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