Step | Hyp | Ref
| Expression |
1 | | metucn.u |
. . . . . 6
β’ π = (metUnifβπΆ) |
2 | | metucn.c |
. . . . . . 7
β’ (π β πΆ β (PsMetβπ)) |
3 | | metuval 23928 |
. . . . . . 7
β’ (πΆ β (PsMetβπ) β (metUnifβπΆ) = ((π Γ π)filGenran (π β β+ β¦ (β‘πΆ β (0[,)π))))) |
4 | 2, 3 | syl 17 |
. . . . . 6
β’ (π β (metUnifβπΆ) = ((π Γ π)filGenran (π β β+ β¦ (β‘πΆ β (0[,)π))))) |
5 | 1, 4 | eqtrid 2785 |
. . . . 5
β’ (π β π = ((π Γ π)filGenran (π β β+ β¦ (β‘πΆ β (0[,)π))))) |
6 | | metucn.v |
. . . . . 6
β’ π = (metUnifβπ·) |
7 | | metucn.d |
. . . . . . 7
β’ (π β π· β (PsMetβπ)) |
8 | | metuval 23928 |
. . . . . . 7
β’ (π· β (PsMetβπ) β (metUnifβπ·) = ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π))))) |
9 | 7, 8 | syl 17 |
. . . . . 6
β’ (π β (metUnifβπ·) = ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π))))) |
10 | 6, 9 | eqtrid 2785 |
. . . . 5
β’ (π β π = ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π))))) |
11 | 5, 10 | oveq12d 7379 |
. . . 4
β’ (π β (π Cnuπ) = (((π Γ π)filGenran (π β β+ β¦ (β‘πΆ β (0[,)π)))) Cnu((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π)))))) |
12 | 11 | eleq2d 2820 |
. . 3
β’ (π β (πΉ β (π Cnuπ) β πΉ β (((π Γ π)filGenran (π β β+ β¦ (β‘πΆ β (0[,)π)))) Cnu((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π))))))) |
13 | | eqid 2733 |
. . . 4
β’ ((π Γ π)filGenran (π β β+ β¦ (β‘πΆ β (0[,)π)))) = ((π Γ π)filGenran (π β β+ β¦ (β‘πΆ β (0[,)π)))) |
14 | | eqid 2733 |
. . . 4
β’ ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π)))) = ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π)))) |
15 | | metucn.x |
. . . . 5
β’ (π β π β β
) |
16 | | oveq2 7369 |
. . . . . . . . 9
β’ (π = π β (0[,)π) = (0[,)π)) |
17 | 16 | imaeq2d 6017 |
. . . . . . . 8
β’ (π = π β (β‘πΆ β (0[,)π)) = (β‘πΆ β (0[,)π))) |
18 | 17 | cbvmptv 5222 |
. . . . . . 7
β’ (π β β+
β¦ (β‘πΆ β (0[,)π))) = (π β β+ β¦ (β‘πΆ β (0[,)π))) |
19 | 18 | rneqi 5896 |
. . . . . 6
β’ ran
(π β
β+ β¦ (β‘πΆ β (0[,)π))) = ran (π β β+ β¦ (β‘πΆ β (0[,)π))) |
20 | 19 | metust 23937 |
. . . . 5
β’ ((π β β
β§ πΆ β (PsMetβπ)) β ((π Γ π)filGenran (π β β+ β¦ (β‘πΆ β (0[,)π)))) β (UnifOnβπ)) |
21 | 15, 2, 20 | syl2anc 585 |
. . . 4
