Step | Hyp | Ref
| Expression |
1 | | fveq2 5517 |
. . . . . . 7
β’ (π€ = 1 β (πΉβπ€) = (πΉβ1)) |
2 | 1 | oveq1d 5892 |
. . . . . 6
β’ (π€ = 1 β ((πΉβπ€)β2) = ((πΉβ1)β2)) |
3 | 2 | oveq1d 5892 |
. . . . 5
β’ (π€ = 1 β (((πΉβπ€)β2) β π΄) = (((πΉβ1)β2) β π΄)) |
4 | | oveq1 5884 |
. . . . . . 7
β’ (π€ = 1 β (π€ β 1) = (1 β 1)) |
5 | 4 | oveq2d 5893 |
. . . . . 6
β’ (π€ = 1 β (4β(π€ β 1)) = (4β(1
β 1))) |
6 | 5 | oveq2d 5893 |
. . . . 5
β’ (π€ = 1 β (((πΉβ1)β2) / (4β(π€ β 1))) = (((πΉβ1)β2) / (4β(1
β 1)))) |
7 | 3, 6 | breq12d 4018 |
. . . 4
β’ (π€ = 1 β ((((πΉβπ€)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π€ β 1))) β (((πΉβ1)β2) β π΄) β€ (((πΉβ1)β2) / (4β(1 β
1))))) |
8 | 7 | imbi2d 230 |
. . 3
β’ (π€ = 1 β ((π β (((πΉβπ€)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π€ β 1)))) β (π β (((πΉβ1)β2) β π΄) β€ (((πΉβ1)β2) / (4β(1 β
1)))))) |
9 | | fveq2 5517 |
. . . . . . 7
β’ (π€ = π β (πΉβπ€) = (πΉβπ)) |
10 | 9 | oveq1d 5892 |
. . . . . 6
β’ (π€ = π β ((πΉβπ€)β2) = ((πΉβπ)β2)) |
11 | 10 | oveq1d 5892 |
. . . . 5
β’ (π€ = π β (((πΉβπ€)β2) β π΄) = (((πΉβπ)β2) β π΄)) |
12 | | oveq1 5884 |
. . . . . . 7
β’ (π€ = π β (π€ β 1) = (π β 1)) |
13 | 12 | oveq2d 5893 |
. . . . . 6
β’ (π€ = π β (4β(π€ β 1)) = (4β(π β 1))) |
14 | 13 | oveq2d 5893 |
. . . . 5
β’ (π€ = π β (((πΉβ1)β2) / (4β(π€ β 1))) = (((πΉβ1)β2) /
(4β(π β
1)))) |
15 | 11, 14 | breq12d 4018 |
. . . 4
β’ (π€ = π β ((((πΉβπ€)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π€ β 1))) β (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1))))) |
16 | 15 | imbi2d 230 |
. . 3
β’ (π€ = π β ((π β (((πΉβπ€)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π€ β 1)))) β (π β (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β
1)))))) |
17 | | fveq2 5517 |
. . . . . . 7
β’ (π€ = (π + 1) β (πΉβπ€) = (πΉβ(π + 1))) |
18 | 17 | oveq1d 5892 |
. . . . . 6
β’ (π€ = (π + 1) β ((πΉβπ€)β2) = ((πΉβ(π + 1))β2)) |
19 | 18 | oveq1d 5892 |
. . . . 5
β’ (π€ = (π + 1) β (((πΉβπ€)β2) β π΄) = (((πΉβ(π + 1))β2) β π΄)) |
20 | | oveq1 5884 |
. . . . . . 7
β’ (π€ = (π + 1) β (π€ β 1) = ((π + 1) β 1)) |
21 | 20 | oveq2d 5893 |
. . . . . 6
β’ (π€ = (π + 1) β (4β(π€ β 1)) = (4β((π + 1) β 1))) |
22 | 21 | oveq2d 5893 |
. . . . 5
β’ (π€ = (π + 1) β (((πΉβ1)β2) / (4β(π€ β 1))) = (((πΉβ1)β2) /
(4β((π + 1) β
1)))) |
23 | 19, 22 | breq12d 4018 |
. . . 4
β’ (π€ = (π + 1) β ((((πΉβπ€)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π€ β 1))) β (((πΉβ(π + 1))β2) β π΄) β€ (((πΉβ1)β2) / (4β((π + 1) β
