Step |
Hyp |
Ref |
Expression |
1 |
|
keridl.1 |
⊢ 𝐺 = ( 1st ‘ 𝑆 ) |
2 |
|
keridl.2 |
⊢ 𝑍 = ( GId ‘ 𝐺 ) |
3 |
|
cnvimass |
⊢ ( ◡ 𝐹 “ { 𝑍 } ) ⊆ dom 𝐹 |
4 |
|
eqid |
⊢ ( 1st ‘ 𝑅 ) = ( 1st ‘ 𝑅 ) |
5 |
|
eqid |
⊢ ran ( 1st ‘ 𝑅 ) = ran ( 1st ‘ 𝑅 ) |
6 |
|
eqid |
⊢ ran 𝐺 = ran 𝐺 |
7 |
4 5 1 6
|
rngohomf |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → 𝐹 : ran ( 1st ‘ 𝑅 ) ⟶ ran 𝐺 ) |
8 |
3 7
|
fssdm |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ◡ 𝐹 “ { 𝑍 } ) ⊆ ran ( 1st ‘ 𝑅 ) ) |
9 |
|
eqid |
⊢ ( GId ‘ ( 1st ‘ 𝑅 ) ) = ( GId ‘ ( 1st ‘ 𝑅 ) ) |
10 |
4 5 9
|
rngo0cl |
⊢ ( 𝑅 ∈ RingOps → ( GId ‘ ( 1st ‘ 𝑅 ) ) ∈ ran ( 1st ‘ 𝑅 ) ) |
11 |
10
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( GId ‘ ( 1st ‘ 𝑅 ) ) ∈ ran ( 1st ‘ 𝑅 ) ) |
12 |
4 9 1 2
|
rngohom0 |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝐹 ‘ ( GId ‘ ( 1st ‘ 𝑅 ) ) ) = 𝑍 ) |
13 |
|
fvex |
⊢ ( 𝐹 ‘ ( GId ‘ ( 1st ‘ 𝑅 ) ) ) ∈ V |
14 |
13
|
elsn |
⊢ ( ( 𝐹 ‘ ( GId ‘ ( 1st ‘ 𝑅 ) ) ) ∈ { 𝑍 } ↔ ( 𝐹 ‘ ( GId ‘ ( 1st ‘ 𝑅 ) ) ) = 𝑍 ) |
15 |
12 14
|
sylibr |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝐹 ‘ ( GId ‘ ( 1st ‘ 𝑅 ) ) ) ∈ { 𝑍 } ) |
16 |
|
ffn |
⊢ ( 𝐹 : ran ( 1st ‘ 𝑅 ) ⟶ ran 𝐺 → 𝐹 Fn ran ( 1st ‘ 𝑅 ) ) |
17 |
|
elpreima |
⊢ ( 𝐹 Fn ran ( 1st ‘ 𝑅 ) → ( ( GId ‘ ( 1st ‘ 𝑅 ) ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ↔ ( ( GId ‘ ( 1st ‘ 𝑅 ) ) ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ ( GId ‘ ( 1st ‘ 𝑅 ) ) ) ∈ { 𝑍 } ) ) ) |
18 |
7 16 17
|
3syl |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( GId ‘ ( 1st ‘ 𝑅 ) ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ↔ ( ( GId ‘ ( 1st ‘ 𝑅 ) ) ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ ( GId ‘ ( 1st ‘ 𝑅 ) ) ) ∈ { 𝑍 } ) ) ) |
19 |
11 15 18
|
mpbir2and |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( GId ‘ ( 1st ‘ 𝑅 ) ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) |
20 |
|
an4 |
⊢ ( ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ { 𝑍 } ) ∧ ( 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ { 𝑍 } ) ) ↔ ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑥 ) ∈ { 𝑍 } ∧ ( 𝐹 ‘ 𝑦 ) ∈ { 𝑍 } ) ) ) |
21 |
4 5 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 ) 𝐺 ( 𝐹 ‘ 𝑦 ) ) ) |
22 |
21
|
adantr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) ∧ ( ( 𝐹 ‘ 𝑥 ) = 𝑍 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑍 ) ) → ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = ( ( 𝐹 ‘ 𝑥 ) 𝐺 ( 𝐹 ‘ 𝑦 ) ) ) |
23 |
|
oveq12 |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) = 𝑍 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑍 ) → ( ( 𝐹 ‘ 𝑥 ) 𝐺 ( 𝐹 ‘ 𝑦 ) ) = ( 𝑍 𝐺 𝑍 ) ) |
24 |
23
|
adantl |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) ∧ ( ( 𝐹 ‘ 𝑥 ) = 𝑍 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑍 ) ) → ( ( 𝐹 ‘ 𝑥 ) 𝐺 ( 𝐹 ‘ 𝑦 ) ) = ( 𝑍 𝐺 𝑍 ) ) |
25 |
1
|
rngogrpo |
⊢ ( 𝑆 ∈ RingOps → 𝐺 ∈ GrpOp ) |
26 |
6 2
|
grpoidcl |
⊢ ( 𝐺 ∈ GrpOp → 𝑍 ∈ ran 𝐺 ) |
27 |
6 2
|
grpolid |
⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝑍 ∈ ran 𝐺 ) → ( 𝑍 𝐺 𝑍 ) = 𝑍 ) |
28 |
25 26 27
|
syl2anc2 |
⊢ ( 𝑆 ∈ RingOps → ( 𝑍 𝐺 𝑍 ) = 𝑍 ) |
29 |
28
|
3ad2ant2 |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝑍 𝐺 𝑍 ) = 𝑍 ) |
30 |
29
|
ad2antrr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) ∧ ( ( 𝐹 ‘ 𝑥 ) = 𝑍 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑍 ) ) → ( 𝑍 𝐺 𝑍 ) = 𝑍 ) |
31 |
22 24 30
|
3eqtrd |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) ∧ ( ( 𝐹 ‘ 𝑥 ) = 𝑍 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑍 ) ) → ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = 𝑍 ) |
32 |
31
|
ex |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( ( ( 𝐹 ‘ 𝑥 ) = 𝑍 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑍 ) → ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = 𝑍 ) ) |
