Step | Hyp | Ref
| Expression |
1 | | dvconstbi.y |
. . . . . . 7
β’ (π β π:πβΆβ) |
2 | | dvconstbi.s |
. . . . . . . . 9
β’ (π β π β {β, β}) |
3 | | elpri 4612 |
. . . . . . . . 9
β’ (π β {β, β}
β (π = β β¨
π =
β)) |
4 | 2, 3 | syl 17 |
. . . . . . . 8
β’ (π β (π = β β¨ π = β)) |
5 | | 0re 11165 |
. . . . . . . . . 10
β’ 0 β
β |
6 | | eleq2 2823 |
. . . . . . . . . 10
β’ (π = β β (0 β
π β 0 β
β)) |
7 | 5, 6 | mpbiri 258 |
. . . . . . . . 9
β’ (π = β β 0 β π) |
8 | | 0cn 11155 |
. . . . . . . . . 10
β’ 0 β
β |
9 | | eleq2 2823 |
. . . . . . . . . 10
β’ (π = β β (0 β
π β 0 β
β)) |
10 | 8, 9 | mpbiri 258 |
. . . . . . . . 9
β’ (π = β β 0 β π) |
11 | 7, 10 | jaoi 856 |
. . . . . . . 8
β’ ((π = β β¨ π = β) β 0 β
π) |
12 | 4, 11 | syl 17 |
. . . . . . 7
β’ (π β 0 β π) |
13 | | ffvelcdm 7036 |
. . . . . . 7
β’ ((π:πβΆβ β§ 0 β π) β (πβ0) β β) |
14 | 1, 12, 13 | syl2anc 585 |
. . . . . 6
β’ (π β (πβ0) β β) |
15 | 14 | adantr 482 |
. . . . 5
β’ ((π β§ (π D π) = (π Γ {0})) β (πβ0) β β) |
16 | 1 | ffnd 6673 |
. . . . . . 7
β’ (π β π Fn π) |
17 | 16 | adantr 482 |
. . . . . 6
β’ ((π β§ (π D π) = (π Γ {0})) β π Fn π) |
18 | | fvex 6859 |
. . . . . . 7
β’ (πβ0) β
V |
19 | | fnconstg 6734 |
. . . . . . 7
β’ ((πβ0) β V β (π Γ {(πβ0)}) Fn π) |
20 | 18, 19 | mp1i 13 |
. . . . . 6
β’ ((π β§ (π D π) = (π Γ {0})) β (π Γ {(πβ0)}) Fn π) |
21 | 18 | fvconst2 7157 |
. . . . . . . 8
β’ (π¦ β π β ((π Γ {(πβ0)})βπ¦) = (πβ0)) |
22 | 21 | adantl 483 |
. . . . . . 7
β’ (((π β§ (π D π) = (π Γ {0})) β§ π¦ β π) β ((π Γ {(πβ0)})βπ¦) = (πβ0)) |
23 | | eqid 2733 |
. . . . . . . . . . . . . . . . . . 19
β’ ((abs
β β ) βΎ (π Γ π)) = ((abs β β ) βΎ (π Γ π)) |
24 | 2, 23 | sblpnf 42682 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β§ 0 β π) β (0(ballβ((abs β β
) βΎ (π Γ π)))+β) = π) |
25 | 12, 24 | mpdan 686 |
. . . . . . . . . . . . . . . . 17
β’ (π β (0(ballβ((abs
β β ) βΎ (π Γ π)))+β) = π) |
26 | 25 | eleq2d 2820 |
. . . . . . . . . . . . . . . 16
β’ (π β (π¦ β (0(ballβ((abs β β )
βΎ (π Γ π)))+β) β π¦ β π)) |
27 | 26 | biimpar 479 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π¦ β π) β π¦ β (0(ballβ((abs β β )
βΎ (π Γ π)))+β)) |
28 | 12, 25 | eleqtrrd 2837 |
. . . . . . . . . . . . . . . . 17
β’ (π β 0 β
(0(ballβ((abs β β ) βΎ (π Γ π)))+β)) |
29 | 2 | adantr 482 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β§ (π D π) = (π Γ {0})) β π β {β, β}) |
30 | | ssidd 3971 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β§ (π D π) = (π Γ {0})) β π β π) |
31 | 1 | adantr 482 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β§ (π D π) = (π Γ {0})) β π:πβΆβ) |