1))))) |
24 | 23 | imbi2d 230 |
. . 3
β’ (π€ = (π + 1) β ((π β (((πΉβπ€)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π€ β 1)))) β (π β (((πΉβ(π + 1))β2) β π΄) β€ (((πΉβ1)β2) / (4β((π + 1) β
1)))))) |
25 | | fveq2 5517 |
. . . . . . 7
β’ (π€ = π β (πΉβπ€) = (πΉβπ)) |
26 | 25 | oveq1d 5892 |
. . . . . 6
β’ (π€ = π β ((πΉβπ€)β2) = ((πΉβπ)β2)) |
27 | 26 | oveq1d 5892 |
. . . . 5
β’ (π€ = π β (((πΉβπ€)β2) β π΄) = (((πΉβπ)β2) β π΄)) |
28 | | oveq1 5884 |
. . . . . . 7
β’ (π€ = π β (π€ β 1) = (π β 1)) |
29 | 28 | oveq2d 5893 |
. . . . . 6
β’ (π€ = π β (4β(π€ β 1)) = (4β(π β 1))) |
30 | 29 | oveq2d 5893 |
. . . . 5
β’ (π€ = π β (((πΉβ1)β2) / (4β(π€ β 1))) = (((πΉβ1)β2) /
(4β(π β
1)))) |
31 | 27, 30 | breq12d 4018 |
. . . 4
β’ (π€ = π β ((((πΉβπ€)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π€ β 1))) β (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1))))) |
32 | 31 | imbi2d 230 |
. . 3
β’ (π€ = π β ((π β (((πΉβπ€)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π€ β 1)))) β (π β (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β
1)))))) |
33 | | resqrexlemex.a |
. . . . . . 7
β’ (π β π΄ β β) |
34 | 33 | renegcld 8339 |
. . . . . 6
β’ (π β -π΄ β β) |
35 | | 0red 7960 |
. . . . . 6
β’ (π β 0 β
β) |
36 | | resqrexlemex.seq |
. . . . . . . . . 10
β’ πΉ = seq1((π¦ β β+, π§ β β+
β¦ ((π¦ + (π΄ / π¦)) / 2)), (β Γ {(1 + π΄)})) |
37 | | resqrexlemex.agt0 |
. . . . . . . . . 10
β’ (π β 0 β€ π΄) |
38 | 36, 33, 37 | resqrexlemf 11018 |
. . . . . . . . 9
β’ (π β πΉ:ββΆβ+) |
39 | | 1nn 8932 |
. . . . . . . . . 10
β’ 1 β
β |
40 | 39 | a1i 9 |
. . . . . . . . 9
β’ (π β 1 β
β) |
41 | 38, 40 | ffvelcdmd 5654 |
. . . . . . . 8
β’ (π β (πΉβ1) β
β+) |
42 | 41 | rpred 9698 |
. . . . . . 7
β’ (π β (πΉβ1) β β) |
43 | 42 | resqcld 10682 |
. . . . . 6
β’ (π β ((πΉβ1)β2) β
β) |
44 | 33 | le0neg2d 8477 |
. . . . . . 7
β’ (π β (0 β€ π΄ β -π΄ β€ 0)) |
45 | 37, 44 | mpbid 147 |
. . . . . 6
β’ (π β -π΄ β€ 0) |
46 | 34, 35, 43, 45 | leadd2dd 8519 |
. . . . 5
β’ (π β (((πΉβ1)β2) + -π΄) β€ (((πΉβ1)β2) + 0)) |
47 | 43 | recnd 7988 |
. . . . . 6
β’ (π β ((πΉβ1)β2) β
β) |
48 | 33 | recnd 7988 |
. . . . . 6
β’ (π β π΄ β β) |
49 | 47, 48 | negsubd 8276 |
. . . . 5
β’ (π β (((πΉβ1)β2) + -π΄) = (((πΉβ1)β2) β π΄)) |
50 | 47 | addid1d 8108 |
. . . . 5
β’ (π β (((πΉβ1)β2) + 0) = ((πΉβ1)β2)) |
51 | 46, 49, 50 | 3brtr3d 4036 |
. . . 4
β’ (π β (((πΉβ1)β2) β π΄) β€ ((πΉβ1)β2)) |
52 | | 1m1e0 8990 |
