Step |
Hyp |
Ref |
Expression |
1 |
|
rhmpreimaprmidl.p |
|- P = ( PrmIdeal ` R ) |
2 |
|
rhmrcl1 |
|- ( F e. ( R RingHom S ) -> R e. Ring ) |
3 |
2
|
ad2antlr |
|- ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) -> R e. Ring ) |
4 |
|
rhmrcl2 |
|- ( F e. ( R RingHom S ) -> S e. Ring ) |
5 |
|
prmidlidl |
|- ( ( S e. Ring /\ J e. ( PrmIdeal ` S ) ) -> J e. ( LIdeal ` S ) ) |
6 |
4 5
|
sylan |
|- ( ( F e. ( R RingHom S ) /\ J e. ( PrmIdeal ` S ) ) -> J e. ( LIdeal ` S ) ) |
7 |
|
eqid |
|- ( LIdeal ` R ) = ( LIdeal ` R ) |
8 |
7
|
rhmpreimaidl |
|- ( ( F e. ( R RingHom S ) /\ J e. ( LIdeal ` S ) ) -> ( `' F " J ) e. ( LIdeal ` R ) ) |
9 |
6 8
|
syldan |
|- ( ( F e. ( R RingHom S ) /\ J e. ( PrmIdeal ` S ) ) -> ( `' F " J ) e. ( LIdeal ` R ) ) |
10 |
9
|
adantll |
|- ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) -> ( `' F " J ) e. ( LIdeal ` R ) ) |
11 |
4
|
adantr |
|- ( ( F e. ( R RingHom S ) /\ J e. ( PrmIdeal ` S ) ) -> S e. Ring ) |
12 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
13 |
|
eqid |
|- ( .r ` S ) = ( .r ` S ) |
14 |
12 13
|
prmidlnr |
|- ( ( S e. Ring /\ J e. ( PrmIdeal ` S ) ) -> J =/= ( Base ` S ) ) |
15 |
4 14
|
sylan |
|- ( ( F e. ( R RingHom S ) /\ J e. ( PrmIdeal ` S ) ) -> J =/= ( Base ` S ) ) |
16 |
|
eqid |
|- ( 1r ` S ) = ( 1r ` S ) |
17 |
12 16
|
pridln1 |
|- ( ( S e. Ring /\ J e. ( LIdeal ` S ) /\ J =/= ( Base ` S ) ) -> -. ( 1r ` S ) e. J ) |
18 |
11 6 15 17
|
syl3anc |
|- ( ( F e. ( R RingHom S ) /\ J e. ( PrmIdeal ` S ) ) -> -. ( 1r ` S ) e. J ) |
19 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
20 |
19 16
|
rhm1 |
|- ( F e. ( R RingHom S ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) ) |
21 |
20
|
ad2antrr |
|- ( ( ( F e. ( R RingHom S ) /\ J e. ( PrmIdeal ` S ) ) /\ ( `' F " J ) = ( Base ` R ) ) -> ( F ` ( 1r ` R ) ) = ( 1r ` S ) ) |
22 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
23 |
22 12
|
rhmf |
|- ( F e. ( R RingHom S ) -> F : ( Base ` R ) --> ( Base ` S ) ) |
24 |
23
|
ffnd |
|- ( F e. ( R RingHom S ) -> F Fn ( Base ` R ) ) |
25 |
24
|
ad2antrr |
|- ( ( ( F e. ( R RingHom S ) /\ J e. ( PrmIdeal ` S ) ) /\ ( `' F " J ) = ( Base ` R ) ) -> F Fn ( Base ` R ) ) |
26 |
22 19
|
ringidcl |
|- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) ) |
27 |
2 26
|
syl |
|- ( F e. ( R RingHom S ) -> ( 1r ` R ) e. ( Base ` R ) ) |
28 |
27
|
ad2antrr |
|- ( ( ( F e. ( R RingHom S ) /\ J e. ( PrmIdeal ` S ) ) /\ ( `' F " J ) = ( Base ` R ) ) -> ( 1r ` R ) e. ( Base ` R ) ) |
29 |
|
simpr |
|- ( ( ( F e. ( R RingHom S ) /\ J e. ( PrmIdeal ` S ) ) /\ ( `' F " J ) = ( Base ` R ) ) -> ( `' F " J ) = ( Base ` R ) ) |
30 |
28 29
|
eleqtrrd |
|- ( ( ( F e. ( R RingHom S ) /\ J e. ( PrmIdeal ` S ) ) /\ ( `' F " J ) = ( Base ` R ) ) -> ( 1r ` R ) e. ( `' F " J ) ) |
31 |
|
elpreima |
|- ( F Fn ( Base ` R ) -> ( ( 1r ` R ) e. ( `' F " J ) <-> ( ( 1r ` R ) e. ( Base ` R ) /\ ( F ` ( 1r ` R ) ) e. J ) ) ) |
32 |
31
|
biimpa |
