Step |
Hyp |
Ref |
Expression |
1 |
|
dochfl1.h |
|- H = ( LHyp ` K ) |
2 |
|
dochfl1.o |
|- ._|_ = ( ( ocH ` K ) ` W ) |
3 |
|
dochfl1.u |
|- U = ( ( DVecH ` K ) ` W ) |
4 |
|
dochfl1.v |
|- V = ( Base ` U ) |
5 |
|
dochfl1.a |
|- .+ = ( +g ` U ) |
6 |
|
dochfl1.t |
|- .x. = ( .s ` U ) |
7 |
|
dochfl1.z |
|- .0. = ( 0g ` U ) |
8 |
|
dochfl1.d |
|- D = ( Scalar ` U ) |
9 |
|
dochfl1.r |
|- R = ( Base ` D ) |
10 |
|
dochfl1.i |
|- .1. = ( 1r ` D ) |
11 |
|
dochfl1.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
12 |
|
dochfl1.x |
|- ( ph -> X e. ( V \ { .0. } ) ) |
13 |
|
dochfl1.g |
|- G = ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) ) |
14 |
12
|
eldifad |
|- ( ph -> X e. V ) |
15 |
|
eqeq1 |
|- ( v = X -> ( v = ( w .+ ( k .x. X ) ) <-> X = ( w .+ ( k .x. X ) ) ) ) |
16 |
15
|
rexbidv |
|- ( v = X -> ( E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) <-> E. w e. ( ._|_ ` { X } ) X = ( w .+ ( k .x. X ) ) ) ) |
17 |
16
|
riotabidv |
|- ( v = X -> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) = ( iota_ k e. R E. w e. ( ._|_ ` { X } ) X = ( w .+ ( k .x. X ) ) ) ) |
18 |
|
riotaex |
|- ( iota_ k e. R E. w e. ( ._|_ ` { X } ) X = ( w .+ ( k .x. X ) ) ) e. _V |
19 |
17 13 18
|
fvmpt |
|- ( X e. V -> ( G ` X ) = ( iota_ k e. R E. w e. ( ._|_ ` { X } ) X = ( w .+ ( k .x. X ) ) ) ) |
20 |
14 19
|
syl |
|- ( ph -> ( G ` X ) = ( iota_ k e. R E. w e. ( ._|_ ` { X } ) X = ( w .+ ( k .x. X ) ) ) ) |
21 |
1 3 11
|
dvhlmod |
|- ( ph -> U e. LMod ) |
22 |
14
|
snssd |
|- ( ph -> { X } C_ V ) |
23 |
|
eqid |
|- ( LSubSp ` U ) = ( LSubSp ` U ) |
24 |
1 3 4 23 2
|
dochlss |
|- ( ( ( K e. HL /\ W e. H ) /\ { X } C_ V ) -> ( ._|_ ` { X } ) e. ( LSubSp ` U ) ) |
25 |
11 22 24
|
syl2anc |
|- ( ph -> ( ._|_ ` { X } ) e. ( LSubSp ` U ) ) |
26 |
7 23
|
lss0cl |
|- ( ( U e. LMod /\ ( ._|_ ` { X } ) e. ( LSubSp ` U ) ) -> .0. e. ( ._|_ ` { X } ) ) |
27 |
21 25 26
|
syl2anc |
|- ( ph -> .0. e. ( ._|_ ` { X } ) ) |
28 |
4 8 6 10
|
lmodvs1 |
|- ( ( U e. LMod /\ X e. V ) -> ( .1. .x. X ) = X ) |
29 |
21 14 28
|
syl2anc |
|- ( ph -> ( .1. .x. X ) = X ) |
30 |
29
|
oveq2d |
|- ( ph -> ( .0. .+ ( .1. .x. X ) ) = ( .0. .+ X ) ) |
31 |
4 5 7
|
lmod0vlid |