. . . . . . . 8
β’ (1
β 1) = 0 |
53 | 52 | oveq2i 5888 |
. . . . . . 7
β’
(4β(1 β 1)) = (4β0) |
54 | | 4cn 8999 |
. . . . . . . 8
β’ 4 β
β |
55 | | exp0 10526 |
. . . . . . . 8
β’ (4 β
β β (4β0) = 1) |
56 | 54, 55 | ax-mp 5 |
. . . . . . 7
β’
(4β0) = 1 |
57 | 53, 56 | eqtri 2198 |
. . . . . 6
β’
(4β(1 β 1)) = 1 |
58 | 57 | oveq2i 5888 |
. . . . 5
β’ (((πΉβ1)β2) / (4β(1
β 1))) = (((πΉβ1)β2) / 1) |
59 | 47 | div1d 8739 |
. . . . 5
β’ (π β (((πΉβ1)β2) / 1) = ((πΉβ1)β2)) |
60 | 58, 59 | eqtrid 2222 |
. . . 4
β’ (π β (((πΉβ1)β2) / (4β(1 β 1)))
= ((πΉβ1)β2)) |
61 | 51, 60 | breqtrrd 4033 |
. . 3
β’ (π β (((πΉβ1)β2) β π΄) β€ (((πΉβ1)β2) / (4β(1 β
1)))) |
62 | 38 | adantr 276 |
. . . . . . . . . . . . 13
β’ ((π β§ π β β) β πΉ:ββΆβ+) |
63 | | peano2nn 8933 |
. . . . . . . . . . . . . 14
β’ (π β β β (π + 1) β
β) |
64 | 63 | adantl 277 |
. . . . . . . . . . . . 13
β’ ((π β§ π β β) β (π + 1) β β) |
65 | 62, 64 | ffvelcdmd 5654 |
. . . . . . . . . . . 12
β’ ((π β§ π β β) β (πΉβ(π + 1)) β
β+) |
66 | 65 | rpred 9698 |
. . . . . . . . . . 11
β’ ((π β§ π β β) β (πΉβ(π + 1)) β β) |
67 | 66 | resqcld 10682 |
. . . . . . . . . 10
β’ ((π β§ π β β) β ((πΉβ(π + 1))β2) β
β) |
68 | 33 | adantr 276 |
. . . . . . . . . 10
β’ ((π β§ π β β) β π΄ β β) |
69 | 67, 68 | resubcld 8340 |
. . . . . . . . 9
β’ ((π β§ π β β) β (((πΉβ(π + 1))β2) β π΄) β β) |
70 | 69 | adantr 276 |
. . . . . . . 8
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β (((πΉβ(π + 1))β2) β π΄) β β) |
71 | 38 | ffvelcdmda 5653 |
. . . . . . . . . . . . 13
β’ ((π β§ π β β) β (πΉβπ) β
β+) |
72 | 71 | rpred 9698 |
. . . . . . . . . . . 12
β’ ((π β§ π β β) β (πΉβπ) β β) |
73 | 72 | resqcld 10682 |
. . . . . . . . . . 11
β’ ((π β§ π β β) β ((πΉβπ)β2) β β) |
74 | 73, 68 | resubcld 8340 |
. . . . . . . . . 10
β’ ((π β§ π β β) β (((πΉβπ)β2) β π΄) β β) |
75 | | 4re 8998 |
. . . . . . . . . . . 12
β’ 4 β
β |
76 | 75 | a1i 9 |
. . . . . . . . . . 11
β’ ((π β§ π β β) β 4 β
β) |
77 | | 4pos 9018 |
. . . . . . . . . . . 12
β’ 0 <
4 |
78 | 77 | a1i 9 |
. . . . . . . . . . 11
β’ ((π β§ π β β) β 0 <
4) |
79 | 76, 78 | elrpd 9695 |
. . . . . . . . . 10
β’ ((π β§ π β β) β 4 β
β+) |
80 | 74, 79 | rerpdivcld 9730 |
. . . . . . . . 9
β’ ((π β§ π β β) β ((((πΉβπ)β2) β π΄) / 4) β β) |
81 | 80 | adantr 276 |
. . . . . . . 8
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β ((((πΉβπ)β2) β π΄) / 4) β β) |
82 | 43 | adantr 276 |
. . . . . . . . . . 11
β’ ((π β§ π β β) β ((πΉβ1)β2) β