β’ (π β ((π Γ π)filGenran (π β β+ β¦ (β‘πΆ β (0[,)π)))) β (UnifOnβπ)) |
22 | | metucn.y |
. . . . 5
β’ (π β π β β
) |
23 | | oveq2 7369 |
. . . . . . . . 9
β’ (π = π β (0[,)π) = (0[,)π)) |
24 | 23 | imaeq2d 6017 |
. . . . . . . 8
β’ (π = π β (β‘π· β (0[,)π)) = (β‘π· β (0[,)π))) |
25 | 24 | cbvmptv 5222 |
. . . . . . 7
β’ (π β β+
β¦ (β‘π· β (0[,)π))) = (π β β+ β¦ (β‘π· β (0[,)π))) |
26 | 25 | rneqi 5896 |
. . . . . 6
β’ ran
(π β
β+ β¦ (β‘π· β (0[,)π))) = ran (π β β+ β¦ (β‘π· β (0[,)π))) |
27 | 26 | metust 23937 |
. . . . 5
β’ ((π β β
β§ π· β (PsMetβπ)) β ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π)))) β (UnifOnβπ)) |
28 | 22, 7, 27 | syl2anc 585 |
. . . 4
β’ (π β ((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π)))) β (UnifOnβπ)) |
29 | | oveq2 7369 |
. . . . . . . . 9
β’ (π = π β (0[,)π) = (0[,)π)) |
30 | 29 | imaeq2d 6017 |
. . . . . . . 8
β’ (π = π β (β‘πΆ β (0[,)π)) = (β‘πΆ β (0[,)π))) |
31 | 30 | cbvmptv 5222 |
. . . . . . 7
β’ (π β β+
β¦ (β‘πΆ β (0[,)π))) = (π β β+ β¦ (β‘πΆ β (0[,)π))) |
32 | 31 | rneqi 5896 |
. . . . . 6
β’ ran
(π β
β+ β¦ (β‘πΆ β (0[,)π))) = ran (π β β+ β¦ (β‘πΆ β (0[,)π))) |
33 | 32 | metustfbas 23936 |
. . . . 5
β’ ((π β β
β§ πΆ β (PsMetβπ)) β ran (π β β+
β¦ (β‘πΆ β (0[,)π))) β (fBasβ(π Γ π))) |
34 | 15, 2, 33 | syl2anc 585 |
. . . 4
β’ (π β ran (π β β+ β¦ (β‘πΆ β (0[,)π))) β (fBasβ(π Γ π))) |
35 | | oveq2 7369 |
. . . . . . . . 9
β’ (π = π β (0[,)π) = (0[,)π)) |
36 | 35 | imaeq2d 6017 |
. . . . . . . 8
β’ (π = π β (β‘π· β (0[,)π)) = (β‘π· β (0[,)π))) |
37 | 36 | cbvmptv 5222 |
. . . . . . 7
β’ (π β β+
β¦ (β‘π· β (0[,)π))) = (π β β+ β¦ (β‘π· β (0[,)π))) |
38 | 37 | rneqi 5896 |
. . . . . 6
β’ ran
(π β
β+ β¦ (β‘π· β (0[,)π))) = ran (π β β+ β¦ (β‘π· β (0[,)π))) |
39 | 38 | metustfbas 23936 |
. . . . 5
β’ ((π β β
β§ π· β (PsMetβπ)) β ran (π β β+
β¦ (β‘π· β (0[,)π))) β (fBasβ(π Γ π))) |
40 | 22, 7, 39 | syl2anc 585 |
. . . 4
β’ (π β ran (π β β+ β¦ (β‘π· β (0[,)π))) β (fBasβ(π Γ π))) |
41 | 13, 14, 21, 28, 34, 40 | isucn2 23654 |
. . 3
β’ (π β (πΉ β (((π Γ π)filGenran (π β β+ β¦ (β‘πΆ β (0[,)π)))) Cnu((π Γ π)filGenran (π β β+ β¦ (β‘π· β (0[,)π))))) β (πΉ:πβΆπ β§ βπ£ β ran (π β β+ β¦ (β‘π· β (0[,)π)))βπ’ β ran (π β β+ β¦ (β‘πΆ β (0[,)π)))βπ₯ β π βπ¦ β π (π₯π’π¦ β (πΉβπ₯)π£(πΉβπ¦))))) |
