Step |
Hyp |
Ref |
Expression |
1 |
|
rmfsuppf2.r |
|- R = ( Base ` M ) |
2 |
|
rmfsupp2.m |
|- ( ph -> M e. Ring ) |
3 |
|
rmfsupp2.v |
|- ( ph -> V e. X ) |
4 |
|
rmfsupp2.c |
|- ( ( ph /\ v e. V ) -> C e. R ) |
5 |
|
rmfsupp2.a |
|- ( ph -> A : V --> R ) |
6 |
|
rmfsupp2.1 |
|- ( ph -> A finSupp ( 0g ` M ) ) |
7 |
|
funmpt |
|- Fun ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) |
8 |
7
|
a1i |
|- ( ph -> Fun ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) ) |
9 |
3
|
mptexd |
|- ( ph -> ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) e. _V ) |
10 |
|
ringgrp |
|- ( M e. Ring -> M e. Grp ) |
11 |
|
eqid |
|- ( 0g ` M ) = ( 0g ` M ) |
12 |
1 11
|
grpidcl |
|- ( M e. Grp -> ( 0g ` M ) e. R ) |
13 |
2 10 12
|
3syl |
|- ( ph -> ( 0g ` M ) e. R ) |
14 |
|
suppval1 |
|- ( ( Fun ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) /\ ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) e. _V /\ ( 0g ` M ) e. R ) -> ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) supp ( 0g ` M ) ) = { u e. dom ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) | ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) ` u ) =/= ( 0g ` M ) } ) |
15 |
8 9 13 14
|
syl3anc |
|- ( ph -> ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) supp ( 0g ` M ) ) = { u e. dom ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) | ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) ` u ) =/= ( 0g ` M ) } ) |
16 |
|
ovex |
|- ( ( A ` v ) ( .r ` M ) C ) e. _V |
17 |
|
eqid |
|- ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) = ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) |
18 |
16 17
|
dmmpti |
|- dom ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) = V |
19 |
18
|
a1i |
|- ( ph -> dom ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) = V ) |
20 |
|
ovex |
|- ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) e. _V |
21 |
|
nfcv |
|- F/_ v u |
22 |
|
nfcv |
|- F/_ v ( A ` u ) |
23 |
|
nfcv |
|- F/_ v ( .r ` M ) |
24 |
|
nfcsb1v |
|- F/_ v [_ u / v ]_ C |
25 |
22 23 24
|
nfov |
|- F/_ v ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) |
26 |
|
fveq2 |
|- ( v = u -> ( A ` v ) = ( A ` u ) ) |
27 |
|
csbeq1a |
|- ( v = u -> C = [_ u / v ]_ C ) |
28 |
26 27
|
oveq12d |
|- ( v = u -> ( ( A ` v ) ( .r ` M ) C ) = ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) ) |
29 |
21 25 28 17
|
fvmptf |
|- ( ( u e. V /\ ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) e. _V ) -> ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) ` u ) = ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) ) |
30 |
20 29
|
mpan2 |
|- ( u e. V -> ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) ` u ) = ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) ) |
31 |
30 18
|
eleq2s |
|- ( u e. dom ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) -> ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) ` u ) = ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) ) |
32 |
31
|
adantl |
|- ( ( ph /\ u e. dom ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) ) -> ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) ` u ) = ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) ) |
33 |
32
|
neeq1d |
|- ( ( ph /\ u e. dom ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) ) -> ( ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) ` u ) =/= ( 0g ` M ) <-> ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) =/= ( 0g ` M ) ) ) |
34 |
19 33
|
rabeqbidva |
|- ( ph -> { u e. dom ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) | ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) ` u ) =/= ( 0g ` M ) } = { u e. V | ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) =/= ( 0g ` M ) } ) |
35 |
5
|
fdmd |
|- ( ph -> dom A = V ) |
36 |
35
|
rabeqdv |
|- ( ph -> { u e. dom A | ( A ` u ) =/= ( 0g ` M ) } = { u e. V | ( A ` u ) =/= ( 0g ` M ) } ) |
