Step |
Hyp |
Ref |
Expression |
1 |
|
frlmup.f |
|- F = ( R freeLMod I ) |
2 |
|
frlmup.b |
|- B = ( Base ` F ) |
3 |
|
frlmup.c |
|- C = ( Base ` T ) |
4 |
|
frlmup.v |
|- .x. = ( .s ` T ) |
5 |
|
frlmup.e |
|- E = ( x e. B |-> ( T gsum ( x oF .x. A ) ) ) |
6 |
|
frlmup.t |
|- ( ph -> T e. LMod ) |
7 |
|
frlmup.i |
|- ( ph -> I e. X ) |
8 |
|
frlmup.r |
|- ( ph -> R = ( Scalar ` T ) ) |
9 |
|
frlmup.a |
|- ( ph -> A : I --> C ) |
10 |
|
frlmup.k |
|- K = ( LSpan ` T ) |
11 |
1 2 3 4 5 6 7 8 9
|
frlmup1 |
|- ( ph -> E e. ( F LMHom T ) ) |
12 |
|
eqid |
|- ( Scalar ` T ) = ( Scalar ` T ) |
13 |
12
|
lmodring |
|- ( T e. LMod -> ( Scalar ` T ) e. Ring ) |
14 |
6 13
|
syl |
|- ( ph -> ( Scalar ` T ) e. Ring ) |
15 |
8 14
|
eqeltrd |
|- ( ph -> R e. Ring ) |
16 |
|
eqid |
|- ( R unitVec I ) = ( R unitVec I ) |
17 |
16 1 2
|
uvcff |
|- ( ( R e. Ring /\ I e. X ) -> ( R unitVec I ) : I --> B ) |
18 |
15 7 17
|
syl2anc |
|- ( ph -> ( R unitVec I ) : I --> B ) |
19 |
18
|
frnd |
|- ( ph -> ran ( R unitVec I ) C_ B ) |
20 |
|
eqid |
|- ( LSpan ` F ) = ( LSpan ` F ) |
21 |
2 20 10
|
lmhmlsp |
|- ( ( E e. ( F LMHom T ) /\ ran ( R unitVec I ) C_ B ) -> ( E " ( ( LSpan ` F ) ` ran ( R unitVec I ) ) ) = ( K ` ( E " ran ( R unitVec I ) ) ) ) |
22 |
11 19 21
|
syl2anc |
|- ( ph -> ( E " ( ( LSpan ` F ) ` ran ( R unitVec I ) ) ) = ( K ` ( E " ran ( R unitVec I ) ) ) ) |
23 |
2 3
|
lmhmf |
|- ( E e. ( F LMHom T ) -> E : B --> C ) |
24 |
11 23
|
syl |
|- ( ph -> E : B --> C ) |
25 |
24
|
ffnd |
|- ( ph -> E Fn B ) |
26 |
|
fnima |
|- ( E Fn B -> ( E " B ) = ran E ) |
27 |
25 26
|
syl |
|- ( ph -> ( E " B ) = ran E ) |
28 |
|
eqid |
|- ( LBasis ` F ) = ( LBasis ` F ) |
29 |
1 16 28
|
frlmlbs |
|- ( ( R e. Ring /\ I e. X ) -> ran ( R unitVec I ) e. ( LBasis ` F ) ) |
30 |
15 7 29
|
syl2anc |
|- ( ph -> ran ( R unitVec I ) e. ( LBasis ` F ) ) |
31 |
2 28 20
|
lbssp |
|- ( ran ( R unitVec I ) e. ( LBasis ` F ) -> ( ( LSpan ` F ) ` ran ( R unitVec I ) ) = B ) |
32 |
30 31
|
syl |
|- ( ph -> ( ( LSpan ` F ) ` ran ( R unitVec I ) ) = B ) |
33 |
32
|
eqcomd |
|- ( ph -> B = ( ( LSpan ` F ) ` ran ( R unitVec I ) ) ) |
34 |
