Step | Hyp | Ref
| Expression |
1 | | minveco.x |
. . 3
β’ π = (BaseSetβπ) |
2 | | minveco.m |
. . 3
β’ π = ( βπ£
βπ) |
3 | | minveco.n |
. . 3
β’ π =
(normCVβπ) |
4 | | minveco.y |
. . 3
β’ π = (BaseSetβπ) |
5 | | minveco.u |
. . 3
β’ (π β π β
CPreHilOLD) |
6 | | minveco.w |
. . 3
β’ (π β π β ((SubSpβπ) β© CBan)) |
7 | | minveco.a |
. . 3
β’ (π β π΄ β π) |
8 | | minveco.d |
. . 3
β’ π· = (IndMetβπ) |
9 | | minveco.j |
. . 3
β’ π½ = (MetOpenβπ·) |
10 | | minveco.r |
. . 3
β’ π
= ran (π¦ β π β¦ (πβ(π΄ππ¦))) |
11 | | minveco.s |
. . 3
β’ π = inf(π
, β, < ) |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11 | minvecolem5 29865 |
. 2
β’ (π β βπ₯ β π βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦))) |
13 | 5 | ad2antrr 725 |
. . . . . 6
β’ (((π β§ (π₯ β π β§ π€ β π)) β§ (((π΄π·π₯)β2) β€ ((πβ2) + 0) β§ ((π΄π·π€)β2) β€ ((πβ2) + 0))) β π β
CPreHilOLD) |
14 | 6 | ad2antrr 725 |
. . . . . 6
β’ (((π β§ (π₯ β π β§ π€ β π)) β§ (((π΄π·π₯)β2) β€ ((πβ2) + 0) β§ ((π΄π·π€)β2) β€ ((πβ2) + 0))) β π β ((SubSpβπ) β© CBan)) |
15 | 7 | ad2antrr 725 |
. . . . . 6
β’ (((π β§ (π₯ β π β§ π€ β π)) β§ (((π΄π·π₯)β2) β€ ((πβ2) + 0) β§ ((π΄π·π€)β2) β€ ((πβ2) + 0))) β π΄ β π) |
16 | | 0re 11164 |
. . . . . . 7
β’ 0 β
β |
17 | 16 | a1i 11 |
. . . . . 6
β’ (((π β§ (π₯ β π β§ π€ β π)) β§ (((π΄π·π₯)β2) β€ ((πβ2) + 0) β§ ((π΄π·π€)β2) β€ ((πβ2) + 0))) β 0 β
β) |
18 | | 0le0 12261 |
. . . . . . 7
β’ 0 β€
0 |
19 | 18 | a1i 11 |
. . . . . 6
β’ (((π β§ (π₯ β π β§ π€ β π)) β§ (((π΄π·π₯)β2) β€ ((πβ2) + 0) β§ ((π΄π·π€)β2) β€ ((πβ2) + 0))) β 0 β€
0) |
20 | | simplrl 776 |
. . . . . 6
β’ (((π β§ (π₯ β π β§ π€ β π)) β§ (((π΄π·π₯)β2) β€ ((πβ2) + 0) β§ ((π΄π·π€)β2) β€ ((πβ2) + 0))) β π₯ β π) |
21 | | simplrr 777 |
. . . . . 6
β’ (((π β§ (π₯ β π β§ π€ β π)) β§ (((π΄π·π₯)β2) β€ ((πβ2) + 0) β§ ((π΄π·π€)β2) β€ ((πβ2) + 0))) β π€ β π) |
22 | | simprl 770 |
. . . . . 6
β’ (((π β§ (π₯ β π β§ π€ β π)) β§ (((π΄π·π₯)β2) β€ ((πβ2) + 0) β§ ((π΄π·π€)β2) β€ ((πβ2) + 0))) β ((π΄π·π₯)β2) β€ ((πβ2) + 0)) |
23 | | simprr 772 |
. . . . . 6
β’ (((π β§ (π₯ β π β§ π€ β π)) β§ (((π΄π·π₯)β2) β€ ((πβ2) + 0) β§ ((π΄π·π€)β2) β€ ((πβ2) + 0))) β ((π΄π·π€)β2) β€ ((πβ2) + 0)) |
24 | 1, 2, 3, 4, 13, 14, 15, 8, 9, 10, 11, 17, 19, 20, 21, 22, 23 | minvecolem2 29859 |
. . . . 5
β’ (((π β§ (π₯ β π β§ π€ β π)) β§ (((π΄π·π₯)β2) β€ ((πβ2) + 0) β§ ((π΄π·π€)β2) β€ ((πβ2) + 0))) β ((π₯π·π€)β2) β€ (4 Β·
0)) |
25 | 24 | ex 414 |
. . . 4
β’ ((π β§ (π₯ β π β§ π€ β π)) β ((((π΄π·π₯)β2) β€ ((πβ2) + 0) β§ ((π΄π·π€)β2) β€ ((πβ2) + 0)) β ((π₯π·π€)β2) β€ (4 Β·
0))) |
