Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- ( Base ` Z ) = ( Base ` Z ) |
2 |
|
eqid |
|- ( 0g ` Z ) = ( 0g ` Z ) |
3 |
|
eqid |
|- ( 1r ` Z ) = ( 1r ` Z ) |
4 |
1 2 3
|
0ring1eq0 |
|- ( Z e. ( Ring \ NzRing ) -> ( 1r ` Z ) = ( 0g ` Z ) ) |
5 |
4
|
adantr |
|- ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) -> ( 1r ` Z ) = ( 0g ` Z ) ) |
6 |
5
|
adantr |
|- ( ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) /\ f e. ( Z RingHom R ) ) -> ( 1r ` Z ) = ( 0g ` Z ) ) |
7 |
6
|
eqcomd |
|- ( ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) /\ f e. ( Z RingHom R ) ) -> ( 0g ` Z ) = ( 1r ` Z ) ) |
8 |
7
|
fveq2d |
|- ( ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) /\ f e. ( Z RingHom R ) ) -> ( f ` ( 0g ` Z ) ) = ( f ` ( 1r ` Z ) ) ) |
9 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
10 |
3 9
|
rhm1 |
|- ( f e. ( Z RingHom R ) -> ( f ` ( 1r ` Z ) ) = ( 1r ` R ) ) |
11 |
10
|
adantl |
|- ( ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) /\ f e. ( Z RingHom R ) ) -> ( f ` ( 1r ` Z ) ) = ( 1r ` R ) ) |
12 |
8 11
|
eqtrd |
|- ( ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) /\ f e. ( Z RingHom R ) ) -> ( f ` ( 0g ` Z ) ) = ( 1r ` R ) ) |
13 |
|
rhmghm |
|- ( f e. ( Z RingHom R ) -> f e. ( Z GrpHom R ) ) |
14 |
13
|
adantl |
|- ( ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) /\ f e. ( Z RingHom R ) ) -> f e. ( Z GrpHom R ) ) |
15 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
16 |
2 15
|
ghmid |
|- ( f e. ( Z GrpHom R ) -> ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) |
17 |
14 16
|
syl |
|- ( ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) /\ f e. ( Z RingHom R ) ) -> ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) |
18 |
12 17
|
jca |
|- ( ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) /\ f e. ( Z RingHom R ) ) -> ( ( f ` ( 0g ` Z ) ) = ( 1r ` R ) /\ ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) ) |
19 |
18
|
ralrimiva |
|- ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) -> A. f e. ( Z RingHom R ) ( ( f ` ( 0g ` Z ) ) = ( 1r ` R ) /\ ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) ) |
20 |
9 15
|
nzrnz |
|- ( R e. NzRing -> ( 1r ` R ) =/= ( 0g ` R ) ) |
21 |
20
|
necomd |
|- ( R e. NzRing -> ( 0g ` R ) =/= ( 1r ` R ) ) |
22 |
21
|
adantl |
|- ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) -> ( 0g ` R ) =/= ( 1r ` R ) ) |
23 |
22
|
adantr |
|- ( ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) /\ ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) -> ( 0g ` R ) =/= ( 1r ` R ) ) |
24 |
|
neeq1 |
|- ( ( f ` ( 0g ` Z ) ) = ( 0g ` R ) -> ( ( f ` ( 0g ` Z ) ) =/= ( 1r ` R ) <-> ( 0g ` R ) =/= ( 1r ` R ) ) ) |
25 |
24
|
adantl |
|- ( ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) /\ ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) -> ( ( f ` ( 0g ` Z ) ) =/= ( 1r ` R ) <-> ( 0g ` R ) =/= ( 1r ` R ) ) ) |
26 |
23 25
|
mpbird |
|- ( ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) /\ ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) -> ( f ` ( 0g ` Z ) ) =/= ( 1r ` R ) ) |
27 |
26
|
orcd |
|- ( ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) /\ ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) -> ( ( f ` ( 0g ` Z ) ) =/= ( 1r ` R ) \/ ( f ` ( 0g ` Z ) ) =/= ( 0g ` R ) ) ) |
28 |
27
|
expcom |
|- ( ( f ` ( 0g ` Z ) ) = ( 0g ` R ) -> ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) -> ( ( f ` ( 0g ` Z ) ) =/= ( 1r ` R ) \/ ( f ` ( 0g ` Z ) ) =/= ( 0g ` R ) ) ) ) |
29 |
|
olc |
|- ( ( f ` ( 0g ` Z ) ) =/= ( 0g ` R ) -> ( ( f ` ( 0g ` Z ) ) =/= ( 1r ` R ) \/ ( f ` ( 0g ` Z ) ) =/= ( 0g ` R ) ) ) |
30 |
29
|
a1d |
|- ( ( f ` ( 0g ` Z ) ) =/= ( 0g ` R ) -> ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) -> ( ( f ` ( 0g ` Z ) ) =/= ( 1r ` R ) \/ ( f ` ( 0g ` Z ) ) =/= ( 0g ` R ) ) ) ) |
31 |
28 30
|
pm2.61ine |
|- ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) -> ( ( f ` ( 0g ` Z ) ) =/= ( 1r ` R ) \/ ( f ` ( 0g ` Z ) ) =/= ( 0g ` R ) ) ) |
32 |
|
neorian |
|- ( ( ( f ` ( 0g ` Z ) ) =/= ( 1r ` R ) \/ ( f ` ( 0g ` Z ) ) =/= ( 0g ` R ) ) <-> -. ( ( f ` ( 0g ` Z ) ) = ( 1r ` R ) /\ ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) ) |
33 |
31 32
|
sylib |
|- ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) -> -. ( ( f ` ( 0g ` Z ) ) = ( 1r ` R ) /\ ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) ) |
34 |
|
con3 |
|- ( ( f e. ( Z RingHom R ) -> ( ( f ` ( 0g ` Z ) ) = ( 1r ` R ) /\ ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) ) -> ( -. ( ( f ` ( 0g ` Z ) ) = ( 1r ` R ) /\ ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) -> -. f e. ( Z RingHom R ) ) ) |
35 |
33 34
|
syl5com |
|- ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) -> ( ( f e. ( Z RingHom R ) -> ( ( f ` ( 0g ` Z ) ) = ( 1r ` R ) /\ ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) ) -> -. f e. ( Z RingHom R ) ) ) |
36 |
35
|
alimdv |
|- ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) -> ( A. f ( f e. ( Z RingHom R ) -> ( ( f ` ( 0g ` Z ) ) = ( 1r ` R ) /\ ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) ) -> A. f -. f e. ( Z RingHom R ) ) ) |
37 |
|
df-ral |
|- ( A. f e. ( Z RingHom R ) ( ( f ` ( 0g ` Z ) ) = ( 1r ` R ) /\ ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) <-> A. f ( f e. ( Z RingHom R ) -> ( ( f ` ( 0g ` Z ) ) = ( 1r ` R ) /\ ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) ) ) |
38 |
|
eq0 |
|- ( ( Z RingHom R ) = (/) <-> A. f -. f e. ( Z RingHom R ) ) |
39 |
36 37 38
|
3imtr4g |
|- ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) -> ( A. f e. ( Z RingHom R ) ( ( f ` ( 0g ` Z ) ) = ( 1r ` R ) /\ ( f ` ( 0g ` Z ) ) = ( 0g ` R ) ) -> ( Z RingHom R ) = (/) ) ) |
40 |
19 39
|
mpd |
|- ( ( Z e. ( Ring \ NzRing ) /\ R e. NzRing ) -> ( Z RingHom R ) = (/) ) |