|- ( ( U e. LMod /\ X e. V ) -> ( .0. .+ X ) = X ) |
32 |
21 14 31
|
syl2anc |
|- ( ph -> ( .0. .+ X ) = X ) |
33 |
30 32
|
eqtr2d |
|- ( ph -> X = ( .0. .+ ( .1. .x. X ) ) ) |
34 |
|
oveq1 |
|- ( w = .0. -> ( w .+ ( .1. .x. X ) ) = ( .0. .+ ( .1. .x. X ) ) ) |
35 |
34
|
rspceeqv |
|- ( ( .0. e. ( ._|_ ` { X } ) /\ X = ( .0. .+ ( .1. .x. X ) ) ) -> E. w e. ( ._|_ ` { X } ) X = ( w .+ ( .1. .x. X ) ) ) |
36 |
27 33 35
|
syl2anc |
|- ( ph -> E. w e. ( ._|_ ` { X } ) X = ( w .+ ( .1. .x. X ) ) ) |
37 |
8
|
lmodring |
|- ( U e. LMod -> D e. Ring ) |
38 |
9 10
|
ringidcl |
|- ( D e. Ring -> .1. e. R ) |
39 |
21 37 38
|
3syl |
|- ( ph -> .1. e. R ) |
40 |
|
eqid |
|- ( LSpan ` U ) = ( LSpan ` U ) |
41 |
|
eqid |
|- ( LSSum ` U ) = ( LSSum ` U ) |
42 |
|
eqid |
|- ( LSHyp ` U ) = ( LSHyp ` U ) |
43 |
1 3 11
|
dvhlvec |
|- ( ph -> U e. LVec ) |
44 |
1 2 3 4 7 42 11 12
|
dochsnshp |
|- ( ph -> ( ._|_ ` { X } ) e. ( LSHyp ` U ) ) |
45 |
1 2 3 4 7 40 41 11 12
|
dochexmidat |
|- ( ph -> ( ( ._|_ ` { X } ) ( LSSum ` U ) ( ( LSpan ` U ) ` { X } ) ) = V ) |
46 |
4 5 40 41 42 43 44 14 14 45 8 9 6
|
lshpsmreu |
|- ( ph -> E! k e. R E. w e. ( ._|_ ` { X } ) X = ( w .+ ( k .x. X ) ) ) |
47 |
|
oveq1 |
|- ( k = .1. -> ( k .x. X ) = ( .1. .x. X ) ) |
48 |
47
|
oveq2d |
|- ( k = .1. -> ( w .+ ( k .x. X ) ) = ( w .+ ( .1. .x. X ) ) ) |
49 |
48
|
eqeq2d |
|- ( k = .1. -> ( X = ( w .+ ( k .x. X ) ) <-> X = ( w .+ ( .1. .x. X ) ) ) ) |
50 |
49
|
rexbidv |
|- ( k = .1. -> ( E. w e. ( ._|_ ` { X } ) X = ( w .+ ( k .x. X ) ) <-> E. w e. ( ._|_ ` { X } ) X = ( w .+ ( .1. .x. X ) ) ) ) |
51 |
50
|
riota2 |
|- ( ( .1. e. R /\ E! k e. R E. w e. ( ._|_ ` { X } ) X = ( w .+ ( k .x. X ) ) ) -> ( E. w e. ( ._|_ ` { X } ) X = ( w .+ ( .1. .x. X ) ) <-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) X = ( w .+ ( k .x. X ) ) ) = .1. ) ) |
52 |
39 46 51
|
syl2anc |
|- ( ph -> ( E. w e. ( ._|_ ` { X } ) X = ( w .+ ( .1. .x. X ) ) <-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) X = ( w .+ ( k .x. X ) ) ) = .1. ) ) |
53 |
36 52
|
mpbid |
|- ( ph -> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) X = ( w .+ ( k .x. X ) ) ) = .1. ) |
54 |
20 53
|
eqtrd |
|- ( ph -> ( G ` X ) = .1. ) |