33 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑥 ) ∈ V |
34 |
33
|
elsn |
⊢ ( ( 𝐹 ‘ 𝑥 ) ∈ { 𝑍 } ↔ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) |
35 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑦 ) ∈ V |
36 |
35
|
elsn |
⊢ ( ( 𝐹 ‘ 𝑦 ) ∈ { 𝑍 } ↔ ( 𝐹 ‘ 𝑦 ) = 𝑍 ) |
37 |
34 36
|
anbi12i |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) ∈ { 𝑍 } ∧ ( 𝐹 ‘ 𝑦 ) ∈ { 𝑍 } ) ↔ ( ( 𝐹 ‘ 𝑥 ) = 𝑍 ∧ ( 𝐹 ‘ 𝑦 ) = 𝑍 ) ) |
38 |
|
fvex |
⊢ ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) ∈ V |
39 |
38
|
elsn |
⊢ ( ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ↔ ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) = 𝑍 ) |
40 |
32 37 39
|
3imtr4g |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( ( ( 𝐹 ‘ 𝑥 ) ∈ { 𝑍 } ∧ ( 𝐹 ‘ 𝑦 ) ∈ { 𝑍 } ) → ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) ) |
41 |
40
|
imdistanda |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑥 ) ∈ { 𝑍 } ∧ ( 𝐹 ‘ 𝑦 ) ∈ { 𝑍 } ) ) → ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) ) ) |
42 |
4 5
|
rngogcl |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
43 |
42
|
3expib |
⊢ ( 𝑅 ∈ RingOps → ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ) ) |
44 |
43
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ) ) |
45 |
44
|
anim1d |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ∧ ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) → ( ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) ) ) |
46 |
41 45
|
syld |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ) ∧ ( ( 𝐹 ‘ 𝑥 ) ∈ { 𝑍 } ∧ ( 𝐹 ‘ 𝑦 ) ∈ { 𝑍 } ) ) → ( ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) ) ) |
47 |
20 46
|
syl5bi |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ { 𝑍 } ) ∧ ( 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ { 𝑍 } ) ) → ( ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) ) ) |
48 |
|
elpreima |
⊢ ( 𝐹 Fn ran ( 1st ‘ 𝑅 ) → ( 𝑥 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ↔ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ { 𝑍 } ) ) ) |
49 |
7 16 48
|
3syl |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝑥 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ↔ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ { 𝑍 } ) ) ) |
50 |
|
elpreima |
⊢ ( 𝐹 Fn ran ( 1st ‘ 𝑅 ) → ( 𝑦 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ↔ ( 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ { 𝑍 } ) ) ) |
51 |
7 16 50
|
3syl |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝑦 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ↔ ( 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ { 𝑍 } ) ) ) |
52 |
49 51
|
anbi12d |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) ↔ ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ { 𝑍 } ) ∧ ( 𝑦 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ { 𝑍 } ) ) ) ) |
53 |
|
elpreima |
⊢ ( 𝐹 Fn ran ( 1st ‘ 𝑅 ) → ( ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) ) ) |
54 |
7 16 53
|
3syl |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ) ∈ { 𝑍 } ) ) ) |
55 |
47 52 54
|
3imtr4d |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) → ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) ) |
56 |
55
|
impl |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) → ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) |
57 |
56
|
ralrimiva |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) → ∀ 𝑦 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) |
58 |
34
|
anbi2i |
⊢ ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ { 𝑍 } ) ↔ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) |
59 |
|
eqid |
⊢ ( 2nd ‘ 𝑅 ) = ( 2nd ‘ 𝑅 ) |
60 |
4 59 5
|
rngocl |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
61 |
60
|
3expb |
⊢ ( ( 𝑅 ∈ RingOps ∧ ( 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
