Step |
Hyp |
Ref |
Expression |
1 |
|
mplsubglem.s |
|- S = ( I mPwSer R ) |
2 |
|
mplsubglem.b |
|- B = ( Base ` S ) |
3 |
|
mplsubglem.z |
|- .0. = ( 0g ` R ) |
4 |
|
mplsubglem.d |
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } |
5 |
|
mplsubglem.i |
|- ( ph -> I e. W ) |
6 |
|
mplsubglem.0 |
|- ( ph -> (/) e. A ) |
7 |
|
mplsubglem.a |
|- ( ( ph /\ ( x e. A /\ y e. A ) ) -> ( x u. y ) e. A ) |
8 |
|
mplsubglem.y |
|- ( ( ph /\ ( x e. A /\ y C_ x ) ) -> y e. A ) |
9 |
|
mplsubglem.u |
|- ( ph -> U = { g e. B | ( g supp .0. ) e. A } ) |
10 |
|
mpllsslem.r |
|- ( ph -> R e. Ring ) |
11 |
1 5 10
|
psrsca |
|- ( ph -> R = ( Scalar ` S ) ) |
12 |
|
eqidd |
|- ( ph -> ( Base ` R ) = ( Base ` R ) ) |
13 |
2
|
a1i |
|- ( ph -> B = ( Base ` S ) ) |
14 |
|
eqidd |
|- ( ph -> ( +g ` S ) = ( +g ` S ) ) |
15 |
|
eqidd |
|- ( ph -> ( .s ` S ) = ( .s ` S ) ) |
16 |
|
eqidd |
|- ( ph -> ( LSubSp ` S ) = ( LSubSp ` S ) ) |
17 |
|
ringgrp |
|- ( R e. Ring -> R e. Grp ) |
18 |
10 17
|
syl |
|- ( ph -> R e. Grp ) |
19 |
1 2 3 4 5 6 7 8 9 18
|
mplsubglem |
|- ( ph -> U e. ( SubGrp ` S ) ) |
20 |
2
|
subgss |
|- ( U e. ( SubGrp ` S ) -> U C_ B ) |
21 |
19 20
|
syl |
|- ( ph -> U C_ B ) |
22 |
|
eqid |
|- ( 0g ` S ) = ( 0g ` S ) |
23 |
22
|
subg0cl |
|- ( U e. ( SubGrp ` S ) -> ( 0g ` S ) e. U ) |
24 |
|
ne0i |
|- ( ( 0g ` S ) e. U -> U =/= (/) ) |
25 |
19 23 24
|
3syl |
|- ( ph -> U =/= (/) ) |
26 |
19
|
adantr |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U /\ w e. U ) ) -> U e. ( SubGrp ` S ) ) |
27 |
|
eqid |
|- ( .s ` S ) = ( .s ` S ) |
28 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
29 |
10
|
adantr |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> R e. Ring ) |
30 |
|
simprl |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> u e. ( Base ` R ) ) |
31 |
|
simprr |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> v e. U ) |
32 |
9
|
adantr |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> U = { g e. B | ( g supp .0. ) e. A } ) |
33 |
32
|
eleq2d |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( v e. U <-> v e. { g e. B | ( g supp .0. ) e. A } ) ) |
34 |
|
oveq1 |
|- ( g = v -> ( g supp .0. ) = ( v supp .0. ) ) |
35 |
34
|
eleq1d |
|- ( g = v -> ( ( g supp .0. ) e. A <-> ( v supp .0. ) e. A ) ) |
36 |
35
|
elrab |
|- ( v e. { g e. B | ( g supp .0. ) e. A } <-> ( v e. B /\ ( v supp .0. ) e. A ) ) |
37 |
33 36
|
bitrdi |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( v e. U <-> ( v e. B /\ ( v supp .0. ) e. A ) ) ) |
38 |
31 37
|
mpbid |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( v e. B /\ ( v supp .0. ) e. A ) ) |
39 |
38
|
simpld |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> v e. B ) |
40 |
1 27 28 2 29 30 39
|
psrvscacl |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( u ( .s ` S ) v ) e. B ) |
41 |
|
ovex |
|- ( ( u ( .s ` S ) v ) supp .0. ) e. _V |
42 |
41
|
a1i |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( ( u ( .s ` S ) v ) supp .0. ) e. _V ) |
43 |
|
sseq2 |
|- ( x = ( v supp .0. ) -> ( y C_ x <-> y C_ ( v supp .0. ) ) ) |
44 |
43
|
imbi1d |
|- ( x = ( v supp .0. ) -> ( ( y C_ x -> y e. A ) <-> ( y C_ ( v supp .0. ) -> y e. A ) ) ) |
45 |
44
|
albidv |
|- ( x = ( v supp .0. ) -> ( A. y ( y C_ x -> y e. A ) <-> A. y ( y C_ ( v supp .0. ) -> y e. A ) ) ) |
46 |
8
|
expr |
|- ( ( ph /\ x e. A ) -> ( y C_ x -> y e. A ) ) |
47 |
46
|
alrimiv |
|- ( ( ph /\ x e. A ) -> A. y ( y C_ x -> y e. A ) ) |
48 |
47
|
ralrimiva |
|- ( ph -> A. x e. A A. y ( y C_ x -> y e. A ) ) |
49 |
48
|
adantr |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> A. x e. A A. y ( y C_ x -> y e. A ) ) |
50 |
38
|
simprd |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( v supp .0. ) e. A ) |
51 |
45 49 50
|
rspcdva |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> A. y ( y C_ ( v supp .0. ) -> y e. A ) ) |
52 |
1 28 4 2 40
|
psrelbas |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( u ( .s ` S ) v ) : D --> ( Base ` R ) ) |
53 |
|
eqid |
|- ( .r ` R ) = ( .r ` R ) |
54 |
30
|
adantr |
|- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> u e. ( Base ` R ) ) |
55 |
39
|
adantr |
|- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> v e. B ) |
56 |
|
eldifi |
|- ( k e. ( D \ ( v supp .0. ) ) -> k e. D ) |
57 |
56
|
adantl |
|- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> k e. D ) |
58 |
1 27 28 2 53 4 54 55 57
|
psrvscaval |
|- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> ( ( u ( .s ` S ) v ) ` k ) = ( u ( .r ` R ) ( v ` k ) ) ) |
59 |
1 28 4 2 39
|
psrelbas |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> v : D --> ( Base ` R ) ) |
60 |
|
ssidd |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( v supp .0. ) C_ ( v supp .0. ) ) |
61 |
|
ovex |
|- ( NN0 ^m I ) e. _V |
62 |
4 61
|
rabex2 |
|- D e. _V |
63 |
62
|
a1i |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> D e. _V ) |
64 |
3
|
fvexi |
|- .0. e. _V |
65 |
64
|
a1i |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> .0. e. _V ) |
66 |
59 60 63 65
|
suppssr |
|- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> ( v ` k ) = .0. ) |
67 |
66
|
oveq2d |
|- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> ( u ( .r ` R ) ( v ` k ) ) = ( u ( .r ` R ) .0. ) ) |
68 |
28 53 3
|
ringrz |
|- ( ( R e. Ring /\ u e. ( Base ` R ) ) -> ( u ( .r ` R ) .0. ) = .0. ) |
69 |
10 30 68
|
syl2an2r |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( u ( .r ` R ) .0. ) = .0. ) |
70 |
69
|
adantr |
|- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> ( u ( .r ` R ) .0. ) = .0. ) |
71 |
58 67 70
|
3eqtrd |
|- ( ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) /\ k e. ( D \ ( v supp .0. ) ) ) -> ( ( u ( .s ` S ) v ) ` k ) = .0. ) |
72 |
52 71
|
suppss |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( ( u ( .s ` S ) v ) supp .0. ) C_ ( v supp .0. ) ) |
73 |
|
sseq1 |
|- ( y = ( ( u ( .s ` S ) v ) supp .0. ) -> ( y C_ ( v supp .0. ) <-> ( ( u ( .s ` S ) v ) supp .0. ) C_ ( v supp .0. ) ) ) |
74 |
|
eleq1 |
|- ( y = ( ( u ( .s ` S ) v ) supp .0. ) -> ( y e. A <-> ( ( u ( .s ` S ) v ) supp .0. ) e. A ) ) |
75 |
73 74
|
imbi12d |
|- ( y = ( ( u ( .s ` S ) v ) supp .0. ) -> ( ( y C_ ( v supp .0. ) -> y e. A ) <-> ( ( ( u ( .s ` S ) v ) supp .0. ) C_ ( v supp .0. ) -> ( ( u ( .s ` S ) v ) supp .0. ) e. A ) ) ) |
76 |
75
|
spcgv |
|- ( ( ( u ( .s ` S ) v ) supp .0. ) e. _V -> ( A. y ( y C_ ( v supp .0. ) -> y e. A ) -> ( ( ( u ( .s ` S ) v ) supp .0. ) C_ ( v supp .0. ) -> ( ( u ( .s ` S ) v ) supp .0. ) e. A ) ) ) |
77 |
42 51 72 76
|
syl3c |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( ( u ( .s ` S ) v ) supp .0. ) e. A ) |
78 |
32
|
eleq2d |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( ( u ( .s ` S ) v ) e. U <-> ( u ( .s ` S ) v ) e. { g e. B | ( g supp .0. ) e. A } ) ) |
79 |
|
oveq1 |
|- ( g = ( u ( .s ` S ) v ) -> ( g supp .0. ) = ( ( u ( .s ` S ) v ) supp .0. ) ) |
80 |
79
|
eleq1d |
|- ( g = ( u ( .s ` S ) v ) -> ( ( g supp .0. ) e. A <-> ( ( u ( .s ` S ) v ) supp .0. ) e. A ) ) |
81 |
80
|
elrab |
|- ( ( u ( .s ` S ) v ) e. { g e. B | ( g supp .0. ) e. A } <-> ( ( u ( .s ` S ) v ) e. B /\ ( ( u ( .s ` S ) v ) supp .0. ) e. A ) ) |
82 |
78 81
|
bitrdi |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( ( u ( .s ` S ) v ) e. U <-> ( ( u ( .s ` S ) v ) e. B /\ ( ( u ( .s ` S ) v ) supp .0. ) e. A ) ) ) |
83 |
40 77 82
|
mpbir2and |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U ) ) -> ( u ( .s ` S ) v ) e. U ) |
84 |
83
|
3adantr3 |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U /\ w e. U ) ) -> ( u ( .s ` S ) v ) e. U ) |
85 |
|
simpr3 |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U /\ w e. U ) ) -> w e. U ) |
86 |
|
eqid |
|- ( +g ` S ) = ( +g ` S ) |
87 |
86
|
subgcl |
|- ( ( U e. ( SubGrp ` S ) /\ ( u ( .s ` S ) v ) e. U /\ w e. U ) -> ( ( u ( .s ` S ) v ) ( +g ` S ) w ) e. U ) |
88 |
26 84 85 87
|
syl3anc |
|- ( ( ph /\ ( u e. ( Base ` R ) /\ v e. U /\ w e. U ) ) -> ( ( u ( .s ` S ) v ) ( +g ` S ) w ) e. U ) |
89 |
11 12 13 14 15 16 21 25 88
|
islssd |
|- ( ph -> U e. ( LSubSp ` S ) ) |