42 | 12, 41 | bitrd 279 |
. 2
β’ (π β (πΉ β (π Cnuπ) β (πΉ:πβΆπ β§ βπ£ β ran (π β β+ β¦ (β‘π· β (0[,)π)))βπ’ β ran (π β β+ β¦ (β‘πΆ β (0[,)π)))βπ₯ β π βπ¦ β π (π₯π’π¦ β (πΉβπ₯)π£(πΉβπ¦))))) |
43 | | eqid 2733 |
. . . . . . . . . 10
β’ (β‘π· β (0[,)π)) = (β‘π· β (0[,)π)) |
44 | | oveq2 7369 |
. . . . . . . . . . . 12
β’ (π = π β (0[,)π) = (0[,)π)) |
45 | 44 | imaeq2d 6017 |
. . . . . . . . . . 11
β’ (π = π β (β‘π· β (0[,)π)) = (β‘π· β (0[,)π))) |
46 | 45 | rspceeqv 3599 |
. . . . . . . . . 10
β’ ((π β β+
β§ (β‘π· β (0[,)π)) = (β‘π· β (0[,)π))) β βπ β β+ (β‘π· β (0[,)π)) = (β‘π· β (0[,)π))) |
47 | 43, 46 | mpan2 690 |
. . . . . . . . 9
β’ (π β β+
β βπ β
β+ (β‘π· β (0[,)π)) = (β‘π· β (0[,)π))) |
48 | 47 | adantl 483 |
. . . . . . . 8
β’ ((π β§ π β β+) β
βπ β
β+ (β‘π· β (0[,)π)) = (β‘π· β (0[,)π))) |
49 | 38 | metustel 23929 |
. . . . . . . . . 10
β’ (π· β (PsMetβπ) β ((β‘π· β (0[,)π)) β ran (π β β+ β¦ (β‘π· β (0[,)π))) β βπ β β+ (β‘π· β (0[,)π)) = (β‘π· β (0[,)π)))) |
50 | 7, 49 | syl 17 |
. . . . . . . . 9
β’ (π β ((β‘π· β (0[,)π)) β ran (π β β+ β¦ (β‘π· β (0[,)π))) β βπ β β+ (β‘π· β (0[,)π)) = (β‘π· β (0[,)π)))) |
51 | 50 | adantr 482 |
. . . . . . . 8
β’ ((π β§ π β β+) β ((β‘π· β (0[,)π)) β ran (π β β+ β¦ (β‘π· β (0[,)π))) β βπ β β+ (β‘π· β (0[,)π)) = (β‘π· β (0[,)π)))) |
52 | 48, 51 | mpbird 257 |
. . . . . . 7
β’ ((π β§ π β β+) β (β‘π· β (0[,)π)) β ran (π β β+ β¦ (β‘π· β (0[,)π)))) |
53 | 26 | metustel 23929 |
. . . . . . . 8
β’ (π· β (PsMetβπ) β (π£ β ran (π β β+ β¦ (β‘π· β (0[,)π))) β βπ β β+ π£ = (β‘π· β (0[,)π)))) |
54 | 7, 53 | syl 17 |
. . . . . . 7
β’ (π β (π£ β ran (π β β+ β¦ (β‘π· β (0[,)π))) β βπ β β+ π£ = (β‘π· β (0[,)π)))) |
55 | | simpr 486 |
. . . . . . . . . . 11
β’ ((π β§ π£ = (β‘π· β (0[,)π))) β π£ = (β‘π· β (0[,)π))) |
56 | 55 | breqd 5120 |
. . . . . . . . . 10
β’ ((π β§ π£ = (β‘π· β (0[,)π))) β ((πΉβπ₯)π£(πΉβπ¦) β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦))) |
57 | 56 | imbi2d 341 |
. . . . . . . . 9
β’ ((π β§ π£ = (β‘π· β (0[,)π))) β ((π₯π’π¦ β (πΉβπ₯)π£(πΉβπ¦)) β (π₯π’π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)))) |