32 | 12 | adantr 482 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β§ (π D π) = (π Γ {0})) β 0 β π) |
33 | | pnfxr 11217 |
. . . . . . . . . . . . . . . . . . 19
β’ +β
β β* |
34 | 33 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β§ (π D π) = (π Γ {0})) β +β β
β*) |
35 | | eqid 2733 |
. . . . . . . . . . . . . . . . . 18
β’
(0(ballβ((abs β β ) βΎ (π Γ π)))+β) = (0(ballβ((abs β
β ) βΎ (π
Γ π)))+β) |
36 | 25 | adantr 482 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π β§ (π D π) = (π Γ {0})) β (0(ballβ((abs
β β ) βΎ (π Γ π)))+β) = π) |
37 | | dvconstbi.dy |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β dom (π D π) = π) |
38 | 37 | adantr 482 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π β§ (π D π) = (π Γ {0})) β dom (π D π) = π) |
39 | 36, 38 | eqtr4d 2776 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π β§ (π D π) = (π Γ {0})) β (0(ballβ((abs
β β ) βΎ (π Γ π)))+β) = dom (π D π)) |
40 | | eqimss 4004 |
. . . . . . . . . . . . . . . . . . 19
β’
((0(ballβ((abs β β ) βΎ (π Γ π)))+β) = dom (π D π) β (0(ballβ((abs β β
) βΎ (π Γ π)))+β) β dom (π D π)) |
41 | 39, 40 | syl 17 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β§ (π D π) = (π Γ {0})) β (0(ballβ((abs
β β ) βΎ (π Γ π)))+β) β dom (π D π)) |
42 | 5 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β§ (π D π) = (π Γ {0})) β 0 β
β) |
43 | 25 | eleq2d 2820 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π β (π₯ β (0(ballβ((abs β β )
βΎ (π Γ π)))+β) β π₯ β π)) |
44 | 43 | biimpa 478 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((π β§ π₯ β (0(ballβ((abs β β )
βΎ (π Γ π)))+β)) β π₯ β π) |
45 | 44 | 3adant2 1132 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π β§ (π D π) = (π Γ {0}) β§ π₯ β (0(ballβ((abs β β )
βΎ (π Γ π)))+β)) β π₯ β π) |
46 | | fveq1 6845 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ ((π D π) = (π Γ {0}) β ((π D π)βπ₯) = ((π Γ {0})βπ₯)) |
47 | | c0ex 11157 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ 0 β
V |
48 | 47 | fvconst2 7157 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ (π₯ β π β ((π Γ {0})βπ₯) = 0) |
49 | 46, 48 | sylan9eq 2793 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (((π D π) = (π Γ {0}) β§ π₯ β π) β ((π D π)βπ₯) = 0) |
50 | 49, 8 | eqeltrdi 2842 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (((π D π) = (π Γ {0}) β§ π₯ β π) β ((π D π)βπ₯) β β) |
51 | 50 | abscld 15330 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((π D π) = (π Γ {0}) β§ π₯ β π) β (absβ((π D π)βπ₯)) β β) |
52 | 49 | abs00bd 15185 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((π D π) = (π Γ {0}) β§ π₯ β π) β (absβ((π D π)βπ₯)) = 0) |