33
|
imaeq2d |
|- ( ph -> ( E " B ) = ( E " ( ( LSpan ` F ) ` ran ( R unitVec I ) ) ) ) |
35 |
27 34
|
eqtr3d |
|- ( ph -> ran E = ( E " ( ( LSpan ` F ) ` ran ( R unitVec I ) ) ) ) |
36 |
|
imaco |
|- ( ( E o. ( R unitVec I ) ) " I ) = ( E " ( ( R unitVec I ) " I ) ) |
37 |
9
|
ffnd |
|- ( ph -> A Fn I ) |
38 |
18
|
ffnd |
|- ( ph -> ( R unitVec I ) Fn I ) |
39 |
|
fnco |
|- ( ( E Fn B /\ ( R unitVec I ) Fn I /\ ran ( R unitVec I ) C_ B ) -> ( E o. ( R unitVec I ) ) Fn I ) |
40 |
25 38 19 39
|
syl3anc |
|- ( ph -> ( E o. ( R unitVec I ) ) Fn I ) |
41 |
|
fvco2 |
|- ( ( ( R unitVec I ) Fn I /\ u e. I ) -> ( ( E o. ( R unitVec I ) ) ` u ) = ( E ` ( ( R unitVec I ) ` u ) ) ) |
42 |
38 41
|
sylan |
|- ( ( ph /\ u e. I ) -> ( ( E o. ( R unitVec I ) ) ` u ) = ( E ` ( ( R unitVec I ) ` u ) ) ) |
43 |
6
|
adantr |
|- ( ( ph /\ u e. I ) -> T e. LMod ) |
44 |
7
|
adantr |
|- ( ( ph /\ u e. I ) -> I e. X ) |
45 |
8
|
adantr |
|- ( ( ph /\ u e. I ) -> R = ( Scalar ` T ) ) |
46 |
9
|
adantr |
|- ( ( ph /\ u e. I ) -> A : I --> C ) |
47 |
|
simpr |
|- ( ( ph /\ u e. I ) -> u e. I ) |
48 |
1 2 3 4 5 43 44 45 46 47 16
|
frlmup2 |
|- ( ( ph /\ u e. I ) -> ( E ` ( ( R unitVec I ) ` u ) ) = ( A ` u ) ) |
49 |
42 48
|
eqtr2d |
|- ( ( ph /\ u e. I ) -> ( A ` u ) = ( ( E o. ( R unitVec I ) ) ` u ) ) |
50 |
37 40 49
|
eqfnfvd |
|- ( ph -> A = ( E o. ( R unitVec I ) ) ) |
51 |
50
|
imaeq1d |
|- ( ph -> ( A " I ) = ( ( E o. ( R unitVec I ) ) " I ) ) |
52 |
|
fnima |
|- ( A Fn I -> ( A " I ) = ran A ) |
53 |
37 52
|
syl |
|- ( ph -> ( A " I ) = ran A ) |
54 |
51 53
|
eqtr3d |
|- ( ph -> ( ( E o. ( R unitVec I ) ) " I ) = ran A ) |
55 |
|
fnima |
|- ( ( R unitVec I ) Fn I -> ( ( R unitVec I ) " I ) = ran ( R unitVec I ) ) |
56 |
38 55
|
syl |
|- ( ph -> ( ( R unitVec I ) " I ) = ran ( R unitVec I ) ) |
57 |
56
|
imaeq2d |
|- ( ph -> ( E " ( ( R unitVec I ) " I ) ) = ( E " ran ( R unitVec I ) ) ) |
58 |
36 54 57
|
3eqtr3a |
|- ( ph -> ran A = ( E " ran ( R unitVec I ) ) ) |
59 |
58
|
fveq2d |
|- ( ph -> ( K ` ran A ) = ( K ` ( E " ran ( R unitVec I ) ) ) ) |
60 |
22 35 59
|
3eqtr4d |
|- ( ph -> ran E = ( K ` ran A ) ) |