58 | 57 | ralbidv 3171 |
. . . . . . . 8
β’ ((π β§ π£ = (β‘π· β (0[,)π))) β (βπ¦ β π (π₯π’π¦ β (πΉβπ₯)π£(πΉβπ¦)) β βπ¦ β π (π₯π’π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)))) |
59 | 58 | rexralbidv 3211 |
. . . . . . 7
β’ ((π β§ π£ = (β‘π· β (0[,)π))) β (βπ’ β ran (π β β+ β¦ (β‘πΆ β (0[,)π)))βπ₯ β π βπ¦ β π (π₯π’π¦ β (πΉβπ₯)π£(πΉβπ¦)) β βπ’ β ran (π β β+ β¦ (β‘πΆ β (0[,)π)))βπ₯ β π βπ¦ β π (π₯π’π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)))) |
60 | 52, 54, 59 | ralxfr2d 5369 |
. . . . . 6
β’ (π β (βπ£ β ran (π β β+ β¦ (β‘π· β (0[,)π)))βπ’ β ran (π β β+ β¦ (β‘πΆ β (0[,)π)))βπ₯ β π βπ¦ β π (π₯π’π¦ β (πΉβπ₯)π£(πΉβπ¦)) β βπ β β+ βπ’ β ran (π β β+ β¦ (β‘πΆ β (0[,)π)))βπ₯ β π βπ¦ β π (π₯π’π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)))) |
61 | | eqid 2733 |
. . . . . . . . . . 11
β’ (β‘πΆ β (0[,)π)) = (β‘πΆ β (0[,)π)) |
62 | | oveq2 7369 |
. . . . . . . . . . . . 13
β’ (π = π β (0[,)π) = (0[,)π)) |
63 | 62 | imaeq2d 6017 |
. . . . . . . . . . . 12
β’ (π = π β (β‘πΆ β (0[,)π)) = (β‘πΆ β (0[,)π))) |
64 | 63 | rspceeqv 3599 |
. . . . . . . . . . 11
β’ ((π β β+
β§ (β‘πΆ β (0[,)π)) = (β‘πΆ β (0[,)π))) β βπ β β+ (β‘πΆ β (0[,)π)) = (β‘πΆ β (0[,)π))) |
65 | 61, 64 | mpan2 690 |
. . . . . . . . . 10
β’ (π β β+
β βπ β
β+ (β‘πΆ β (0[,)π)) = (β‘πΆ β (0[,)π))) |
66 | 65 | adantl 483 |
. . . . . . . . 9
β’ ((π β§ π β β+) β
βπ β
β+ (β‘πΆ β (0[,)π)) = (β‘πΆ β (0[,)π))) |
67 | 32 | metustel 23929 |
. . . . . . . . . . 11
β’ (πΆ β (PsMetβπ) β ((β‘πΆ β (0[,)π)) β ran (π β β+ β¦ (β‘πΆ β (0[,)π))) β βπ β β+ (β‘πΆ β (0[,)π)) = (β‘πΆ β (0[,)π)))) |
68 | 2, 67 | syl 17 |
. . . . . . . . . 10
β’ (π β ((β‘πΆ β (0[,)π)) β ran (π β β+ β¦ (β‘πΆ β (0[,)π))) β βπ β β+ (β‘πΆ β (0[,)π)) = (β‘πΆ β (0[,)π)))) |
69 | 68 | adantr 482 |
. . . . . . . . 9
β’ ((π β§ π β β+) β ((β‘πΆ β (0[,)π)) β ran (π β β+ β¦ (β‘πΆ β (0[,)π))) β βπ β β+ (β‘πΆ β (0[,)π)) = (β‘πΆ β (0[,)π)))) |
70 | 66, 69 | mpbird 257 |
. . . . . . . 8
β’ ((π β§ π β β+) β (β‘πΆ β (0[,)π)) β ran (π β β+ β¦ (β‘πΆ β (0[,)π)))) |
71 | 19 | metustel 23929 |
. . . . . . . . 9
β’ (πΆ β (PsMetβπ) β (π’ β ran (π β β+ β¦ (β‘πΆ β (0[,)π))) β βπ β β+ π’ = (β‘πΆ β (0[,)π)))) |