53 | | eqle 11265 |
. . . . . . . . . . . . . . . . . . . . . 22
β’
(((absβ((π D
π)βπ₯)) β β β§ (absβ((π D π)βπ₯)) = 0) β (absβ((π D π)βπ₯)) β€ 0) |
54 | 51, 52, 53 | syl2anc 585 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((π D π) = (π Γ {0}) β§ π₯ β π) β (absβ((π D π)βπ₯)) β€ 0) |
55 | 54 | 3adant1 1131 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π β§ (π D π) = (π Γ {0}) β§ π₯ β π) β (absβ((π D π)βπ₯)) β€ 0) |
56 | 45, 55 | syld3an3 1410 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π β§ (π D π) = (π Γ {0}) β§ π₯ β (0(ballβ((abs β β )
βΎ (π Γ π)))+β)) β
(absβ((π D π)βπ₯)) β€ 0) |
57 | 56 | 3expa 1119 |
. . . . . . . . . . . . . . . . . 18
β’ (((π β§ (π D π) = (π Γ {0})) β§ π₯ β (0(ballβ((abs β β )
βΎ (π Γ π)))+β)) β
(absβ((π D π)βπ₯)) β€ 0) |
58 | 29, 23, 30, 31, 32, 34, 35, 41, 42, 57 | dvlip2 25382 |
. . . . . . . . . . . . . . . . 17
β’ (((π β§ (π D π) = (π Γ {0})) β§ (0 β
(0(ballβ((abs β β ) βΎ (π Γ π)))+β) β§ π¦ β (0(ballβ((abs β β )
βΎ (π Γ π)))+β))) β
(absβ((πβ0)
β (πβπ¦))) β€ (0 Β·
(absβ(0 β π¦)))) |
59 | 28, 58 | sylanr1 681 |
. . . . . . . . . . . . . . . 16
β’ (((π β§ (π D π) = (π Γ {0})) β§ (π β§ π¦ β (0(ballβ((abs β β )
βΎ (π Γ π)))+β))) β
(absβ((πβ0)
β (πβπ¦))) β€ (0 Β·
(absβ(0 β π¦)))) |
60 | 59 | 3impdi 1351 |
. . . . . . . . . . . . . . 15
β’ ((π β§ (π D π) = (π Γ {0}) β§ π¦ β (0(ballβ((abs β β )
βΎ (π Γ π)))+β)) β
(absβ((πβ0)
β (πβπ¦))) β€ (0 Β·
(absβ(0 β π¦)))) |
61 | 27, 60 | syl3an3 1166 |
. . . . . . . . . . . . . 14
β’ ((π β§ (π D π) = (π Γ {0}) β§ (π β§ π¦ β π)) β (absβ((πβ0) β (πβπ¦))) β€ (0 Β· (absβ(0 β
π¦)))) |
62 | 61 | 3expa 1119 |
. . . . . . . . . . . . 13
β’ (((π β§ (π D π) = (π Γ {0})) β§ (π β§ π¦ β π)) β (absβ((πβ0) β (πβπ¦))) β€ (0 Β· (absβ(0 β
π¦)))) |
63 | 62 | 3impdi 1351 |
. . . . . . . . . . . 12
β’ ((π β§ (π D π) = (π Γ {0}) β§ π¦ β π) β (absβ((πβ0) β (πβπ¦))) β€ (0 Β· (absβ(0 β
π¦)))) |
64 | | recnprss 25291 |
. . . . . . . . . . . . . . . . . . 19
β’ (π β {β, β}
β π β
β) |
65 | 2, 64 | syl 17 |
. . . . . . . . . . . . . . . . . 18
β’ (π β π β β) |
66 | 65 | sseld 3947 |
. . . . . . . . . . . . . . . . 17
β’ (π β (π¦ β π β π¦ β β)) |
67 | | subcl 11408 |
. . . . . . . . . . . . . . . . . . 19
β’ ((0
β β β§ π¦
β β) β (0 β π¦) β β) |
68 | 67 | abscld 15330 |
. . . . . . . . . . . . . . . . . 18
β’ ((0
β β β§ π¦
β β) β (absβ(0 β π¦)) β β) |
69 | 8, 68 | mpan 689 |
. . . . . . . . . . . . . . . . 17
β’ (π¦ β β β
(absβ(0 β π¦))
β β) |
70 | 66, 69 | syl6 35 |
. . . . . . . . . . . . . . . 16
β’ (π β (π¦ β π β (absβ(0 β π¦)) β
β)) |