62 |
61
|
3ad2antl1 |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
63 |
62
|
anass1rs |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
64 |
63
|
adantlrr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
65 |
|
eqid |
⊢ ( 2nd ‘ 𝑆 ) = ( 2nd ‘ 𝑆 ) |
66 |
4 5 59 65
|
rngohommul |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝐹 ‘ ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑥 ) ) ) |
67 |
66
|
anass1rs |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑥 ) ) ) |
68 |
67
|
adantlrr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑥 ) ) ) |
69 |
|
oveq2 |
⊢ ( ( 𝐹 ‘ 𝑥 ) = 𝑍 → ( ( 𝐹 ‘ 𝑧 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( 2nd ‘ 𝑆 ) 𝑍 ) ) |
70 |
69
|
adantl |
⊢ ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) → ( ( 𝐹 ‘ 𝑧 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( 2nd ‘ 𝑆 ) 𝑍 ) ) |
71 |
70
|
ad2antlr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐹 ‘ 𝑧 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑥 ) ) = ( ( 𝐹 ‘ 𝑧 ) ( 2nd ‘ 𝑆 ) 𝑍 ) ) |
72 |
4 5 1 6
|
rngohomcl |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ ran 𝐺 ) |
73 |
2 6 1 65
|
rngorz |
⊢ ( ( 𝑆 ∈ RingOps ∧ ( 𝐹 ‘ 𝑧 ) ∈ ran 𝐺 ) → ( ( 𝐹 ‘ 𝑧 ) ( 2nd ‘ 𝑆 ) 𝑍 ) = 𝑍 ) |
74 |
73
|
3ad2antl2 |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝐹 ‘ 𝑧 ) ∈ ran 𝐺 ) → ( ( 𝐹 ‘ 𝑧 ) ( 2nd ‘ 𝑆 ) 𝑍 ) = 𝑍 ) |
75 |
72 74
|
syldan |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐹 ‘ 𝑧 ) ( 2nd ‘ 𝑆 ) 𝑍 ) = 𝑍 ) |
76 |
75
|
adantlr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐹 ‘ 𝑧 ) ( 2nd ‘ 𝑆 ) 𝑍 ) = 𝑍 ) |
77 |
68 71 76
|
3eqtrd |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ) = 𝑍 ) |
78 |
|
fvex |
⊢ ( 𝐹 ‘ ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ) ∈ V |
79 |
78
|
elsn |
⊢ ( ( 𝐹 ‘ ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ) ∈ { 𝑍 } ↔ ( 𝐹 ‘ ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ) = 𝑍 ) |
80 |
77 79
|
sylibr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ) ∈ { 𝑍 } ) |
81 |
|
elpreima |
⊢ ( 𝐹 Fn ran ( 1st ‘ 𝑅 ) → ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ) ∈ { 𝑍 } ) ) ) |
82 |
7 16 81
|
3syl |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ) ∈ { 𝑍 } ) ) ) |
83 |
82
|
ad2antrr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ) ∈ { 𝑍 } ) ) ) |
84 |
64 80 83
|
mpbir2and |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) |
85 |
4 59 5
|
rngocl |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
86 |
85
|
3expb |
⊢ ( ( 𝑅 ∈ RingOps ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
87 |
86
|
3ad2antl1 |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
88 |
87
|
anassrs |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
89 |
88
|
adantlrr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ran ( 1st ‘ 𝑅 ) ) |
90 |
4 5 59 65
|
rngohommul |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) ) |
91 |
90
|
anassrs |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) ) |
92 |
91
|
adantlrr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ) = ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) ) |
93 |
|
oveq1 |
⊢ ( ( 𝐹 ‘ 𝑥 ) = 𝑍 → ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝑍 ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) ) |
94 |
93
|
adantl |
⊢ ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) → ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝑍 ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) ) |
95 |
94
|
ad2antlr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝐹 ‘ 𝑥 ) ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) = ( 𝑍 ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) ) |
96 |
2 6 1 65
|
rngolz |
⊢ ( ( 𝑆 ∈ RingOps ∧ ( 𝐹 ‘ 𝑧 ) ∈ ran 𝐺 ) → ( 𝑍 ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) = 𝑍 ) |
97 |
96