26 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11 | minvecolem6 29866 |
. . . . . 6
β’ ((π β§ π₯ β π) β (((π΄π·π₯)β2) β€ ((πβ2) + 0) β βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦)))) |
27 | 26 | adantrr 716 |
. . . . 5
β’ ((π β§ (π₯ β π β§ π€ β π)) β (((π΄π·π₯)β2) β€ ((πβ2) + 0) β βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦)))) |
28 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11 | minvecolem6 29866 |
. . . . . 6
β’ ((π β§ π€ β π) β (((π΄π·π€)β2) β€ ((πβ2) + 0) β βπ¦ β π (πβ(π΄ππ€)) β€ (πβ(π΄ππ¦)))) |
29 | 28 | adantrl 715 |
. . . . 5
β’ ((π β§ (π₯ β π β§ π€ β π)) β (((π΄π·π€)β2) β€ ((πβ2) + 0) β βπ¦ β π (πβ(π΄ππ€)) β€ (πβ(π΄ππ¦)))) |
30 | 27, 29 | anbi12d 632 |
. . . 4
β’ ((π β§ (π₯ β π β§ π€ β π)) β ((((π΄π·π₯)β2) β€ ((πβ2) + 0) β§ ((π΄π·π€)β2) β€ ((πβ2) + 0)) β (βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦)) β§ βπ¦ β π (πβ(π΄ππ€)) β€ (πβ(π΄ππ¦))))) |
31 | | 4cn 12245 |
. . . . . . 7
β’ 4 β
β |
32 | 31 | mul01i 11352 |
. . . . . 6
β’ (4
Β· 0) = 0 |
33 | 32 | breq2i 5118 |
. . . . 5
β’ (((π₯π·π€)β2) β€ (4 Β· 0) β ((π₯π·π€)β2) β€ 0) |
34 | | phnv 29798 |
. . . . . . . . . . . 12
β’ (π β CPreHilOLD
β π β
NrmCVec) |
35 | 5, 34 | syl 17 |
. . . . . . . . . . 11
β’ (π β π β NrmCVec) |
36 | 35 | adantr 482 |
. . . . . . . . . 10
β’ ((π β§ (π₯ β π β§ π€ β π)) β π β NrmCVec) |
37 | 1, 8 | imsmet 29675 |
. . . . . . . . . 10
β’ (π β NrmCVec β π· β (Metβπ)) |
38 | 36, 37 | syl 17 |
. . . . . . . . 9
β’ ((π β§ (π₯ β π β§ π€ β π)) β π· β (Metβπ)) |
39 | | inss1 4193 |
. . . . . . . . . . . . 13
β’
((SubSpβπ)
β© CBan) β (SubSpβπ) |
40 | 39, 6 | sselid 3947 |
. . . . . . . . . . . 12
β’ (π β π β (SubSpβπ)) |
41 | | eqid 2737 |
. . . . . . . . . . . . 13
β’
(SubSpβπ) =
(SubSpβπ) |
42 | 1, 4, 41 | sspba 29711 |
. . . . . . . . . . . 12
β’ ((π β NrmCVec β§ π β (SubSpβπ)) β π β π) |
43 | 35, 40, 42 | syl2anc 585 |
. . . . . . . . . . 11
β’ (π β π β π) |
44 | 43 | adantr 482 |
. . . . . . . . . 10
β’ ((π β§ (π₯ β π β§ π€ β π)) β π β π) |
45 | | simprl 770 |
. . . . . . . . . 10
β’ ((π β§ (π₯ β π β§ π€ β π)) β π₯ β π) |
46 | 44, 45 | sseldd 3950 |
. . . . . . . . 9
β’ ((π β§ (π₯ β π β§ π€ β π)) β π₯ β π) |
47 | | simprr 772 |
. . . . . . . . . 10
β’ ((π β§ (π₯ β π β§ π€ β π)) β π€ β π) |
48 | 44, 47 | sseldd 3950 |
. . . . . . . . 9
β’ ((π β§ (π₯ β π β§ π€ β π)) β π€ β π) |
49 | | metcl 23701 |
. . . . . . . . 9
β’ ((π· β (Metβπ) β§ π₯ β π β§ π€ β π) β (π₯π·π€) β β) |
50 | 38, 46, 48, 49 | syl3anc 1372 |
. . . . . . . 8
β’ ((π β§ (π₯ β π β§ π€ β π)) β (π₯π·π€) β β) |
51 | 50 | sqge0d 14049 |
. . . . . . 7
