Step | Hyp | Ref
| Expression |
1 | | eqid 2726 |
. . . . 5
β’
(Baseβπ) =
(Baseβπ) |
2 | 1 | linds1 21705 |
. . . 4
β’ (πΉ β (LIndSβπ) β πΉ β (Baseβπ)) |
3 | | eldifi 4121 |
. . . . 5
β’ (π β ((Baseβπ) β ((LSpanβπ)βπΉ)) β π β (Baseβπ)) |
4 | 3 | snssd 4807 |
. . . 4
β’ (π β ((Baseβπ) β ((LSpanβπ)βπΉ)) β {π} β (Baseβπ)) |
5 | | unss 4179 |
. . . . 5
β’ ((πΉ β (Baseβπ) β§ {π} β (Baseβπ)) β (πΉ βͺ {π}) β (Baseβπ)) |
6 | 5 | biimpi 215 |
. . . 4
β’ ((πΉ β (Baseβπ) β§ {π} β (Baseβπ)) β (πΉ βͺ {π}) β (Baseβπ)) |
7 | 2, 4, 6 | syl2an 595 |
. . 3
β’ ((πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β (πΉ βͺ {π}) β (Baseβπ)) |
8 | 7 | 3adant1 1127 |
. 2
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β (πΉ βͺ {π}) β (Baseβπ)) |
9 | | eldifn 4122 |
. . . . . . 7
β’ (π β ((Baseβπ) β ((LSpanβπ)βπΉ)) β Β¬ π β ((LSpanβπ)βπΉ)) |
10 | 9 | 3ad2ant3 1132 |
. . . . . 6
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β Β¬ π β ((LSpanβπ)βπΉ)) |
11 | 10 | adantr 480 |
. . . . 5
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β Β¬ π β ((LSpanβπ)βπΉ)) |
12 | | simpll1 1209 |
. . . . . . . 8
β’ ((((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β§ π₯ β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π}))) β π β LVec) |
13 | 2 | ssdifssd 4137 |
. . . . . . . . . 10
β’ (πΉ β (LIndSβπ) β (πΉ β {π₯}) β (Baseβπ)) |
14 | 13 | 3ad2ant2 1131 |
. . . . . . . . 9
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β (πΉ β {π₯}) β (Baseβπ)) |
15 | 14 | ad2antrr 723 |
. . . . . . . 8
β’ ((((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β§ π₯ β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π}))) β (πΉ β {π₯}) β (Baseβπ)) |
16 | 3 | 3ad2ant3 1132 |
. . . . . . . . 9
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β π β (Baseβπ)) |
17 | 16 | ad2antrr 723 |
. . . . . . . 8
β’ ((((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β§ π₯ β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π}))) β π β (Baseβπ)) |
18 | | simpr 484 |
. . . . . . . . 9
β’ ((((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β§ π₯ β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π}))) β π₯ β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π}))) |
19 | | lveclmod 20954 |
. . . . . . . . . . . . 13
β’ (π β LVec β π β LMod) |
20 | 19 | ad2antrr 723 |
. . . . . . . . . . . 12
β’ (((π β LVec β§ πΉ β (LIndSβπ)) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β π β LMod) |
21 | | eqid 2726 |
. . . . . . . . . . . . . . . 16
β’
(Scalarβπ) =
(Scalarβπ) |
22 | 21 | lmodring 20714 |
. . . . . . . . . . . . . . 15
β’ (π β LMod β
(Scalarβπ) β
Ring) |
23 | 19, 22 | syl 17 |
. . . . . . . . . . . . . 14
β’ (π β LVec β
(Scalarβπ) β
Ring) |
24 | | eqid 2726 |
. . . . . . . . . . . . . . 15
β’
(0gβ(Scalarβπ)) =
(0gβ(Scalarβπ)) |
25 | | eqid 2726 |
. . . . . . . . . . . . . . 15
β’
(Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) |
26 | 24, 25 | ringelnzr 20423 |
. . . . . . . . . . . . . 14
β’
(((Scalarβπ)
β Ring β§ π β
((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β (Scalarβπ) β NzRing) |
27 | 23, 26 | sylan 579 |
. . . . . . . . . . . . 13
β’ ((π β LVec β§ π β
((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β (Scalarβπ) β NzRing) |
28 | 27 | ad2ant2rl 746 |