71 | 70 | imp 408 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π¦ β π) β (absβ(0 β π¦)) β
β) |
72 | 71 | recnd 11191 |
. . . . . . . . . . . . . 14
β’ ((π β§ π¦ β π) β (absβ(0 β π¦)) β
β) |
73 | 72 | mul02d 11361 |
. . . . . . . . . . . . 13
β’ ((π β§ π¦ β π) β (0 Β· (absβ(0 β
π¦))) = 0) |
74 | 73 | 3adant2 1132 |
. . . . . . . . . . . 12
β’ ((π β§ (π D π) = (π Γ {0}) β§ π¦ β π) β (0 Β· (absβ(0 β
π¦))) = 0) |
75 | 63, 74 | breqtrd 5135 |
. . . . . . . . . . 11
β’ ((π β§ (π D π) = (π Γ {0}) β§ π¦ β π) β (absβ((πβ0) β (πβπ¦))) β€ 0) |
76 | | ffvelcdm 7036 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π:πβΆβ β§ π¦ β π) β (πβπ¦) β β) |
77 | 13, 76 | anim12dan 620 |
. . . . . . . . . . . . . . . . . 18
β’ ((π:πβΆβ β§ (0 β π β§ π¦ β π)) β ((πβ0) β β β§ (πβπ¦) β β)) |
78 | 1, 77 | sylan 581 |
. . . . . . . . . . . . . . . . 17
β’ ((π β§ (0 β π β§ π¦ β π)) β ((πβ0) β β β§ (πβπ¦) β β)) |
79 | 78 | 3impb 1116 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ 0 β π β§ π¦ β π) β ((πβ0) β β β§ (πβπ¦) β β)) |
80 | 12, 79 | syl3an2 1165 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β§ π¦ β π) β ((πβ0) β β β§ (πβπ¦) β β)) |
81 | 80 | 3anidm12 1420 |
. . . . . . . . . . . . . 14
β’ ((π β§ π¦ β π) β ((πβ0) β β β§ (πβπ¦) β β)) |
82 | | subcl 11408 |
. . . . . . . . . . . . . 14
β’ (((πβ0) β β β§
(πβπ¦) β β) β ((πβ0) β (πβπ¦)) β β) |
83 | 81, 82 | syl 17 |
. . . . . . . . . . . . 13
β’ ((π β§ π¦ β π) β ((πβ0) β (πβπ¦)) β β) |
84 | 83 | absge0d 15338 |
. . . . . . . . . . . 12
β’ ((π β§ π¦ β π) β 0 β€ (absβ((πβ0) β (πβπ¦)))) |
85 | 84 | 3adant2 1132 |
. . . . . . . . . . 11
β’ ((π β§ (π D π) = (π Γ {0}) β§ π¦ β π) β 0 β€ (absβ((πβ0) β (πβπ¦)))) |
86 | 83 | abscld 15330 |
. . . . . . . . . . . . 13
β’ ((π β§ π¦ β π) β (absβ((πβ0) β (πβπ¦))) β β) |
87 | | letri3 11248 |
. . . . . . . . . . . . 13
β’
(((absβ((πβ0) β (πβπ¦))) β β β§ 0 β β)
β ((absβ((πβ0) β (πβπ¦))) = 0 β ((absβ((πβ0) β (πβπ¦))) β€ 0 β§ 0 β€ (absβ((πβ0) β (πβπ¦)))))) |
88 | 86, 5, 87 | sylancl 587 |
. . . . . . . . . . . 12
β’ ((π β§ π¦ β π) β ((absβ((πβ0) β (πβπ¦))) = 0 β ((absβ((πβ0) β (πβπ¦))) β€ 0 β§ 0 β€ (absβ((πβ0) β (πβπ¦)))))) |
89 | 88 | 3adant2 1132 |
. . . . . . . . . . 11
β’ ((π β§ (π D π) = (π Γ {0}) β§ π¦ β π) β ((absβ((πβ0) β (πβπ¦))) = 0 β ((absβ((πβ0) β (πβπ¦))) β€ 0 β§ 0 β€ (absβ((πβ0) β (πβπ¦)))))) |
90 | 75, 85, 89 | mpbir2and 712 |
. . . . . . . . . 10
β’ ((π β§ (π D π) = (π Γ {0}) β§ π¦ β π) β (absβ((πβ0) β (πβπ¦))) = 0) |
91 | 83 | abs00ad 15184 |
. . . . . . . . . . 11
β’ ((π β§ π¦ β π) β ((absβ((πβ0) β (πβπ¦))) = 0 β ((πβ0) β (πβπ¦)) = 0)) |
92 | 91 | 3adant2 1132 |