β’ ((π β§ (π₯ β π β§ π€ β π)) β 0 β€ ((π₯π·π€)β2)) |
52 | 51 | biantrud 533 |
. . . . . 6
β’ ((π β§ (π₯ β π β§ π€ β π)) β (((π₯π·π€)β2) β€ 0 β (((π₯π·π€)β2) β€ 0 β§ 0 β€ ((π₯π·π€)β2)))) |
53 | 50 | resqcld 14037 |
. . . . . . 7
β’ ((π β§ (π₯ β π β§ π€ β π)) β ((π₯π·π€)β2) β β) |
54 | | letri3 11247 |
. . . . . . 7
β’ ((((π₯π·π€)β2) β β β§ 0 β
β) β (((π₯π·π€)β2) = 0 β (((π₯π·π€)β2) β€ 0 β§ 0 β€ ((π₯π·π€)β2)))) |
55 | 53, 16, 54 | sylancl 587 |
. . . . . 6
β’ ((π β§ (π₯ β π β§ π€ β π)) β (((π₯π·π€)β2) = 0 β (((π₯π·π€)β2) β€ 0 β§ 0 β€ ((π₯π·π€)β2)))) |
56 | 50 | recnd 11190 |
. . . . . . . 8
β’ ((π β§ (π₯ β π β§ π€ β π)) β (π₯π·π€) β β) |
57 | | sqeq0 14032 |
. . . . . . . 8
β’ ((π₯π·π€) β β β (((π₯π·π€)β2) = 0 β (π₯π·π€) = 0)) |
58 | 56, 57 | syl 17 |
. . . . . . 7
β’ ((π β§ (π₯ β π β§ π€ β π)) β (((π₯π·π€)β2) = 0 β (π₯π·π€) = 0)) |
59 | | meteq0 23708 |
. . . . . . . 8
β’ ((π· β (Metβπ) β§ π₯ β π β§ π€ β π) β ((π₯π·π€) = 0 β π₯ = π€)) |
60 | 38, 46, 48, 59 | syl3anc 1372 |
. . . . . . 7
β’ ((π β§ (π₯ β π β§ π€ β π)) β ((π₯π·π€) = 0 β π₯ = π€)) |
61 | 58, 60 | bitrd 279 |
. . . . . 6
β’ ((π β§ (π₯ β π β§ π€ β π)) β (((π₯π·π€)β2) = 0 β π₯ = π€)) |
62 | 52, 55, 61 | 3bitr2d 307 |
. . . . 5
β’ ((π β§ (π₯ β π β§ π€ β π)) β (((π₯π·π€)β2) β€ 0 β π₯ = π€)) |
63 | 33, 62 | bitrid 283 |
. . . 4
β’ ((π β§ (π₯ β π β§ π€ β π)) β (((π₯π·π€)β2) β€ (4 Β· 0) β π₯ = π€)) |
64 | 25, 30, 63 | 3imtr3d 293 |
. . 3
β’ ((π β§ (π₯ β π β§ π€ β π)) β ((βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦)) β§ βπ¦ β π (πβ(π΄ππ€)) β€ (πβ(π΄ππ¦))) β π₯ = π€)) |
65 | 64 | ralrimivva 3198 |
. 2
β’ (π β βπ₯ β π βπ€ β π ((βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦)) β§ βπ¦ β π (πβ(π΄ππ€)) β€ (πβ(π΄ππ¦))) β π₯ = π€)) |
66 | | oveq2 7370 |
. . . . . 6
β’ (π₯ = π€ β (π΄ππ₯) = (π΄ππ€)) |
67 | 66 | fveq2d 6851 |
. . . . 5
β’ (π₯ = π€ β (πβ(π΄ππ₯)) = (πβ(π΄ππ€))) |
68 | 67 | breq1d 5120 |
. . . 4
β’ (π₯ = π€ β ((πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦)) β (πβ(π΄ππ€)) β€ (πβ(π΄ππ¦)))) |
69 | 68 | ralbidv 3175 |
. . 3
β’ (π₯ = π€ β (βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦)) β βπ¦ β π (πβ(π΄ππ€)) β€ (πβ(π΄ππ¦)))) |
70 | 69 | reu4 3694 |
. 2
β’
(β!π₯ β
π βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦)) β (βπ₯ β π βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦)) β§ βπ₯ β π βπ€ β π ((βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦)) β§ βπ¦ β π (πβ(π΄ππ€)) β€ (πβ(π΄ππ¦))) β π₯ = π€))) |
71 | 12, 65, 70 | sylanbrc 584 |
1
β’ (π β β!π₯ β π βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦))) |