. . . . . . . . . . . 12
β’ (((π β LVec β§ πΉ β (LIndSβπ)) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β (Scalarβπ) β
NzRing) |
29 | | simplr 766 |
. . . . . . . . . . . 12
β’ (((π β LVec β§ πΉ β (LIndSβπ)) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β πΉ β (LIndSβπ)) |
30 | | simprl 768 |
. . . . . . . . . . . 12
β’ (((π β LVec β§ πΉ β (LIndSβπ)) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β π₯ β πΉ) |
31 | | eqid 2726 |
. . . . . . . . . . . . 13
β’
(LSpanβπ) =
(LSpanβπ) |
32 | 31, 21 | lindsind2 21714 |
. . . . . . . . . . . 12
β’ (((π β LMod β§
(Scalarβπ) β
NzRing) β§ πΉ β
(LIndSβπ) β§ π₯ β πΉ) β Β¬ π₯ β ((LSpanβπ)β(πΉ β {π₯}))) |
33 | 20, 28, 29, 30, 32 | syl211anc 1373 |
. . . . . . . . . . 11
β’ (((π β LVec β§ πΉ β (LIndSβπ)) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β Β¬ π₯ β ((LSpanβπ)β(πΉ β {π₯}))) |
34 | 33 | 3adantl3 1165 |
. . . . . . . . . 10
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β Β¬ π₯ β ((LSpanβπ)β(πΉ β {π₯}))) |
35 | 34 | adantr 480 |
. . . . . . . . 9
β’ ((((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β§ π₯ β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π}))) β Β¬ π₯ β ((LSpanβπ)β(πΉ β {π₯}))) |
36 | 18, 35 | eldifd 3954 |
. . . . . . . 8
β’ ((((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β§ π₯ β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π}))) β π₯ β (((LSpanβπ)β((πΉ β {π₯}) βͺ {π})) β ((LSpanβπ)β(πΉ β {π₯})))) |
37 | | eqid 2726 |
. . . . . . . . 9
β’
(LSubSpβπ) =
(LSubSpβπ) |
38 | 1, 37, 31 | lspsolv 20994 |
. . . . . . . 8
β’ ((π β LVec β§ ((πΉ β {π₯}) β (Baseβπ) β§ π β (Baseβπ) β§ π₯ β (((LSpanβπ)β((πΉ β {π₯}) βͺ {π})) β ((LSpanβπ)β(πΉ β {π₯}))))) β π β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π₯}))) |
39 | 12, 15, 17, 36, 38 | syl13anc 1369 |
. . . . . . 7
β’ ((((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β§ π₯ β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π}))) β π β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π₯}))) |
40 | 39 | ex 412 |
. . . . . 6
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β (π₯ β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π})) β π β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π₯})))) |
41 | | eldif 3953 |
. . . . . . . . . . . . . . . . . 18
β’ (π β ((Baseβπ) β ((LSpanβπ)βπΉ)) β (π β (Baseβπ) β§ Β¬ π β ((LSpanβπ)βπΉ))) |
42 | | snssi 4806 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ (π β πΉ β {π} β πΉ) |
43 | 1, 31 | lspss 20831 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ ((π β LMod β§ πΉ β (Baseβπ) β§ {π} β πΉ) β ((LSpanβπ)β{π}) β ((LSpanβπ)βπΉ)) |
44 | 19, 2, 42, 43 | syl3an 1157 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β πΉ) β ((LSpanβπ)β{π}) β ((LSpanβπ)βπΉ)) |
45 | 44 | ad4ant124 1170 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((((π β LVec β§ πΉ β (LIndSβπ)) β§ π β (Baseβπ)) β§ π β πΉ) β ((LSpanβπ)β{π}) β ((LSpanβπ)βπΉ)) |
46 | 1, 31 | lspsnid 20840 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ ((π β LMod β§ π β (Baseβπ)) β π β ((LSpanβπ)β{π})) |
47 | 19, 46 | sylan 579 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((π β LVec β§ π β (Baseβπ)) β π β ((LSpanβπ)β{π})) |
