = ( 𝐺 ‘ ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) ) ) |
97 |
14 96
|
sylan |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) = ( 𝐺 ‘ ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) ) ) |
98 |
95 97
|
syldan |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) = ( 𝐺 ‘ ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) ) ) |
99 |
|
oveq12 |
⊢ ( ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) = ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ∧ ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) = ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) → ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 2nd ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 2nd ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
100 |
73 99
|
syl |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 2nd ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) = ( ( 𝐺 ‘ ( 𝐹 ‘ 𝑥 ) ) ( 2nd ‘ 𝑇 ) ( 𝐺 ‘ ( 𝐹 ‘ 𝑦 ) ) ) ) |
101 |
91 98 100
|
3eqtr4d |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) = ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 2nd ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) ) |
102 |
76 101
|
jca |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 1st ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) ∧ ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) = ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 2nd ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) ) ) |
103 |
102
|
ralrimivva |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) → ∀ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∀ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ( ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 1st ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) ∧ ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) = ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 2nd ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) ) ) |
104 |
9 17 10 18 3 31 4 32
|
isrngohom |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑇 ∈ RingOps ) → ( ( 𝐺 ∘ 𝐹 ) ∈ ( 𝑅 RngHom 𝑇 ) ↔ ( ( 𝐺 ∘ 𝐹 ) : ran ( 1st ‘ 𝑅 ) ⟶ ran ( 1st ‘ 𝑇 ) ∧ ( ( 𝐺 ∘ 𝐹 ) ‘ ( GId ‘ ( 2nd ‘ 𝑅 ) ) ) = ( GId ‘ ( 2nd ‘ 𝑇 ) ) ∧ ∀ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∀ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ( ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 1st ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) ∧ ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) = ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 2nd ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) ) ) ) ) |
105 |
104
|
3adant2 |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) → ( ( 𝐺 ∘ 𝐹 ) ∈ ( 𝑅 RngHom 𝑇 ) ↔ ( ( 𝐺 ∘ 𝐹 ) : ran ( 1st ‘ 𝑅 ) ⟶ ran ( 1st ‘ 𝑇 ) ∧ ( ( 𝐺 ∘ 𝐹 ) ‘ ( GId ‘ ( 2nd ‘ 𝑅 ) ) ) = ( GId ‘ ( 2nd ‘ 𝑇 ) ) ∧ ∀ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∀ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ( ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 1st ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) ∧ ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) = ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 2nd ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) ) ) ) ) |
106 |
105
|
adantr |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) → ( ( 𝐺 ∘ 𝐹 ) ∈ ( 𝑅 RngHom 𝑇 ) ↔ ( ( 𝐺 ∘ 𝐹 ) : ran ( 1st ‘ 𝑅 ) ⟶ ran ( 1st ‘ 𝑇 ) ∧ ( ( 𝐺 ∘ 𝐹 ) ‘ ( GId ‘ ( 2nd ‘ 𝑅 ) ) ) = ( GId ‘ ( 2nd ‘ 𝑇 ) ) ∧ ∀ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∀ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ( ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 1st ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) ∧ ( ( 𝐺 ∘ 𝐹 ) ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑦 ) ) = ( ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑥 ) ( 2nd ‘ 𝑇 ) ( ( 𝐺 ∘ 𝐹 ) ‘ 𝑦 ) ) ) ) ) ) |
107 |
16 38 103 106
|
mpbir3and |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ) ∧ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RngHom 𝑇 ) ) ) → ( 𝐺 ∘ 𝐹 ) ∈ ( 𝑅 RngHom 𝑇 ) ) |