. . . . . . . . . 10
β’ ((π β§ (π D π) = (π Γ {0}) β§ π¦ β π) β ((absβ((πβ0) β (πβπ¦))) = 0 β ((πβ0) β (πβπ¦)) = 0)) |
93 | 90, 92 | mpbid 231 |
. . . . . . . . 9
β’ ((π β§ (π D π) = (π Γ {0}) β§ π¦ β π) β ((πβ0) β (πβπ¦)) = 0) |
94 | | subeq0 11435 |
. . . . . . . . . . 11
β’ (((πβ0) β β β§
(πβπ¦) β β) β (((πβ0) β (πβπ¦)) = 0 β (πβ0) = (πβπ¦))) |
95 | 81, 94 | syl 17 |
. . . . . . . . . 10
β’ ((π β§ π¦ β π) β (((πβ0) β (πβπ¦)) = 0 β (πβ0) = (πβπ¦))) |
96 | 95 | 3adant2 1132 |
. . . . . . . . 9
β’ ((π β§ (π D π) = (π Γ {0}) β§ π¦ β π) β (((πβ0) β (πβπ¦)) = 0 β (πβ0) = (πβπ¦))) |
97 | 93, 96 | mpbid 231 |
. . . . . . . 8
β’ ((π β§ (π D π) = (π Γ {0}) β§ π¦ β π) β (πβ0) = (πβπ¦)) |
98 | 97 | 3expa 1119 |
. . . . . . 7
β’ (((π β§ (π D π) = (π Γ {0})) β§ π¦ β π) β (πβ0) = (πβπ¦)) |
99 | 22, 98 | eqtr2d 2774 |
. . . . . 6
β’ (((π β§ (π D π) = (π Γ {0})) β§ π¦ β π) β (πβπ¦) = ((π Γ {(πβ0)})βπ¦)) |
100 | 17, 20, 99 | eqfnfvd 6989 |
. . . . 5
β’ ((π β§ (π D π) = (π Γ {0})) β π = (π Γ {(πβ0)})) |
101 | | sneq 4600 |
. . . . . . 7
β’ (π₯ = (πβ0) β {π₯} = {(πβ0)}) |
102 | 101 | xpeq2d 5667 |
. . . . . 6
β’ (π₯ = (πβ0) β (π Γ {π₯}) = (π Γ {(πβ0)})) |
103 | 102 | rspceeqv 3599 |
. . . . 5
β’ (((πβ0) β β β§
π = (π Γ {(πβ0)})) β βπ₯ β β π = (π Γ {π₯})) |
104 | 15, 100, 103 | syl2anc 585 |
. . . 4
β’ ((π β§ (π D π) = (π Γ {0})) β βπ₯ β β π = (π Γ {π₯})) |
105 | 104 | ex 414 |
. . 3
β’ (π β ((π D π) = (π Γ {0}) β βπ₯ β β π = (π Γ {π₯}))) |
106 | | oveq2 7369 |
. . . . . 6
β’ (π = (π Γ {π₯}) β (π D π) = (π D (π Γ {π₯}))) |
107 | 106 | 3ad2ant3 1136 |
. . . . 5
β’ ((π β§ π₯ β β β§ π = (π Γ {π₯})) β (π D π) = (π D (π Γ {π₯}))) |
108 | | dvsconst 42702 |
. . . . . . 7
β’ ((π β {β, β} β§
π₯ β β) β
(π D (π Γ {π₯})) = (π Γ {0})) |
109 | 2, 108 | sylan 581 |
. . . . . 6
β’ ((π β§ π₯ β β) β (π D (π Γ {π₯})) = (π Γ {0})) |
110 | 109 | 3adant3 1133 |
. . . . 5
β’ ((π β§ π₯ β β β§ π = (π Γ {π₯})) β (π D (π Γ {π₯})) = (π Γ {0})) |
111 | 107, 110 | eqtrd 2773 |
. . . 4
β’ ((π β§ π₯ β β β§ π = (π Γ {π₯})) β (π D π) = (π Γ {0})) |
112 | 111 | rexlimdv3a 3153 |
. . 3
β’ (π β (βπ₯ β β π = (π Γ {π₯}) β (π D π) = (π Γ {0}))) |
113 | 105, 112 | impbid 211 |
. 2
β’ (π β ((π D π) = (π Γ {0}) β βπ₯ β β π = (π Γ {π₯}))) |
114 | | sneq 4600 |
. . . . 5
β’ (π = π₯ β {π} = {π₯}) |
115 | 114 | xpeq2d 5667 |
. . . 4
β’ (π = π₯ β (π Γ {π}) = (π Γ {π₯})) |
116 | 115 | eqeq2d 2744 |
. . 3
β’ (π = π₯ β (π = (π Γ {π}) β π = (π Γ {π₯}))) |
117 | 116 | cbvrexvw 3225 |
. 2
β’
(βπ β
β π = (π Γ {π}) β βπ₯ β β π = (π Γ {π₯})) |
118 | 113, 117 | bitr4di 289 |
1
β’ (π β ((π D π) = (π Γ {0}) β βπ β β π = (π Γ {π}))) |