Step | Hyp | Ref
| Expression |
1 | | fveq2 5515 |
. . . . . 6
β’ (π€ = 1 β (πΉβπ€) = (πΉβ1)) |
2 | 1 | oveq1d 5889 |
. . . . 5
β’ (π€ = 1 β ((πΉβπ€)β2) = ((πΉβ1)β2)) |
3 | 2 | breq2d 4015 |
. . . 4
β’ (π€ = 1 β (π΄ < ((πΉβπ€)β2) β π΄ < ((πΉβ1)β2))) |
4 | 3 | imbi2d 230 |
. . 3
β’ (π€ = 1 β ((π β π΄ < ((πΉβπ€)β2)) β (π β π΄ < ((πΉβ1)β2)))) |
5 | | fveq2 5515 |
. . . . . 6
β’ (π€ = π β (πΉβπ€) = (πΉβπ)) |
6 | 5 | oveq1d 5889 |
. . . . 5
β’ (π€ = π β ((πΉβπ€)β2) = ((πΉβπ)β2)) |
7 | 6 | breq2d 4015 |
. . . 4
β’ (π€ = π β (π΄ < ((πΉβπ€)β2) β π΄ < ((πΉβπ)β2))) |
8 | 7 | imbi2d 230 |
. . 3
β’ (π€ = π β ((π β π΄ < ((πΉβπ€)β2)) β (π β π΄ < ((πΉβπ)β2)))) |
9 | | fveq2 5515 |
. . . . . 6
β’ (π€ = (π + 1) β (πΉβπ€) = (πΉβ(π + 1))) |
10 | 9 | oveq1d 5889 |
. . . . 5
β’ (π€ = (π + 1) β ((πΉβπ€)β2) = ((πΉβ(π + 1))β2)) |
11 | 10 | breq2d 4015 |
. . . 4
β’ (π€ = (π + 1) β (π΄ < ((πΉβπ€)β2) β π΄ < ((πΉβ(π + 1))β2))) |
12 | 11 | imbi2d 230 |
. . 3
β’ (π€ = (π + 1) β ((π β π΄ < ((πΉβπ€)β2)) β (π β π΄ < ((πΉβ(π + 1))β2)))) |
13 | | fveq2 5515 |
. . . . . 6
β’ (π€ = π β (πΉβπ€) = (πΉβπ)) |
14 | 13 | oveq1d 5889 |
. . . . 5
β’ (π€ = π β ((πΉβπ€)β2) = ((πΉβπ)β2)) |
15 | 14 | breq2d 4015 |
. . . 4
β’ (π€ = π β (π΄ < ((πΉβπ€)β2) β π΄ < ((πΉβπ)β2))) |
16 | 15 | imbi2d 230 |
. . 3
β’ (π€ = π β ((π β π΄ < ((πΉβπ€)β2)) β (π β π΄ < ((πΉβπ)β2)))) |
17 | | resqrexlemex.a |
. . . . 5
β’ (π β π΄ β β) |
18 | 17 | resqcld 10679 |
. . . . . 6
β’ (π β (π΄β2) β β) |
19 | | 2re 8988 |
. . . . . . . 8
β’ 2 β
β |
20 | 19 | a1i 9 |
. . . . . . 7
β’ (π β 2 β
β) |
21 | 20, 17 | remulcld 7987 |
. . . . . 6
β’ (π β (2 Β· π΄) β
β) |
22 | 18, 21 | readdcld 7986 |
. . . . 5
β’ (π β ((π΄β2) + (2 Β· π΄)) β β) |
23 | | 1red 7971 |
. . . . . 6
β’ (π β 1 β
β) |
24 | 22, 23 | readdcld 7986 |
. . . . 5
β’ (π β (((π΄β2) + (2 Β· π΄)) + 1) β β) |
25 | 17 | recnd 7985 |
. . . . . . . 8
β’ (π β π΄ β β) |
26 | 25 | mulid2d 7975 |
. . . . . . 7
β’ (π β (1 Β· π΄) = π΄) |
27 | | resqrexlemex.agt0 |
. . . . . . . 8
β’ (π β 0 β€ π΄) |
28 | | 1le2 9126 |
. . . . . . . . 9
β’ 1 β€
2 |
29 | | lemul1a 8814 |
. . . . . . . . 9
β’ (((1
β β β§ 2 β β β§ (π΄ β β β§ 0 β€ π΄)) β§ 1 β€ 2) β (1
Β· π΄) β€ (2
Β· π΄)) |
30 | 28, 29 | mpan2 425 |
. . . . . . . 8
β’ ((1
β β β§ 2 β β β§ (π΄ β β β§ 0 β€ π΄)) β (1 Β· π΄) β€ (2 Β· π΄)) |
31 | 23, 20, 17, 27, 30 | syl112anc 1242 |
. . . . . . 7
β’ (π β (1 Β· π΄) β€ (2 Β· π΄)) |
32 | 26, 31 | eqbrtrrd 4027 |
. . . . . 6
β’ (π β π΄ β€ (2 Β· π΄)) |
33 | 17 | sqge0d 10680 |
. . . . . . 7
β’ (π β 0 β€ (π΄β2)) |
34 | 21, 18 | addge02d 8490 |
. . . . . . 7
β’ (π β (0 β€ (π΄β2) β (2 Β· π΄) β€ ((π΄β2) + (2 Β· π΄)))) |
35 | 33, 34 | mpbid 147 |
. . . . . 6
β’ (π β (2 Β· π΄) β€ ((π΄β2) + (2 Β· π΄))) |
36 | 17, 21, 22, 32, 35 | letrd 8080 |
. . . . 5
β’ (π β π΄ β€ ((π΄β2) + (2 Β· π΄))) |
37 | 22 | ltp1d 8886 |
. . . . 5
β’ (π β ((π΄β2) + (2 Β· π΄)) < (((π΄β2) + (2 Β· π΄)) + 1)) |
38 | 17, 22, 24, 36, 37 | lelttrd 8081 |
. . . 4
β’ (π β π΄ < (((π΄β2) + (2 Β· π΄)) + 1)) |
39 | | resqrexlemex.seq |
. . . . . . . 8
β’ πΉ = seq1((π¦ β β+, π§ β β+
β¦ ((π¦ + (π΄ / π¦)) / 2)), (β Γ {(1 + π΄)})) |
40 | 39, 17, 27 | resqrexlemf1 11016 |
. . . . . . 7
β’ (π β (πΉβ1) = (1 + π΄)) |
41 | | 1cnd 7972 |
. . . . . . . 8
β’ (π β 1 β
β) |
42 | 41, 25 | addcomd 8107 |
. . . . . . 7
β’ (π β (1 + π΄) = (π΄ + 1)) |
43 | 40, 42 | eqtrd 2210 |
. . . . . 6
β’ (π β (πΉβ1) = (π΄ + 1)) |
44 | 43 | oveq1d 5889 |
. . . . 5