48 | 47 | ad4ant13 748 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((((π β LVec β§ πΉ β (LIndSβπ)) β§ π β (Baseβπ)) β§ π β πΉ) β π β ((LSpanβπ)β{π})) |
49 | 45, 48 | sseldd 3978 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((((π β LVec β§ πΉ β (LIndSβπ)) β§ π β (Baseβπ)) β§ π β πΉ) β π β ((LSpanβπ)βπΉ)) |
50 | 49 | ex 412 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((π β LVec β§ πΉ β (LIndSβπ)) β§ π β (Baseβπ)) β (π β πΉ β π β ((LSpanβπ)βπΉ))) |
51 | 50 | con3d 152 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((π β LVec β§ πΉ β (LIndSβπ)) β§ π β (Baseβπ)) β (Β¬ π β ((LSpanβπ)βπΉ) β Β¬ π β πΉ)) |
52 | 51 | expimpd 453 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π β LVec β§ πΉ β (LIndSβπ)) β ((π β (Baseβπ) β§ Β¬ π β ((LSpanβπ)βπΉ)) β Β¬ π β πΉ)) |
53 | 52 | 3impia 1114 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ (π β (Baseβπ) β§ Β¬ π β ((LSpanβπ)βπΉ))) β Β¬ π β πΉ) |
54 | 41, 53 | syl3an3b 1402 |
. . . . . . . . . . . . . . . . 17
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β Β¬ π β πΉ) |
55 | | eleq1 2815 |
. . . . . . . . . . . . . . . . . 18
β’ (π = π₯ β (π β πΉ β π₯ β πΉ)) |
56 | 55 | notbid 318 |
. . . . . . . . . . . . . . . . 17
β’ (π = π₯ β (Β¬ π β πΉ β Β¬ π₯ β πΉ)) |
57 | 54, 56 | syl5ibcom 244 |
. . . . . . . . . . . . . . . 16
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β (π = π₯ β Β¬ π₯ β πΉ)) |
58 | 57 | necon2ad 2949 |
. . . . . . . . . . . . . . 15
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β (π₯ β πΉ β π β π₯)) |
59 | 58 | imp 406 |
. . . . . . . . . . . . . 14
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ π₯ β πΉ) β π β π₯) |
60 | | disjsn2 4711 |
. . . . . . . . . . . . . 14
β’ (π β π₯ β ({π} β© {π₯}) = β
) |
61 | 59, 60 | syl 17 |
. . . . . . . . . . . . 13
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ π₯ β πΉ) β ({π} β© {π₯}) = β
) |
62 | | disj3 4448 |
. . . . . . . . . . . . 13
β’ (({π} β© {π₯}) = β
β {π} = ({π} β {π₯})) |
63 | 61, 62 | sylib 217 |
. . . . . . . . . . . 12
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ π₯ β πΉ) β {π} = ({π} β {π₯})) |
64 | 63 | uneq2d 4158 |
. . . . . . . . . . 11
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ π₯ β πΉ) β ((πΉ β {π₯}) βͺ {π}) = ((πΉ β {π₯}) βͺ ({π} β {π₯}))) |
65 | | difundir 4275 |
. . . . . . . . . . 11
β’ ((πΉ βͺ {π}) β {π₯}) = ((πΉ β {π₯}) βͺ ({π} β {π₯})) |
66 | 64, 65 | eqtr4di 2784 |
. . . . . . . . . 10
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ π₯ β πΉ) β ((πΉ β {π₯}) βͺ {π}) = ((πΉ βͺ {π}) β {π₯})) |
67 | 66 | fveq2d 6889 |
. . . . . . . . 9
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ π₯ β πΉ) β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π})) = ((LSpanβπ)β((πΉ βͺ {π}) β {π₯}))) |
68 | 67 | eleq2d 2813 |
. . . . . . . 8
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ π₯ β πΉ) β (π₯ β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π})) β π₯ β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})))) |
69 | 68 | adantrr 714 |
. . . . . . 7
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β (π₯ β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π})) β π₯ β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})))) |
70 | | simpl 482 |
. . . . . . . . . . . . 13
β’ ((π β LVec β§ π β
((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β π β LVec) |
71 | | eldifsn 4785 |
. . . . . . . . . . . . . . 15
β’ (π β
((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) β (π β (Baseβ(Scalarβπ)) β§ π β
(0gβ(Scalarβπ)))) |
72 | 71 | biimpi 215 |
. . . . . . . . . . . . . 14
β’ (π β
((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) β (π β (Baseβ(Scalarβπ)) β§ π β
(0gβ(Scalarβπ)))) |
73 | 72 | adantl 481 |
. . . . . . . . . . . . 13
β’ ((π β LVec β§ π β
((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β (π β (Baseβ(Scalarβπ)) β§ π β
(0gβ(Scalarβπ)))) |
74 | 2 | sselda 3977 |
. . . . . . . . . . . . 13
β’ ((πΉ β (LIndSβπ) β§ π₯ β πΉ) β π₯ β (Baseβπ)) |
75 | | eqid 2726 |
. . . . . . . . . . . . . 14
β’ (
Β·π βπ) = ( Β·π
βπ) |
76 | 1, 21, 75, 25, 24, 31 | lspsnvs 20965 |
. . . . . . . . . . . . 13
β’ ((π β LVec β§ (π β
(Baseβ(Scalarβπ)) β§ π β
(0gβ(Scalarβπ))) β§ π₯ β (Baseβπ)) β ((LSpanβπ)β{(π( Β·π
βπ)π₯)}) = ((LSpanβπ)β{π₯})) |
77 | 70, 73, 74, 76 | syl2an3an 1419 |
. . . . . . . . . . . 12
β’ (((π β LVec β§ π β
((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β§ (πΉ β (LIndSβπ) β§ π₯ β πΉ)) β ((LSpanβπ)β{(π( Β·π
βπ)π₯)}) = ((LSpanβπ)β{π₯})) |
78 | 77 | an42s 658 |
. . . . . . . . . . 11
β’ (((π β LVec β§ πΉ β (LIndSβπ)) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β ((LSpanβπ)β{(π( Β·π
βπ)π₯)}) = ((LSpanβπ)β{π₯})) |
79 | 78 | sseq1d 4008 |
. . . . . . . . . 10
β’ (((π β LVec β§ πΉ β (LIndSβπ)) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β (((LSpanβπ)β{(π( Β·π
βπ)π₯)}) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β ((LSpanβπ)β{π₯}) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})))) |
80 | 79 | 3adantl3 1165 |
. . . . . . . . 9
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β (((LSpanβπ)β{(π( Β·π
βπ)π₯)}) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β ((LSpanβπ)β{π₯}) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})))) |
81 | | eldifi 4121 |
. . . . . . . . . 10
β’ (π β
((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) β π β (Baseβ(Scalarβπ))) |
82 | 19 | 3ad2ant1 1130 |
. . . . . . . . . . . 12
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β π β LMod) |
83 | 82 | adantr 480 |
. . . . . . . . . . 11
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ (π₯ β πΉ β§ π β (Baseβ(Scalarβπ)))) β π β LMod) |
84 | | snssi 4806 |
. . . . . . . . . . . . . . . 16
β’ (π β (Baseβπ) β {π} β (Baseβπ)) |
85 | 2, 84, 6 | syl2an 595 |
. . . . . . . . . . . . . . 15
β’ ((πΉ β (LIndSβπ) β§ π β (Baseβπ)) β (πΉ βͺ {π}) β (Baseβπ)) |
86 | 85 | ssdifssd 4137 |
. . . . . . . . . . . . . 14
β’ ((πΉ β (LIndSβπ) β§ π β (Baseβπ)) β ((πΉ βͺ {π}) β {π₯}) β (Baseβπ)) |
87 | 1, 37, 31 | lspcl 20823 |
. . . . . . . . . . . . . 14
β’ ((π β LMod β§ ((πΉ βͺ {π}) β {π₯}) β (Baseβπ)) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β (LSubSpβπ)) |
88 | 19, 86, 87 | syl2an 595 |
. . . . . . . . . . . . 13
β’ ((π β LVec β§ (πΉ β (LIndSβπ) β§ π β (Baseβπ))) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β (LSubSpβπ)) |
89 | 88 | 3impb 1112 |
. . . . . . . . . . . 12
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β (LSubSpβπ)) |
90 | 89 | adantr 480 |
. . . . . . . . . . 11
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ (π₯ β πΉ β§ π β (Baseβ(Scalarβπ)))) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β (LSubSpβπ)) |
91 | 19 | anim1i 614 |
. . . . . . . . . . . . . 14
β’ ((π β LVec β§ π β
(Baseβ(Scalarβπ))) β (π β LMod β§ π β (Baseβ(Scalarβπ)))) |
92 | 1, 21, 75, 25 | lmodvscl 20724 |
. . . . . . . . . . . . . . 15
β’ ((π β LMod β§ π β
(Baseβ(Scalarβπ)) β§ π₯ β (Baseβπ)) β (π( Β·π
βπ)π₯) β (Baseβπ)) |
93 | 92 | 3expa 1115 |
. . . . . . . . . . . . . 14
β’ (((π β LMod β§ π β
(Baseβ(Scalarβπ))) β§ π₯ β (Baseβπ)) β (π( Β·π
βπ)π₯) β (Baseβπ)) |
94 | 91, 74, 93 | syl2an 595 |
. . . . . . . . . . . . 13
β’ (((π β LVec β§ π β
(Baseβ(Scalarβπ))) β§ (πΉ β (LIndSβπ) β§ π₯ β πΉ)) β (π( Β·π
βπ)π₯) β (Baseβπ)) |
95 | 94 | an42s 658 |
. . . . . . . . . . . 12
β’ (((π β LVec β§ πΉ β (LIndSβπ)) β§ (π₯ β πΉ β§ π β (Baseβ(Scalarβπ)))) β (π( Β·π
βπ)π₯) β (Baseβπ)) |
96 | 95 | 3adantl3 1165 |
. . . . . . . . . . 11
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ (π₯ β πΉ β§ π β (Baseβ(Scalarβπ)))) β (π( Β·π
βπ)π₯) β (Baseβπ)) |
97 | 1, 37, 31, 83, 90, 96 | lspsnel5 20842 |
. . . . . . . . . 10
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ (π₯ β πΉ β§ π β (Baseβ(Scalarβπ)))) β ((π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β ((LSpanβπ)β{(π( Β·π
βπ)π₯)}) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})))) |
98 | 81, 97 | sylanr2 680 |
. . . . . . . . 9
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β ((π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β ((LSpanβπ)β{(π( Β·π
βπ)π₯)}) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})))) |
99 | 82 | adantr 480 |
. . . . . . . . . . 11
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ π₯ β πΉ) β π β LMod) |
100 | 89 | adantr 480 |
. . . . . . . . . . 11
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ π₯ β πΉ) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β (LSubSpβπ)) |
101 | 74 | 3ad2antl2 1183 |
. . . . . . . . . . 11
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ π₯ β πΉ) β π₯ β (Baseβπ)) |
102 | 1, 37, 31, 99, 100, 101 | lspsnel5 20842 |
. . . . . . . . . 10
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ π₯ β πΉ) β (π₯ β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β ((LSpanβπ)β{π₯}) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})))) |
103 | 102 | adantrr 714 |
. . . . . . . . 9
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β (π₯ β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β ((LSpanβπ)β{π₯}) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})))) |
104 | 80, 98, 103 | 3bitr4rd 312 |
. . . . . . . 8
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β (π₯ β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})))) |
105 | 3, 104 | syl3anl3 1411 |
. . . . . . 7
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β (π₯ β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})))) |
106 | 69, 105 | bitrd 279 |
. . . . . 6
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β (π₯ β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π})) β (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})))) |
107 | | difsnid 4808 |
. . . . . . . . 9
β’ (π₯ β πΉ β ((πΉ β {π₯}) βͺ {π₯}) = πΉ) |