37 |
5
|
ffund |
|- ( ph -> Fun A ) |
38 |
1
|
fvexi |
|- R e. _V |
39 |
38
|
a1i |
|- ( ph -> R e. _V ) |
40 |
39 3
|
elmapd |
|- ( ph -> ( A e. ( R ^m V ) <-> A : V --> R ) ) |
41 |
5 40
|
mpbird |
|- ( ph -> A e. ( R ^m V ) ) |
42 |
|
suppval1 |
|- ( ( Fun A /\ A e. ( R ^m V ) /\ ( 0g ` M ) e. R ) -> ( A supp ( 0g ` M ) ) = { u e. dom A | ( A ` u ) =/= ( 0g ` M ) } ) |
43 |
37 41 13 42
|
syl3anc |
|- ( ph -> ( A supp ( 0g ` M ) ) = { u e. dom A | ( A ` u ) =/= ( 0g ` M ) } ) |
44 |
6
|
fsuppimpd |
|- ( ph -> ( A supp ( 0g ` M ) ) e. Fin ) |
45 |
43 44
|
eqeltrrd |
|- ( ph -> { u e. dom A | ( A ` u ) =/= ( 0g ` M ) } e. Fin ) |
46 |
36 45
|
eqeltrrd |
|- ( ph -> { u e. V | ( A ` u ) =/= ( 0g ` M ) } e. Fin ) |
47 |
|
simpr |
|- ( ( ( ph /\ u e. V ) /\ ( A ` u ) = ( 0g ` M ) ) -> ( A ` u ) = ( 0g ` M ) ) |
48 |
47
|
oveq1d |
|- ( ( ( ph /\ u e. V ) /\ ( A ` u ) = ( 0g ` M ) ) -> ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) = ( ( 0g ` M ) ( .r ` M ) [_ u / v ]_ C ) ) |
49 |
2
|
ad2antrr |
|- ( ( ( ph /\ u e. V ) /\ ( A ` u ) = ( 0g ` M ) ) -> M e. Ring ) |
50 |
|
simplr |
|- ( ( ( ph /\ u e. V ) /\ ( A ` u ) = ( 0g ` M ) ) -> u e. V ) |
51 |
4
|
ralrimiva |
|- ( ph -> A. v e. V C e. R ) |
52 |
51
|
ad2antrr |
|- ( ( ( ph /\ u e. V ) /\ ( A ` u ) = ( 0g ` M ) ) -> A. v e. V C e. R ) |
53 |
|
rspcsbela |
|- ( ( u e. V /\ A. v e. V C e. R ) -> [_ u / v ]_ C e. R ) |
54 |
50 52 53
|
syl2anc |
|- ( ( ( ph /\ u e. V ) /\ ( A ` u ) = ( 0g ` M ) ) -> [_ u / v ]_ C e. R ) |
55 |
|
eqid |
|- ( .r ` M ) = ( .r ` M ) |
56 |
1 55 11
|
ringlz |
|- ( ( M e. Ring /\ [_ u / v ]_ C e. R ) -> ( ( 0g ` M ) ( .r ` M ) [_ u / v ]_ C ) = ( 0g ` M ) ) |
57 |
49 54 56
|
syl2anc |
|- ( ( ( ph /\ u e. V ) /\ ( A ` u ) = ( 0g ` M ) ) -> ( ( 0g ` M ) ( .r ` M ) [_ u / v ]_ C ) = ( 0g ` M ) ) |
58 |
48 57
|
eqtrd |
|- ( ( ( ph /\ u e. V ) /\ ( A ` u ) = ( 0g ` M ) ) -> ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) = ( 0g ` M ) ) |
59 |
58
|
ex |
|- ( ( ph /\ u e. V ) -> ( ( A ` u ) = ( 0g ` M ) -> ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) = ( 0g ` M ) ) ) |
60 |
59
|
necon3d |
|- ( ( ph /\ u e. V ) -> ( ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) =/= ( 0g ` M ) -> ( A ` u ) =/= ( 0g ` M ) ) ) |
61 |
60
|
ss2rabdv |
|- ( ph -> { u e. V | ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) =/= ( 0g ` M ) } C_ { u e. V | ( A ` u ) =/= ( 0g ` M ) } ) |
62 |
|
ssfi |
|- ( ( { u e. V | ( A ` u ) =/= ( 0g ` M ) } e. Fin /\ { u e. V | ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) =/= ( 0g ` M ) } C_ { u e. V | ( A ` u ) =/= ( 0g ` M ) } ) -> { u e. V | ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) =/= ( 0g ` M ) } e. Fin ) |
63 |
46 61 62
|
syl2anc |
|- ( ph -> { u e. V | ( ( A ` u ) ( .r ` M ) [_ u / v ]_ C ) =/= ( 0g ` M ) } e. Fin ) |
64 |
34 63
|
eqeltrd |
|- ( ph -> { u e. dom ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) | ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) ` u ) =/= ( 0g ` M ) } e. Fin ) |
65 |
15 64
|
eqeltrd |
|- ( ph -> ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) supp ( 0g ` M ) ) e. Fin ) |
66 |
|
isfsupp |
|- ( ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) e. _V /\ ( 0g ` M ) e. R ) -> ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) finSupp ( 0g ` M ) <-> ( Fun ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) /\ ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) supp ( 0g ` M ) ) e. Fin ) ) ) |
67 |
9 13 66
|
syl2anc |
|- ( ph -> ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) finSupp ( 0g ` M ) <-> ( Fun ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) /\ ( ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) supp ( 0g ` M ) ) e. Fin ) ) ) |
68 |
8 65 67
|
mpbir2and |
|- ( ph -> ( v e. V |-> ( ( A ` v ) ( .r ` M ) C ) ) finSupp ( 0g ` M ) ) |