72 | 2, 71 | syl 17 |
. . . . . . . 8
β’ (π β (π’ β ran (π β β+ β¦ (β‘πΆ β (0[,)π))) β βπ β β+ π’ = (β‘πΆ β (0[,)π)))) |
73 | | simpr 486 |
. . . . . . . . . . 11
β’ ((π β§ π’ = (β‘πΆ β (0[,)π))) β π’ = (β‘πΆ β (0[,)π))) |
74 | 73 | breqd 5120 |
. . . . . . . . . 10
β’ ((π β§ π’ = (β‘πΆ β (0[,)π))) β (π₯π’π¦ β π₯(β‘πΆ β (0[,)π))π¦)) |
75 | 74 | imbi1d 342 |
. . . . . . . . 9
β’ ((π β§ π’ = (β‘πΆ β (0[,)π))) β ((π₯π’π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)) β (π₯(β‘πΆ β (0[,)π))π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)))) |
76 | 75 | 2ralbidv 3209 |
. . . . . . . 8
β’ ((π β§ π’ = (β‘πΆ β (0[,)π))) β (βπ₯ β π βπ¦ β π (π₯π’π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)) β βπ₯ β π βπ¦ β π (π₯(β‘πΆ β (0[,)π))π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)))) |
77 | 70, 72, 76 | rexxfr2d 5370 |
. . . . . . 7
β’ (π β (βπ’ β ran (π β β+ β¦ (β‘πΆ β (0[,)π)))βπ₯ β π βπ¦ β π (π₯π’π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)) β βπ β β+ βπ₯ β π βπ¦ β π (π₯(β‘πΆ β (0[,)π))π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)))) |
78 | 77 | ralbidv 3171 |
. . . . . 6
β’ (π β (βπ β β+
βπ’ β ran (π β β+
β¦ (β‘πΆ β (0[,)π)))βπ₯ β π βπ¦ β π (π₯π’π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)) β βπ β β+ βπ β β+
βπ₯ β π βπ¦ β π (π₯(β‘πΆ β (0[,)π))π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)))) |
79 | 60, 78 | bitrd 279 |
. . . . 5
β’ (π β (βπ£ β ran (π β β+ β¦ (β‘π· β (0[,)π)))βπ’ β ran (π β β+ β¦ (β‘πΆ β (0[,)π)))βπ₯ β π βπ¦ β π (π₯π’π¦ β (πΉβπ₯)π£(πΉβπ¦)) β βπ β β+ βπ β β+
βπ₯ β π βπ¦ β π (π₯(β‘πΆ β (0[,)π))π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)))) |
80 | 79 | adantr 482 |
. . . 4
β’ ((π β§ πΉ:πβΆπ) β (βπ£ β ran (π β β+ β¦ (β‘π· β (0[,)π)))βπ’ β ran (π β β+ β¦ (β‘πΆ β (0[,)π)))βπ₯ β π βπ¦ β π (π₯π’π¦ β (πΉβπ₯)π£(πΉβπ¦)) β βπ β β+ βπ β β+
βπ₯ β π βπ¦ β π (π₯(β‘πΆ β (0[,)π))π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)))) |
81 | 2 | ad4antr 731 |
. . . . . . . . 9
β’
(((((π β§ πΉ:πβΆπ) β§ π β β+) β§ π β β+)
β§ (π₯ β π β§ π¦ β π)) β πΆ β (PsMetβπ)) |
82 | | simplr 768 |
. . . . . . . . 9
β’
(((((π β§ πΉ:πβΆπ) β§ π β β+) β§ π β β+)