β) |
83 | | nnz 9274 |
. . . . . . . . . . . . . 14
β’ (π β β β π β
β€) |
84 | | peano2zm 9293 |
. . . . . . . . . . . . . 14
β’ (π β β€ β (π β 1) β
β€) |
85 | 83, 84 | syl 14 |
. . . . . . . . . . . . 13
β’ (π β β β (π β 1) β
β€) |
86 | 85 | adantl 277 |
. . . . . . . . . . . 12
β’ ((π β§ π β β) β (π β 1) β β€) |
87 | 79, 86 | rpexpcld 10680 |
. . . . . . . . . . 11
β’ ((π β§ π β β) β (4β(π β 1)) β
β+) |
88 | 82, 87 | rerpdivcld 9730 |
. . . . . . . . . 10
β’ ((π β§ π β β) β (((πΉβ1)β2) / (4β(π β 1))) β
β) |
89 | 88, 79 | rerpdivcld 9730 |
. . . . . . . . 9
β’ ((π β§ π β β) β ((((πΉβ1)β2) / (4β(π β 1))) / 4) β
β) |
90 | 89 | adantr 276 |
. . . . . . . 8
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β ((((πΉβ1)β2) /
(4β(π β 1))) /
4) β β) |
91 | 36, 33, 37 | resqrexlemcalc2 11026 |
. . . . . . . . 9
β’ ((π β§ π β β) β (((πΉβ(π + 1))β2) β π΄) β€ ((((πΉβπ)β2) β π΄) / 4)) |
92 | 91 | adantr 276 |
. . . . . . . 8
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β (((πΉβ(π + 1))β2) β π΄) β€ ((((πΉβπ)β2) β π΄) / 4)) |
93 | 74, 88, 79 | lediv1d 9745 |
. . . . . . . . 9
β’ ((π β§ π β β) β ((((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1))) β ((((πΉβπ)β2) β π΄) / 4) β€ ((((πΉβ1)β2) / (4β(π β 1))) /
4))) |
94 | 93 | biimpa 296 |
. . . . . . . 8
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β ((((πΉβπ)β2) β π΄) / 4) β€ ((((πΉβ1)β2) / (4β(π β 1))) /
4)) |
95 | 70, 81, 90, 92, 94 | letrd 8083 |
. . . . . . 7
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β (((πΉβ(π + 1))β2) β π΄) β€ ((((πΉβ1)β2) / (4β(π β 1))) /
4)) |
96 | 47 | ad2antrr 488 |
. . . . . . . . 9
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β ((πΉβ1)β2) β
β) |
97 | 87 | adantr 276 |
. . . . . . . . . 10
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β
(4β(π β 1))
β β+) |
98 | 97 | rpcnd 9700 |
. . . . . . . . 9
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β
(4β(π β 1))
β β) |
99 | 54 | a1i 9 |
. . . . . . . . 9
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β 4 β
β) |
100 | 97 | rpap0d 9704 |
. . . . . . . . 9
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β
(4β(π β 1)) #
0) |
101 | 79 | adantr 276 |
. . . . . . . . . 10
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β 4 β
β+) |
102 | 101 | rpap0d 9704 |
. . . . . . . . 9
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β 4 #
0) |
103 | 96, 98, 99, 100, 102 | divdivap1d 8781 |
. . . . . . . 8