β’ (π β ((πΉβ1)β2) = ((π΄ + 1)β2)) |
45 | | binom21 10632 |
. . . . . 6
β’ (π΄ β β β ((π΄ + 1)β2) = (((π΄β2) + (2 Β· π΄)) + 1)) |
46 | 25, 45 | syl 14 |
. . . . 5
β’ (π β ((π΄ + 1)β2) = (((π΄β2) + (2 Β· π΄)) + 1)) |
47 | 44, 46 | eqtrd 2210 |
. . . 4
β’ (π β ((πΉβ1)β2) = (((π΄β2) + (2 Β· π΄)) + 1)) |
48 | 38, 47 | breqtrrd 4031 |
. . 3
β’ (π β π΄ < ((πΉβ1)β2)) |
49 | 39, 17, 27 | resqrexlemf 11015 |
. . . . . . . . . . . . . 14
β’ (π β πΉ:ββΆβ+) |
50 | 49 | ffvelcdmda 5651 |
. . . . . . . . . . . . 13
β’ ((π β§ π β β) β (πΉβπ) β
β+) |
51 | 50 | rpred 9695 |
. . . . . . . . . . . 12
β’ ((π β§ π β β) β (πΉβπ) β β) |
52 | 17 | adantr 276 |
. . . . . . . . . . . . 13
β’ ((π β§ π β β) β π΄ β β) |
53 | 52, 50 | rerpdivcld 9727 |
. . . . . . . . . . . 12
β’ ((π β§ π β β) β (π΄ / (πΉβπ)) β β) |
54 | 51, 53 | resubcld 8337 |
. . . . . . . . . . 11
β’ ((π β§ π β β) β ((πΉβπ) β (π΄ / (πΉβπ))) β β) |
55 | 54 | adantr 276 |
. . . . . . . . . 10
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β ((πΉβπ) β (π΄ / (πΉβπ))) β β) |
56 | 55 | resqcld 10679 |
. . . . . . . . 9
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β (((πΉβπ) β (π΄ / (πΉβπ)))β2) β β) |
57 | | 4re 8995 |
. . . . . . . . . 10
β’ 4 β
β |
58 | 57 | a1i 9 |
. . . . . . . . 9
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β 4 β
β) |
59 | 51 | resqcld 10679 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β β) β ((πΉβπ)β2) β β) |
60 | 59, 52 | resubcld 8337 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β β) β (((πΉβπ)β2) β π΄) β β) |
61 | 60 | adantr 276 |
. . . . . . . . . . . . 13
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β (((πΉβπ)β2) β π΄) β β) |
62 | 51 | adantr 276 |
. . . . . . . . . . . . 13
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β (πΉβπ) β β) |
63 | 52, 59 | posdifd 8488 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β β) β (π΄ < ((πΉβπ)β2) β 0 < (((πΉβπ)β2) β π΄))) |
64 | 63 | biimpa 296 |
. . . . . . . . . . . . 13
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β 0 < (((πΉβπ)β2) β π΄)) |
65 | 50 | rpgt0d 9698 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β β) β 0 < (πΉβπ)) |
66 | 65 | adantr 276 |
. . . . . . . . . . . . 13
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β 0 < (πΉβπ)) |
67 | 61, 62, 64, 66 | divgt0d 8891 |
. . . . . . . . . . . 12
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β 0 < ((((πΉβπ)β2) β π΄) / (πΉβπ))) |
68 | 51 | recnd 7985 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ π β β) β (πΉβπ) β β) |
69 | 68 | sqcld 10651 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β β) β ((πΉβπ)β2) β β) |
70 | 69 | adantr 276 |
. . . . . . . . . . . . . 14
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β ((πΉβπ)β2) β β) |
71 | 25 | adantr 276 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β β) β π΄ β β) |
72 | 71 | adantr 276 |
. . . . . . . . . . . . . 14
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β π΄ β β) |
73 | 68 | adantr 276 |
. . . . . . . . . . . . . 14
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β (πΉβπ) β β) |
74 | 50 | rpap0d 9701 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β β) β (πΉβπ) # 0) |
75 | 74 | adantr 276 |
. . . . . . . . . . . . . 14
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β (πΉβπ) # 0) |
76 | 70, 72, 73, 75 | divsubdirapd 8786 |
. . . . . . . . . . . . 13
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β ((((πΉβπ)β2) β π΄) / (πