Step | Hyp | Ref
| Expression |
1 | | elex 3462 |
. . . 4
β’ (π β π β π β V) |
2 | | islbs.j |
. . . . 5
β’ π½ = (LBasisβπ) |
3 | | fveq2 6843 |
. . . . . . . . 9
β’ (π€ = π β (Baseβπ€) = (Baseβπ)) |
4 | | islbs.v |
. . . . . . . . 9
β’ π = (Baseβπ) |
5 | 3, 4 | eqtr4di 2791 |
. . . . . . . 8
β’ (π€ = π β (Baseβπ€) = π) |
6 | 5 | pweqd 4578 |
. . . . . . 7
β’ (π€ = π β π« (Baseβπ€) = π« π) |
7 | | fvexd 6858 |
. . . . . . . 8
β’ (π€ = π β (LSpanβπ€) β V) |
8 | | fveq2 6843 |
. . . . . . . . 9
β’ (π€ = π β (LSpanβπ€) = (LSpanβπ)) |
9 | | islbs.n |
. . . . . . . . 9
β’ π = (LSpanβπ) |
10 | 8, 9 | eqtr4di 2791 |
. . . . . . . 8
β’ (π€ = π β (LSpanβπ€) = π) |
11 | | fvexd 6858 |
. . . . . . . . 9
β’ ((π€ = π β§ π = π) β (Scalarβπ€) β V) |
12 | | fveq2 6843 |
. . . . . . . . . . 11
β’ (π€ = π β (Scalarβπ€) = (Scalarβπ)) |
13 | 12 | adantr 482 |
. . . . . . . . . 10
β’ ((π€ = π β§ π = π) β (Scalarβπ€) = (Scalarβπ)) |
14 | | islbs.f |
. . . . . . . . . 10
β’ πΉ = (Scalarβπ) |
15 | 13, 14 | eqtr4di 2791 |
. . . . . . . . 9
β’ ((π€ = π β§ π = π) β (Scalarβπ€) = πΉ) |
16 | | simplr 768 |
. . . . . . . . . . . 12
β’ (((π€ = π β§ π = π) β§ π = πΉ) β π = π) |
17 | 16 | fveq1d 6845 |
. . . . . . . . . . 11
β’ (((π€ = π β§ π = π) β§ π = πΉ) β (πβπ) = (πβπ)) |
18 | 5 | ad2antrr 725 |
. . . . . . . . . . 11
β’ (((π€ = π β§ π = π) β§ π = πΉ) β (Baseβπ€) = π) |
19 | 17, 18 | eqeq12d 2749 |
. . . . . . . . . 10
β’ (((π€ = π β§ π = π) β§ π = πΉ) β ((πβπ) = (Baseβπ€) β (πβπ) = π)) |
20 | | simpr 486 |
. . . . . . . . . . . . . . 15
β’ (((π€ = π β§ π = π) β§ π = πΉ) β π = πΉ) |
21 | 20 | fveq2d 6847 |
. . . . . . . . . . . . . 14
β’ (((π€ = π β§ π = π) β§ π = πΉ) β (Baseβπ) = (BaseβπΉ)) |
22 | | islbs.k |
. . . . . . . . . . . . . 14
β’ πΎ = (BaseβπΉ) |
23 | 21, 22 | eqtr4di 2791 |
. . . . . . . . . . . . 13
β’ (((π€ = π β§ π = π) β§ π = πΉ) β (Baseβπ) = πΎ) |
24 | 20 | fveq2d 6847 |
. . . . . . . . . . . . . . 15
β’ (((π€ = π β§ π = π) β§ π = πΉ) β (0gβπ) = (0gβπΉ)) |
25 | | islbs.z |
. . . . . . . . . . . . . . 15
β’ 0 =
(0gβπΉ) |
26 | 24, 25 | eqtr4di 2791 |
. . . . . . . . . . . . . 14
β’ (((π€ = π β§ π = π) β§ π = πΉ) β (0gβπ) = 0 ) |
27 | 26 | sneqd 4599 |
. . . . . . . . . . . . 13
β’ (((π€ = π β§ π = π) β§ π = πΉ) β {(0gβπ)} = { 0 }) |
28 | 23, 27 | difeq12d 4084 |
. . . . . . . . . . . 12
β’ (((π€ = π β§ π = π) β§ π = πΉ) β ((Baseβπ) β {(0gβπ)}) = (πΎ β { 0 })) |
29 | | fveq2 6843 |
. . . . . . . . . . . . . . . . 17
β’ (π€ = π β (
Β·π βπ€) = ( Β·π
βπ)) |
30 | | islbs.s |
. . . . . . . . . . . . . . . . 17
β’ Β· = (
Β·π βπ) |
31 | 29, 30 | eqtr4di 2791 |
. . . . . . . . . . . . . . . 16
β’ (π€ = π β (
Β·π βπ€) = Β· ) |
32 | 31 | ad2antrr 725 |
. . . . . . . . . . . . . . 15
β’ (((π€ = π β§ π = π) β§ π = πΉ) β (
Β·π βπ€) = Β· ) |
33 | 32 | oveqd 7375 |
. . . . . . . . . . . . . 14
β’ (((π€ = π β§ π = π) β§ π = πΉ) β (π¦( Β·π
βπ€)π₯) = (π¦ Β· π₯)) |
34 | 16 | fveq1d 6845 |
. . . . . . . . . . . . . 14
β’ (((π€ = π β§ π = π) β§ π = πΉ) β (πβ(π β {π₯})) = (πβ(π β {π₯}))) |
35 | 33, 34 | eleq12d 2828 |
. . . . . . . . . . . . 13
β’ (((π€ = π β§ π = π) β§ π = πΉ) β ((π¦( Β·π
βπ€)π₯) β (πβ(π β {π₯})) β (π¦ Β· π₯) β (πβ(π β {π₯})))) |
36 | 35 | notbid 318 |
. . . . . . . . . . . 12
β’ (((π€ = π β§ π = π) β§ π = πΉ) β (Β¬ (π¦( Β·π
βπ€)π₯) β (πβ(π β {π₯})) β Β¬ (π¦ Β· π₯) β (πβ(π β {π₯})))) |
37 | 28, 36 | raleqbidv 3318 |
. . . . . . . . . . 11
β’ (((π€ = π β§ π = π) β§ π = πΉ) β (βπ¦ β ((Baseβπ) β {(0gβπ)}) Β¬ (π¦( Β·π
βπ€)π₯) β (πβ(π β {π₯})) β βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯})))) |
38 | 37 | ralbidv 3171 |
. . . . . . . . . 10
β’ (((π€ = π β§ π = π) β§ π = πΉ) β (βπ₯ β π βπ¦ β ((Baseβπ) β {(0gβπ)}) Β¬ (π¦( Β·π
βπ€)π₯) β (πβ(π β {π₯})) β βπ₯ β π βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯})))) |
39 | 19, 38 | anbi12d 632 |
. . . . . . . . 9
β’ (((π€ = π β§ π = π) β§ π = πΉ) β (((πβπ) = (Baseβπ€) β§ βπ₯ β π βπ¦ β ((Baseβπ) β {(0gβπ)}) Β¬ (π¦( Β·π
βπ€)π₯) β (πβ(π β {π₯}))) β ((πβπ) = π β§ βπ₯ β π βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯}))))) |
40 | 11, 15, 39 | sbcied2 3787 |
. . . . . . . 8
β’ ((π€ = π β§ π = π) β ([(Scalarβπ€) / π]((πβπ) = (Baseβπ€) β§ βπ₯ β π βπ¦ β ((Baseβπ) β {(0gβπ)}) Β¬ (π¦( Β·π
βπ€)π₯) β (πβ(π β {π₯}))) β ((πβπ) = π β§ βπ₯ β π βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯}))))) |
41 | 7, 10, 40 | sbcied2 3787 |
. . . . . . 7
β’ (π€ = π β ([(LSpanβπ€) / π][(Scalarβπ€) / π]((πβπ) = (Baseβπ€) β§ βπ₯ β π βπ¦ β ((Baseβπ) β {(0gβπ)}) Β¬ (π¦( Β·π
βπ€)π₯) β (πβ(π β {π₯}))) β ((πβπ) = π β§ βπ₯ β π βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯}))))) |
42 | 6, 41 | rabeqbidv 3423 |
. . . . . 6
β’ (π€ = π β {π β π« (Baseβπ€) β£
[(LSpanβπ€) /
π][(Scalarβπ€) / π]((πβπ) = (Baseβπ€) β§ βπ₯ β π βπ¦ β ((Baseβπ) β {(0gβπ)}) Β¬ (π¦( Β·π
βπ€)π₯) β (πβ(π β {π₯})))} = {π β π« π β£ ((πβπ) = π β§ βπ₯ β π βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯})))}) |
43 | | df-lbs 20551 |
. . . . . 6
β’ LBasis =
(π€ β V β¦ {π β π«
(Baseβπ€) β£
[(LSpanβπ€) /
π][(Scalarβπ€) / π]((πβπ) = (Baseβπ€) β§ βπ₯ β π βπ¦ β ((Baseβπ) β {(0gβπ)}) Β¬ (π¦( Β·π
βπ€)π₯) β (πβ(π β {π₯})))}) |
44 | 4 | fvexi 6857 |
. . . . . . . 8
β’ π β V |
45 | 44 | pwex 5336 |
. . . . . . 7
β’ π«
π β V |
46 | 45 | rabex 5290 |
. . . . . 6
β’ {π β π« π β£ ((πβπ) = π β§ βπ₯ β π βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯})))} β V |
47 | 42, 43, 46 | fvmpt 6949 |
. . . . 5
β’ (π β V β
(LBasisβπ) = {π β π« π β£ ((πβπ) = π β§ βπ₯ β π βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯})))}) |
48 | 2, 47 | eqtrid 2785 |
. . . 4
β’ (π β V β π½ = {π β π« π β£ ((πβπ) = π β§ βπ₯ β π βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯})))}) |
49 | 1, 48 | syl 17 |
. . 3
β’ (π β π β π½ = {π β π« π β£ ((πβπ) = π β§ βπ₯ β π βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯})))}) |
50 | 49 | eleq2d 2820 |
. 2
β’ (π β π β (π΅ β π½ β π΅ β {π β π« π β£ ((πβπ) = π β§ βπ₯ β π βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯})))})) |
51 | 44 | elpw2 5303 |
. . . 4
β’ (π΅ β π« π β π΅ β π) |
52 | 51 | anbi1i 625 |
. . 3
β’ ((π΅ β π« π β§ ((πβπ΅) = π β§ βπ₯ β π΅ βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π΅ β {π₯})))) β (π΅ β π β§ ((πβπ΅) = π β§ βπ₯ β π΅ βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π΅ β {π₯}))))) |
53 | | fveqeq2 6852 |
. . . . 5
β’ (π = π΅ β ((πβπ) = π β (πβπ΅) = π)) |
54 | | difeq1 4076 |
. . . . . . . . . 10
β’ (π = π΅ β (π β {π₯}) = (π΅ β {π₯})) |
55 | 54 | fveq2d 6847 |
. . . . . . . . 9
β’ (π = π΅ β (πβ(π β {π₯})) = (πβ(π΅ β {π₯}))) |
56 | 55 | eleq2d 2820 |
. . . . . . . 8
β’ (π = π΅ β ((π¦ Β· π₯) β (πβ(π β {π₯})) β (π¦ Β· π₯) β (πβ(π΅ β {π₯})))) |
57 | 56 | notbid 318 |
. . . . . . 7
β’ (π = π΅ β (Β¬ (π¦ Β· π₯) β (πβ(π β {π₯})) β Β¬ (π¦ Β· π₯) β (πβ(π΅ β {π₯})))) |
58 | 57 | ralbidv 3171 |
. . . . . 6
β’ (π = π΅ β (βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯})) β βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π΅ β {π₯})))) |
59 | 58 | raleqbi1dv 3306 |
. . . . 5
β’ (π = π΅ β (βπ₯ β π βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯})) β βπ₯ β π΅ βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π΅ β {π₯})))) |
60 | 53, 59 | anbi12d 632 |
. . . 4
β’ (π = π΅ β (((πβπ) = π β§ βπ₯ β π βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯}))) β ((πβπ΅) = π β§ βπ₯ β π΅ βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π΅ β {π₯}))))) |
61 | 60 | elrab 3646 |
. . 3
β’ (π΅ β {π β π« π β£ ((πβπ) = π β§ βπ₯ β π βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯})))} β (π΅ β π« π β§ ((πβπ΅) = π β§ βπ₯ β π΅ βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π΅ β {π₯}))))) |
62 | | 3anass 1096 |
. . 3
β’ ((π΅ β π β§ (πβπ΅) = π β§ βπ₯ β π΅ βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π΅ β {π₯}))) β (π΅ β π β§ ((πβπ΅) = π β§ βπ₯ β π΅ βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π΅ β {π₯}))))) |
63 | 52, 61, 62 | 3bitr4i 303 |
. 2
β’ (π΅ β {π β π« π β£ ((πβπ) = π β§ βπ₯ β π βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π β {π₯})))} β (π΅ β π β§ (πβπ΅) = π β§ βπ₯ β π΅ βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π΅ β {π₯})))) |
64 | 50, 63 | bitrdi 287 |
1
β’ (π β π β (π΅ β π½ β (π΅ β π β§ (πβπ΅) = π β§ βπ₯ β π΅ βπ¦ β (πΎ β { 0 }) Β¬ (π¦ Β· π₯) β (πβ(π΅ β {π₯}))))) |