|- ( ( F Fn ( Base ` R ) /\ ( 1r ` R ) e. ( `' F " J ) ) -> ( ( 1r ` R ) e. ( Base ` R ) /\ ( F ` ( 1r ` R ) ) e. J ) ) |
33 |
25 30 32
|
syl2anc |
|- ( ( ( F e. ( R RingHom S ) /\ J e. ( PrmIdeal ` S ) ) /\ ( `' F " J ) = ( Base ` R ) ) -> ( ( 1r ` R ) e. ( Base ` R ) /\ ( F ` ( 1r ` R ) ) e. J ) ) |
34 |
33
|
simprd |
|- ( ( ( F e. ( R RingHom S ) /\ J e. ( PrmIdeal ` S ) ) /\ ( `' F " J ) = ( Base ` R ) ) -> ( F ` ( 1r ` R ) ) e. J ) |
35 |
21 34
|
eqeltrrd |
|- ( ( ( F e. ( R RingHom S ) /\ J e. ( PrmIdeal ` S ) ) /\ ( `' F " J ) = ( Base ` R ) ) -> ( 1r ` S ) e. J ) |
36 |
18 35
|
mtand |
|- ( ( F e. ( R RingHom S ) /\ J e. ( PrmIdeal ` S ) ) -> -. ( `' F " J ) = ( Base ` R ) ) |
37 |
36
|
neqned |
|- ( ( F e. ( R RingHom S ) /\ J e. ( PrmIdeal ` S ) ) -> ( `' F " J ) =/= ( Base ` R ) ) |
38 |
37
|
adantll |
|- ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) -> ( `' F " J ) =/= ( Base ` R ) ) |
39 |
|
simp-5l |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> S e. CRing ) |
40 |
|
simp-4r |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> J e. ( PrmIdeal ` S ) ) |
41 |
|
simp-5r |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> F e. ( R RingHom S ) ) |
42 |
41 23
|
syl |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> F : ( Base ` R ) --> ( Base ` S ) ) |
43 |
|
simpllr |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> a e. ( Base ` R ) ) |
44 |
42 43
|
ffvelrnd |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> ( F ` a ) e. ( Base ` S ) ) |
45 |
|
simplr |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> b e. ( Base ` R ) ) |
46 |
42 45
|
ffvelrnd |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> ( F ` b ) e. ( Base ` S ) ) |
47 |
|
eqid |
|- ( .r ` R ) = ( .r ` R ) |
48 |
22 47 13
|
rhmmul |
|- ( ( F e. ( R RingHom S ) /\ a e. ( Base ` R ) /\ b e. ( Base ` R ) ) -> ( F ` ( a ( .r ` R ) b ) ) = ( ( F ` a ) ( .r ` S ) ( F ` b ) ) ) |
49 |
41 43 45 48
|
syl3anc |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> ( F ` ( a ( .r ` R ) b ) ) = ( ( F ` a ) ( .r ` S ) ( F ` b ) ) ) |
50 |
24
|
ad5antlr |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> F Fn ( Base ` R ) ) |
51 |
|
simpr |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> ( a ( .r ` R ) b ) e. ( `' F " J ) ) |
52 |
|
elpreima |
|- ( F Fn ( Base ` R ) -> ( ( a ( .r ` R ) b ) e. ( `' F " J ) <-> ( ( a ( .r ` R ) b ) e. ( Base ` R ) /\ ( F ` ( a ( .r ` R ) b ) ) e. J ) ) ) |
53 |
52
|
simplbda |
|- ( ( F Fn ( Base ` R ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> ( F ` ( a ( .r ` R ) b ) ) e. J ) |
54 |
50 51 53
|
syl2anc |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> ( F ` ( a ( .r ` R ) b ) ) e. J ) |
55 |
49 54
|
eqeltrrd |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> ( ( F ` a ) ( .r ` S ) ( F ` b ) ) e. J ) |
56 |
12 13
|
prmidlc |
|- ( ( ( S e. CRing /\ J e. ( PrmIdeal ` S ) ) /\ ( ( F ` a ) e. ( Base ` S ) /\ ( F ` b ) e. ( Base ` S ) /\ ( ( F ` a ) ( .r ` S ) ( F ` b ) ) e. J ) ) -> ( ( F ` a ) e. J \/ ( F ` b ) e. J ) ) |