β§ (π₯ β π β§ π¦ β π)) β π β β+) |
83 | | simprr 772 |
. . . . . . . . 9
β’
(((((π β§ πΉ:πβΆπ) β§ π β β+) β§ π β β+)
β§ (π₯ β π β§ π¦ β π)) β π¦ β π) |
84 | | simprl 770 |
. . . . . . . . 9
β’
(((((π β§ πΉ:πβΆπ) β§ π β β+) β§ π β β+)
β§ (π₯ β π β§ π¦ β π)) β π₯ β π) |
85 | | elbl4 23942 |
. . . . . . . . . 10
β’ (((πΆ β (PsMetβπ) β§ π β β+) β§ (π¦ β π β§ π₯ β π)) β (π₯ β (π¦(ballβπΆ)π) β π₯(β‘πΆ β (0[,)π))π¦)) |
86 | | rpxr 12932 |
. . . . . . . . . . 11
β’ (π β β+
β π β
β*) |
87 | | elbl3ps 23767 |
. . . . . . . . . . 11
β’ (((πΆ β (PsMetβπ) β§ π β β*) β§ (π¦ β π β§ π₯ β π)) β (π₯ β (π¦(ballβπΆ)π) β (π₯πΆπ¦) < π)) |
88 | 86, 87 | sylanl2 680 |
. . . . . . . . . 10
β’ (((πΆ β (PsMetβπ) β§ π β β+) β§ (π¦ β π β§ π₯ β π)) β (π₯ β (π¦(ballβπΆ)π) β (π₯πΆπ¦) < π)) |
89 | 85, 88 | bitr3d 281 |
. . . . . . . . 9
β’ (((πΆ β (PsMetβπ) β§ π β β+) β§ (π¦ β π β§ π₯ β π)) β (π₯(β‘πΆ β (0[,)π))π¦ β (π₯πΆπ¦) < π)) |
90 | 81, 82, 83, 84, 89 | syl22anc 838 |
. . . . . . . 8
β’
(((((π β§ πΉ:πβΆπ) β§ π β β+) β§ π β β+)
β§ (π₯ β π β§ π¦ β π)) β (π₯(β‘πΆ β (0[,)π))π¦ β (π₯πΆπ¦) < π)) |
91 | 7 | ad4antr 731 |
. . . . . . . . 9
β’
(((((π β§ πΉ:πβΆπ) β§ π β β+) β§ π β β+)
β§ (π₯ β π β§ π¦ β π)) β π· β (PsMetβπ)) |
92 | | simpllr 775 |
. . . . . . . . 9
β’
(((((π β§ πΉ:πβΆπ) β§ π β β+) β§ π β β+)
β§ (π₯ β π β§ π¦ β π)) β π β β+) |
93 | | simp-4r 783 |
. . . . . . . . . 10
β’
(((((π β§ πΉ:πβΆπ) β§ π β β+) β§ π β β+)
β§ (π₯ β π β§ π¦ β π)) β πΉ:πβΆπ) |
94 | 93, 83 | ffvelcdmd 7040 |
. . . . . . . . 9
β’
(((((π β§ πΉ:πβΆπ) β§ π β β+) β§ π β β+)
β§ (π₯ β π β§ π¦ β π)) β (πΉβπ¦) β π) |
95 | 93, 84 | ffvelcdmd 7040 |
. . . . . . . . 9
β’
(((((π β§ πΉ:πβΆπ) β§ π β β+) β§ π β β+)
β§ (π₯ β π β§ π¦ β π)) β (πΉβπ₯) β π) |
96 | | elbl4 23942 |
. . . . . . . . . 10
β’ (((π· β (PsMetβπ) β§ π β β+) β§ ((πΉβπ¦) β π β§ (πΉβπ₯) β π)) β ((πΉβπ₯) β ((πΉβπ¦)(ballβπ·)π) β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦))) |
97 | | rpxr 12932 |
. . . . . . . . . . 11
β’ (π β β+
β π β
β*) |
98 | | elbl3ps 23767 |
. . . . . . . . . . 11
β’ (((π· β (PsMetβπ) β§ π β β*) β§ ((πΉβπ¦) β π β§ (πΉβπ₯) β π)) β ((πΉβπ₯) β ((πΉβπ¦)(ballβπ·)π) β ((πΉβπ₯)π·(πΉβπ¦)) < π)) |
99 | 97, 98 | sylanl2 680 |
