Step |
Hyp |
Ref |
Expression |
1 |
|
frlmplusgvalb.f |
|- F = ( R freeLMod I ) |
2 |
|
frlmplusgvalb.b |
|- B = ( Base ` F ) |
3 |
|
frlmplusgvalb.i |
|- ( ph -> I e. W ) |
4 |
|
frlmplusgvalb.x |
|- ( ph -> X e. B ) |
5 |
|
frlmplusgvalb.z |
|- ( ph -> Z e. B ) |
6 |
|
frlmplusgvalb.r |
|- ( ph -> R e. Ring ) |
7 |
|
frlmvscavalb.k |
|- K = ( Base ` R ) |
8 |
|
frlmvscavalb.a |
|- ( ph -> A e. K ) |
9 |
|
frlmvscavalb.v |
|- .xb = ( .s ` F ) |
10 |
|
frlmvscavalb.t |
|- .x. = ( .r ` R ) |
11 |
1 7 2
|
frlmbasmap |
|- ( ( I e. W /\ Z e. B ) -> Z e. ( K ^m I ) ) |
12 |
3 5 11
|
syl2anc |
|- ( ph -> Z e. ( K ^m I ) ) |
13 |
7
|
fvexi |
|- K e. _V |
14 |
13
|
a1i |
|- ( ph -> K e. _V ) |
15 |
14 3
|
elmapd |
|- ( ph -> ( Z e. ( K ^m I ) <-> Z : I --> K ) ) |
16 |
12 15
|
mpbid |
|- ( ph -> Z : I --> K ) |
17 |
16
|
ffnd |
|- ( ph -> Z Fn I ) |
18 |
1
|
frlmlmod |
|- ( ( R e. Ring /\ I e. W ) -> F e. LMod ) |
19 |
6 3 18
|
syl2anc |
|- ( ph -> F e. LMod ) |
20 |
8 7
|
eleqtrdi |
|- ( ph -> A e. ( Base ` R ) ) |
21 |
1
|
frlmsca |
|- ( ( R e. Ring /\ I e. W ) -> R = ( Scalar ` F ) ) |
22 |
6 3 21
|
syl2anc |
|- ( ph -> R = ( Scalar ` F ) ) |
23 |
22
|
fveq2d |
|- ( ph -> ( Base ` R ) = ( Base ` ( Scalar ` F ) ) ) |
24 |
20 23
|
eleqtrd |
|- ( ph -> A e. ( Base ` ( Scalar ` F ) ) ) |
25 |
|
eqid |
|- ( Scalar ` F ) = ( Scalar ` F ) |
26 |
|
eqid |
|- ( Base ` ( Scalar ` F ) ) = ( Base ` ( Scalar ` F ) ) |
27 |
2 25 9 26
|
lmodvscl |
|- ( ( F e. LMod /\ A e. ( Base ` ( Scalar ` F ) ) /\ X e. B ) -> ( A .xb X ) e. B ) |
28 |
19 24 4 27
|
syl3anc |
|- ( ph -> ( A .xb X ) e. B ) |
29 |
1 7 2
|
frlmbasmap |
|- ( ( I e. W /\ ( A .xb X ) e. B ) -> ( A .xb X ) e. ( K ^m I ) ) |
30 |
3 28 29
|
syl2anc |
|- ( ph -> ( A .xb X ) e. ( K ^m I ) ) |
31 |
14 3
|
elmapd |
|- ( ph -> ( ( A .xb X ) e. ( K ^m I ) <-> ( A .xb X ) : I --> K ) ) |
32 |
30 31
|
mpbid |
|- ( ph -> ( A .xb X ) : I --> K ) |
33 |
32
|
ffnd |
|- ( ph -> ( A .xb X ) Fn I ) |
34 |
|
eqfnfv |
|- ( ( Z Fn I /\ ( A .xb X ) Fn I ) -> ( Z = ( A .xb X ) <-> A. i e. I ( Z ` i ) = ( ( A .xb X ) ` i ) ) ) |
35 |
17 33 34
|
syl2anc |
|- ( ph -> ( Z = ( A .xb X ) <-> A. i e. I ( Z ` i ) = ( ( A .xb X ) ` i ) ) ) |
36 |
3
|
adantr |
|- ( ( ph /\ i e. I ) -> I e. W ) |
37 |
8
|
adantr |
|- ( ( ph /\ i e. I ) -> A e. K ) |
38 |
4
|
adantr |
|- ( ( ph /\ i e. I ) -> X e. B ) |
39 |
|
simpr |
|- ( ( ph /\ i e. I ) -> i e. I ) |
40 |
1 2 7 36 37 38 39 9 10
|
frlmvscaval |
|- ( ( ph /\ i e. I ) -> ( ( A .xb X ) ` i ) = ( A .x. ( X ` i ) ) ) |
41 |
40
|
eqeq2d |
|- ( ( ph /\ i e. I ) -> ( ( Z ` i ) = ( ( A .xb X ) ` i ) <-> ( Z ` i ) = ( A .x. ( X ` i ) ) ) ) |
42 |
41
|
ralbidva |
|- ( ph -> ( A. i e. I ( Z ` i ) = ( ( A .xb X ) ` i ) <-> A. i e. I ( Z ` i ) = ( A .x. ( X ` i ) ) ) ) |
43 |
35 42
|
bitrd |
|- ( ph -> ( Z = ( A .xb X ) <-> A. i e. I ( Z ` i ) = ( A .x. ( X ` i ) ) ) ) |