57 |
39 40 44 46 55 56
|
syl23anc |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> ( ( F ` a ) e. J \/ ( F ` b ) e. J ) ) |
58 |
50
|
adantr |
|- ( ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) /\ ( F ` a ) e. J ) -> F Fn ( Base ` R ) ) |
59 |
43
|
adantr |
|- ( ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) /\ ( F ` a ) e. J ) -> a e. ( Base ` R ) ) |
60 |
|
simpr |
|- ( ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) /\ ( F ` a ) e. J ) -> ( F ` a ) e. J ) |
61 |
58 59 60
|
elpreimad |
|- ( ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) /\ ( F ` a ) e. J ) -> a e. ( `' F " J ) ) |
62 |
61
|
ex |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> ( ( F ` a ) e. J -> a e. ( `' F " J ) ) ) |
63 |
50
|
adantr |
|- ( ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) /\ ( F ` b ) e. J ) -> F Fn ( Base ` R ) ) |
64 |
|
simpllr |
|- ( ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) /\ ( F ` b ) e. J ) -> b e. ( Base ` R ) ) |
65 |
|
simpr |
|- ( ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) /\ ( F ` b ) e. J ) -> ( F ` b ) e. J ) |
66 |
63 64 65
|
elpreimad |
|- ( ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) /\ ( F ` b ) e. J ) -> b e. ( `' F " J ) ) |
67 |
66
|
ex |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> ( ( F ` b ) e. J -> b e. ( `' F " J ) ) ) |
68 |
62 67
|
orim12d |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> ( ( ( F ` a ) e. J \/ ( F ` b ) e. J ) -> ( a e. ( `' F " J ) \/ b e. ( `' F " J ) ) ) ) |
69 |
57 68
|
mpd |
|- ( ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) /\ ( a ( .r ` R ) b ) e. ( `' F " J ) ) -> ( a e. ( `' F " J ) \/ b e. ( `' F " J ) ) ) |
70 |
69
|
ex |
|- ( ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ a e. ( Base ` R ) ) /\ b e. ( Base ` R ) ) -> ( ( a ( .r ` R ) b ) e. ( `' F " J ) -> ( a e. ( `' F " J ) \/ b e. ( `' F " J ) ) ) ) |
71 |
70
|
anasss |
|- ( ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) /\ ( a e. ( Base ` R ) /\ b e. ( Base ` R ) ) ) -> ( ( a ( .r ` R ) b ) e. ( `' F " J ) -> ( a e. ( `' F " J ) \/ b e. ( `' F " J ) ) ) ) |
72 |
71
|
ralrimivva |
|- ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) -> A. a e. ( Base ` R ) A. b e. ( Base ` R ) ( ( a ( .r ` R ) b ) e. ( `' F " J ) -> ( a e. ( `' F " J ) \/ b e. ( `' F " J ) ) ) ) |
73 |
22 47
|
prmidl2 |
|- ( ( ( R e. Ring /\ ( `' F " J ) e. ( LIdeal ` R ) ) /\ ( ( `' F " J ) =/= ( Base ` R ) /\ A. a e. ( Base ` R ) A. b e. ( Base ` R ) ( ( a ( .r ` R ) b ) e. ( `' F " J ) -> ( a e. ( `' F " J ) \/ b e. ( `' F " J ) ) ) ) ) -> ( `' F " J ) e. ( PrmIdeal ` R ) ) |
74 |
3 10 38 72 73
|
syl22anc |
|- ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) -> ( `' F " J ) e. ( PrmIdeal ` R ) ) |
75 |
74 1
|
eleqtrrdi |
|- ( ( ( S e. CRing /\ F e. ( R RingHom S ) ) /\ J e. ( PrmIdeal ` S ) ) -> ( `' F " J ) e. P ) |