|
3ad2antl2 |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝐹 ‘ 𝑧 ) ∈ ran 𝐺 ) → ( 𝑍 ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) = 𝑍 ) |
98 |
72 97
|
syldan |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝑍 ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) = 𝑍 ) |
99 |
98
|
adantlr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝑍 ( 2nd ‘ 𝑆 ) ( 𝐹 ‘ 𝑧 ) ) = 𝑍 ) |
100 |
92 95 99
|
3eqtrd |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ) = 𝑍 ) |
101 |
|
fvex |
⊢ ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ) ∈ V |
102 |
101
|
elsn |
⊢ ( ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ) ∈ { 𝑍 } ↔ ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ) = 𝑍 ) |
103 |
100 102
|
sylibr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ) ∈ { 𝑍 } ) |
104 |
|
elpreima |
⊢ ( 𝐹 Fn ran ( 1st ‘ 𝑅 ) → ( ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ) ∈ { 𝑍 } ) ) ) |
105 |
7 16 104
|
3syl |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ) ∈ { 𝑍 } ) ) ) |
106 |
105
|
ad2antrr |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ↔ ( ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ) ∈ { 𝑍 } ) ) ) |
107 |
89 103 106
|
mpbir2and |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) |
108 |
84 107
|
jca |
⊢ ( ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) ∧ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ) → ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) ) |
109 |
108
|
ralrimiva |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) ) → ∀ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) ) |
110 |
109
|
ex |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝑍 ) → ∀ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) ) ) |
111 |
58 110
|
syl5bi |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( 𝑥 ∈ ran ( 1st ‘ 𝑅 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ { 𝑍 } ) → ∀ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) ) ) |
112 |
49 111
|
sylbid |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( 𝑥 ∈ ( ◡ 𝐹 “ { 𝑍 } ) → ∀ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) ) ) |
113 |
112
|
imp |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) → ∀ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) ) |
114 |
57 113
|
jca |
⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) ∧ 𝑥 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) → ( ∀ 𝑦 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ∀ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) ) ) |
115 |
114
|
ralrimiva |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ∀ 𝑥 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ( ∀ 𝑦 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ∀ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) ) ) |
116 |
4 59 5 9
|
isidl |
⊢ ( 𝑅 ∈ RingOps → ( ( ◡ 𝐹 “ { 𝑍 } ) ∈ ( Idl ‘ 𝑅 ) ↔ ( ( ◡ 𝐹 “ { 𝑍 } ) ⊆ ran ( 1st ‘ 𝑅 ) ∧ ( GId ‘ ( 1st ‘ 𝑅 ) ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ∀ 𝑥 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ( ∀ 𝑦 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ∀ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) ) ) ) ) |
117 |
116
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ( ◡ 𝐹 “ { 𝑍 } ) ∈ ( Idl ‘ 𝑅 ) ↔ ( ( ◡ 𝐹 “ { 𝑍 } ) ⊆ ran ( 1st ‘ 𝑅 ) ∧ ( GId ‘ ( 1st ‘ 𝑅 ) ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ∀ 𝑥 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ( ∀ 𝑦 ∈ ( ◡ 𝐹 “ { 𝑍 } ) ( 𝑥 ( 1st ‘ 𝑅 ) 𝑦 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ∀ 𝑧 ∈ ran ( 1st ‘ 𝑅 ) ( ( 𝑧 ( 2nd ‘ 𝑅 ) 𝑥 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ∧ ( 𝑥 ( 2nd ‘ 𝑅 ) 𝑧 ) ∈ ( ◡ 𝐹 “ { 𝑍 } ) ) ) ) ) ) |
118 |
8 19 115 117
|
mpbir3and |
⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ) → ( ◡ 𝐹 “ { 𝑍 } ) ∈ ( Idl ‘ 𝑅 ) ) |