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β ((((πΉβ1)β2) /
(4β(π β 1))) /
4) = (((πΉβ1)β2)
/ ((4β(π β 1))
Β· 4))) |
104 | | simpr 110 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β β) β π β β) |
105 | 104 | nncnd 8935 |
. . . . . . . . . . . . 13
β’ ((π β§ π β β) β π β β) |
106 | | pncan1 8336 |
. . . . . . . . . . . . 13
β’ (π β β β ((π + 1) β 1) = π) |
107 | 105, 106 | syl 14 |
. . . . . . . . . . . 12
β’ ((π β§ π β β) β ((π + 1) β 1) = π) |
108 | 107 | oveq2d 5893 |
. . . . . . . . . . 11
β’ ((π β§ π β β) β (4β((π + 1) β 1)) =
(4βπ)) |
109 | 108 | adantr 276 |
. . . . . . . . . 10
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β
(4β((π + 1) β
1)) = (4βπ)) |
110 | | simplr 528 |
. . . . . . . . . . 11
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β π β
β) |
111 | | expm1t 10550 |
. . . . . . . . . . 11
β’ ((4
β β β§ π
β β) β (4βπ) = ((4β(π β 1)) Β· 4)) |
112 | 54, 110, 111 | sylancr 414 |
. . . . . . . . . 10
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β
(4βπ) =
((4β(π β 1))
Β· 4)) |
113 | 109, 112 | eqtrd 2210 |
. . . . . . . . 9
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β
(4β((π + 1) β
1)) = ((4β(π β
1)) Β· 4)) |
114 | 113 | oveq2d 5893 |
. . . . . . . 8
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β (((πΉβ1)β2) /
(4β((π + 1) β
1))) = (((πΉβ1)β2) / ((4β(π β 1)) Β·
4))) |
115 | 103, 114 | eqtr4d 2213 |
. . . . . . 7
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β ((((πΉβ1)β2) /
(4β(π β 1))) /
4) = (((πΉβ1)β2)
/ (4β((π + 1) β
1)))) |
116 | 95, 115 | breqtrd 4031 |
. . . . . 6
β’ (((π β§ π β β) β§ (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β (((πΉβ(π + 1))β2) β π΄) β€ (((πΉβ1)β2) / (4β((π + 1) β
1)))) |
117 | 116 | ex 115 |
. . . . 5
β’ ((π β§ π β β) β ((((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1))) β (((πΉβ(π + 1))β2) β π΄) β€ (((πΉβ1)β2) / (4β((π + 1) β
1))))) |
118 | 117 | expcom 116 |
. . . 4
β’ (π β β β (π β ((((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1))) β (((πΉβ(π + 1))β2) β π΄) β€ (((πΉβ1)β2) / (4β((π + 1) β
1)))))) |
119 | 118 | a2d 26 |
. . 3
β’ (π β β β ((π β (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) β (π β (((πΉβ(π + 1))β2) β π΄) β€ (((πΉβ1)β2) / (4β((π + 1) β
1)))))) |
120 | 8, 16, 24, 32, 61, 119 | nnind 8937 |
. 2
β’ (π β β β (π β (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1))))) |
121 | 120 | impcom 125 |
1
β’ ((π β§ π β β) β (((πΉβπ)β2) β π΄) β€ (((πΉβ1)β2) / (4β(π β 1)))) |