108 | 107 | fveq2d 6889 |
. . . . . . . 8
β’ (π₯ β πΉ β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π₯})) = ((LSpanβπ)βπΉ)) |
109 | 108 | eleq2d 2813 |
. . . . . . 7
β’ (π₯ β πΉ β (π β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π₯})) β π β ((LSpanβπ)βπΉ))) |
110 | 109 | ad2antrl 725 |
. . . . . 6
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β (π β ((LSpanβπ)β((πΉ β {π₯}) βͺ {π₯})) β π β ((LSpanβπ)βπΉ))) |
111 | 40, 106, 110 | 3imtr3d 293 |
. . . . 5
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β ((π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β π β ((LSpanβπ)βπΉ))) |
112 | 11, 111 | mtod 197 |
. . . 4
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ (π₯ β πΉ β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}))) β Β¬ (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯}))) |
113 | 112 | ralrimivva 3194 |
. . 3
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β βπ₯ β πΉ βπ β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) Β¬ (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯}))) |
114 | 10 | adantr 480 |
. . . . . 6
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β Β¬ π β ((LSpanβπ)βπΉ)) |
115 | | difsn 4796 |
. . . . . . . . . . 11
β’ (Β¬
π β πΉ β (πΉ β {π}) = πΉ) |
116 | 54, 115 | syl 17 |
. . . . . . . . . 10
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β (πΉ β {π}) = πΉ) |
117 | 116 | fveq2d 6889 |
. . . . . . . . 9
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β ((LSpanβπ)β(πΉ β {π})) = ((LSpanβπ)βπΉ)) |
118 | 117 | eleq2d 2813 |
. . . . . . . 8
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β ((π( Β·π
βπ)π) β ((LSpanβπ)β(πΉ β {π})) β (π( Β·π
βπ)π) β ((LSpanβπ)βπΉ))) |
119 | 118 | adantr 480 |
. . . . . . 7
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β ((π( Β·π
βπ)π) β ((LSpanβπ)β(πΉ β {π})) β (π( Β·π
βπ)π) β ((LSpanβπ)βπΉ))) |
120 | 1, 21, 75, 25, 24, 31 | lspsnvs 20965 |
. . . . . . . . . . . . . 14
β’ ((π β LVec β§ (π β
(Baseβ(Scalarβπ)) β§ π β
(0gβ(Scalarβπ))) β§ π β (Baseβπ)) β ((LSpanβπ)β{(π( Β·π
βπ)π)}) = ((LSpanβπ)β{π})) |
121 | 120 | 3expa 1115 |
. . . . . . . . . . . . 13
β’ (((π β LVec β§ (π β
(Baseβ(Scalarβπ)) β§ π β
(0gβ(Scalarβπ)))) β§ π β (Baseβπ)) β ((LSpanβπ)β{(π( Β·π
βπ)π)}) = ((LSpanβπ)β{π})) |
122 | 121 | an32s 649 |
. . . . . . . . . . . 12
β’ (((π β LVec β§ π β (Baseβπ)) β§ (π β (Baseβ(Scalarβπ)) β§ π β
(0gβ(Scalarβπ)))) β ((LSpanβπ)β{(π( Β·π
βπ)π)}) = ((LSpanβπ)β{π})) |
123 | 71, 122 | sylan2b 593 |
. . . . . . . . . . 11
β’ (((π β LVec β§ π β (Baseβπ)) β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β ((LSpanβπ)β{(π( Β·π
βπ)π)}) = ((LSpanβπ)β{π})) |
124 | 123 | sseq1d 4008 |
. . . . . . . . . 10
β’ (((π β LVec β§ π β (Baseβπ)) β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β (((LSpanβπ)β{(π( Β·π
βπ)π)}) β ((LSpanβπ)βπΉ) β ((LSpanβπ)β{π}) β ((LSpanβπ)βπΉ))) |
125 | 124 | 3adantl2 1164 |
. . . . . . . . 9
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β (((LSpanβπ)β{(π( Β·π
βπ)π)}) β ((LSpanβπ)βπΉ) β ((LSpanβπ)β{π}) β ((LSpanβπ)βπΉ))) |
126 | 82 | adantr 480 |
. . . . . . . . . . 11
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ π β (Baseβ(Scalarβπ))) β π β LMod) |
127 | 1, 37, 31 | lspcl 20823 |
. . . . . . . . . . . . . 14
β’ ((π β LMod β§ πΉ β (Baseβπ)) β ((LSpanβπ)βπΉ) β (LSubSpβπ)) |
128 | 19, 2, 127 | syl2an 595 |
. . . . . . . . . . . . 13
β’ ((π β LVec β§ πΉ β (LIndSβπ)) β ((LSpanβπ)βπΉ) β (LSubSpβπ)) |
129 | 128 | 3adant3 1129 |
. . . . . . . . . . . 12
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β ((LSpanβπ)βπΉ) β (LSubSpβπ)) |
130 | 129 | adantr 480 |
. . . . . . . . . . 11
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ π β (Baseβ(Scalarβπ))) β ((LSpanβπ)βπΉ) β (LSubSpβπ)) |
131 | 1, 21, 75, 25 | lmodvscl 20724 |
. . . . . . . . . . . . . . 15
β’ ((π β LMod β§ π β
(Baseβ(Scalarβπ)) β§ π β (Baseβπ)) β (π( Β·π
βπ)π) β (Baseβπ)) |
132 | 131 | 3expa 1115 |
. . . . . . . . . . . . . 14
β’ (((π β LMod β§ π β
(Baseβ(Scalarβπ))) β§ π β (Baseβπ)) β (π( Β·π
βπ)π) β (Baseβπ)) |
133 | 132 | an32s 649 |
. . . . . . . . . . . . 13
β’ (((π β LMod β§ π β (Baseβπ)) β§ π β (Baseβ(Scalarβπ))) β (π( Β·π
βπ)π) β (Baseβπ)) |
134 | 19, 133 | sylanl1 677 |
. . . . . . . . . . . 12
β’ (((π β LVec β§ π β (Baseβπ)) β§ π β (Baseβ(Scalarβπ))) β (π( Β·π
βπ)π) β (Baseβπ)) |
135 | 134 | 3adantl2 1164 |
. . . . . . . . . . 11
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ π β (Baseβ(Scalarβπ))) β (π( Β·π
βπ)π) β (Baseβπ)) |
136 | 1, 37, 31, 126, 130, 135 | lspsnel5 20842 |
. . . . . . . . . 10
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ π β (Baseβ(Scalarβπ))) β ((π( Β·π
βπ)π) β ((LSpanβπ)βπΉ) β ((LSpanβπ)β{(π( Β·π
βπ)π)}) β ((LSpanβπ)βπΉ))) |
137 | 81, 136 | sylan2 592 |
. . . . . . . . 9
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β ((π( Β·π
βπ)π) β ((LSpanβπ)βπΉ) β ((LSpanβπ)β{(π( Β·π
βπ)π)}) β ((LSpanβπ)βπΉ))) |
138 | | simp3 1135 |
. . . . . . . . . . 11
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β π β (Baseβπ)) |
139 | 1, 37, 31, 82, 129, 138 | lspsnel5 20842 |
. . . . . . . . . 10
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β (π β ((LSpanβπ)βπΉ) β ((LSpanβπ)β{π}) β ((LSpanβπ)βπΉ))) |
140 | 139 | adantr 480 |
. . . . . . . . 9
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β (π β ((LSpanβπ)βπΉ) β ((LSpanβπ)β{π}) β ((LSpanβπ)βπΉ))) |
141 | 125, 137,
140 | 3bitr4d 311 |
. . . . . . . 8
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β (Baseβπ)) β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β ((π( Β·π
βπ)π) β ((LSpanβπ)βπΉ) β π β ((LSpanβπ)βπΉ))) |
142 | 3, 141 | syl3anl3 1411 |
. . . . . . 7
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β ((π( Β·π
βπ)π) β ((LSpanβπ)βπΉ) β π β ((LSpanβπ)βπΉ))) |
143 | 119, 142 | bitrd 279 |
. . . . . 6
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β ((π( Β·π
βπ)π) β ((LSpanβπ)β(πΉ β {π})) β π β ((LSpanβπ)βπΉ))) |
144 | 114, 143 | mtbird 325 |
. . . . 5
β’ (((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β§ π β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))})) β Β¬ (π( Β·π
βπ)π) β ((LSpanβπ)β(πΉ β {π}))) |
145 | 144 | ralrimiva 3140 |
. . . 4
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β βπ β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) Β¬ (π( Β·π
βπ)π) β ((LSpanβπ)β(πΉ β {π}))) |
146 | | oveq2 7413 |
. . . . . . . . 9
β’ (π₯ = π β (π( Β·π
βπ)π₯) = (π( Β·π
βπ)π)) |
147 | | sneq 4633 |
. . . . . . . . . . . 12
β’ (π₯ = π β {π₯} = {π}) |