. . . . . . . . . 10
β’ (((π· β (PsMetβπ) β§ π β β+) β§ ((πΉβπ¦) β π β§ (πΉβπ₯) β π)) β ((πΉβπ₯) β ((πΉβπ¦)(ballβπ·)π) β ((πΉβπ₯)π·(πΉβπ¦)) < π)) |
100 | 96, 99 | bitr3d 281 |
. . . . . . . . 9
β’ (((π· β (PsMetβπ) β§ π β β+) β§ ((πΉβπ¦) β π β§ (πΉβπ₯) β π)) β ((πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦) β ((πΉβπ₯)π·(πΉβπ¦)) < π)) |
101 | 91, 92, 94, 95, 100 | syl22anc 838 |
. . . . . . . 8
β’
(((((π β§ πΉ:πβΆπ) β§ π β β+) β§ π β β+)
β§ (π₯ β π β§ π¦ β π)) β ((πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦) β ((πΉβπ₯)π·(πΉβπ¦)) < π)) |
102 | 90, 101 | imbi12d 345 |
. . . . . . 7
β’
(((((π β§ πΉ:πβΆπ) β§ π β β+) β§ π β β+)
β§ (π₯ β π β§ π¦ β π)) β ((π₯(β‘πΆ β (0[,)π))π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)) β ((π₯πΆπ¦) < π β ((πΉβπ₯)π·(πΉβπ¦)) < π))) |
103 | 102 | 2ralbidva 3207 |
. . . . . 6
β’ ((((π β§ πΉ:πβΆπ) β§ π β β+) β§ π β β+)
β (βπ₯ β
π βπ¦ β π (π₯(β‘πΆ β (0[,)π))π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)) β βπ₯ β π βπ¦ β π ((π₯πΆπ¦) < π β ((πΉβπ₯)π·(πΉβπ¦)) < π))) |
104 | 103 | rexbidva 3170 |
. . . . 5
β’ (((π β§ πΉ:πβΆπ) β§ π β β+) β
(βπ β
β+ βπ₯ β π βπ¦ β π (π₯(β‘πΆ β (0[,)π))π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)) β βπ β β+ βπ₯ β π βπ¦ β π ((π₯πΆπ¦) < π β ((πΉβπ₯)π·(πΉβπ¦)) < π))) |
105 | 104 | ralbidva 3169 |
. . . 4
β’ ((π β§ πΉ:πβΆπ) β (βπ β β+ βπ β β+
βπ₯ β π βπ¦ β π (π₯(β‘πΆ β (0[,)π))π¦ β (πΉβπ₯)(β‘π· β (0[,)π))(πΉβπ¦)) β βπ β β+ βπ β β+
βπ₯ β π βπ¦ β π ((π₯πΆπ¦) < π β ((πΉβπ₯)π·(πΉβπ¦)) < π))) |
106 | 80, 105 | bitrd 279 |
. . 3
β’ ((π β§ πΉ:πβΆπ) β (βπ£ β ran (π β β+ β¦ (β‘π· β (0[,)π)))βπ’ β ran (π β β+ β¦ (β‘πΆ β (0[,)π)))βπ₯ β π βπ¦ β π (π₯π’π¦ β (πΉβπ₯)π£(πΉβπ¦)) β βπ β β+ βπ β β+
βπ₯ β π βπ¦ β π ((π₯πΆπ¦) < π β ((πΉβπ₯)π·(πΉβπ¦)) < π))) |
107 | 106 | pm5.32da 580 |
. 2
β’ (π β ((πΉ:πβΆπ β§ βπ£ β ran (π β β+ β¦ (β‘π· β (0[,)π)))βπ’ β ran (π β β+ β¦ (β‘πΆ β (0[,)π)))βπ₯ β π βπ¦ β π (π₯π’π¦ β (πΉβπ₯)π£(πΉβπ¦))) β (πΉ:πβΆπ β§ βπ β β+ βπ β β+
βπ₯ β π βπ¦ β π ((π₯πΆπ¦) < π β ((πΉβπ₯)π·(πΉβπ¦)) < π)))) |
108 | 42, 107 | bitrd 279 |
1
β’ (π β (πΉ β (π Cnuπ) β (πΉ:πβΆπ β§ βπ β β+ βπ β β+
βπ₯ β π βπ¦ β π ((π₯πΆπ¦) < π β ((πΉβπ₯)π·(πΉβπ¦)) < π)))) |