148 | 147 | difeq2d 4117 |
. . . . . . . . . . 11
β’ (π₯ = π β ((πΉ βͺ {π}) β {π₯}) = ((πΉ βͺ {π}) β {π})) |
149 | | difun2 4475 |
. . . . . . . . . . 11
β’ ((πΉ βͺ {π}) β {π}) = (πΉ β {π}) |
150 | 148, 149 | eqtrdi 2782 |
. . . . . . . . . 10
β’ (π₯ = π β ((πΉ βͺ {π}) β {π₯}) = (πΉ β {π})) |
151 | 150 | fveq2d 6889 |
. . . . . . . . 9
β’ (π₯ = π β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) = ((LSpanβπ)β(πΉ β {π}))) |
152 | 146, 151 | eleq12d 2821 |
. . . . . . . 8
β’ (π₯ = π β ((π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β (π( Β·π
βπ)π) β ((LSpanβπ)β(πΉ β {π})))) |
153 | 152 | notbid 318 |
. . . . . . 7
β’ (π₯ = π β (Β¬ (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β Β¬ (π( Β·π
βπ)π) β ((LSpanβπ)β(πΉ β {π})))) |
154 | 153 | ralbidv 3171 |
. . . . . 6
β’ (π₯ = π β (βπ β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) Β¬ (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β βπ β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) Β¬ (π( Β·π
βπ)π) β ((LSpanβπ)β(πΉ β {π})))) |
155 | 154 | ralsng 4672 |
. . . . 5
β’ (π β ((Baseβπ) β ((LSpanβπ)βπΉ)) β (βπ₯ β {π}βπ β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) Β¬ (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β βπ β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) Β¬ (π( Β·π
βπ)π) β ((LSpanβπ)β(πΉ β {π})))) |
156 | 155 | 3ad2ant3 1132 |
. . . 4
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β (βπ₯ β {π}βπ β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) Β¬ (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β βπ β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) Β¬ (π( Β·π
βπ)π) β ((LSpanβπ)β(πΉ β {π})))) |
157 | 145, 156 | mpbird 257 |
. . 3
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β βπ₯ β {π}βπ β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) Β¬ (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯}))) |
158 | | ralunb 4186 |
. . 3
β’
(βπ₯ β
(πΉ βͺ {π})βπ β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) Β¬ (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β (βπ₯ β πΉ βπ β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) Β¬ (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})) β§ βπ₯ β {π}βπ β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) Β¬ (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯})))) |
159 | 113, 157,
158 | sylanbrc 582 |
. 2
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β βπ₯ β (πΉ βͺ {π})βπ β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) Β¬ (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯}))) |
160 | 1, 75, 31, 21, 25, 24 | islinds2 21708 |
. . 3
β’ (π β LVec β ((πΉ βͺ {π}) β (LIndSβπ) β ((πΉ βͺ {π}) β (Baseβπ) β§ βπ₯ β (πΉ βͺ {π})βπ β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) Β¬ (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯}))))) |
161 | 160 | 3ad2ant1 1130 |
. 2
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β ((πΉ βͺ {π}) β (LIndSβπ) β ((πΉ βͺ {π}) β (Baseβπ) β§ βπ₯ β (πΉ βͺ {π})βπ β ((Baseβ(Scalarβπ)) β
{(0gβ(Scalarβπ))}) Β¬ (π( Β·π
βπ)π₯) β ((LSpanβπ)β((πΉ βͺ {π}) β {π₯}))))) |
162 | 8, 159, 161 | mpbir2and 710 |
1
β’ ((π β LVec β§ πΉ β (LIndSβπ) β§ π β ((Baseβπ) β ((LSpanβπ)βπΉ))) β (πΉ βͺ {π}) β (LIndSβπ)) |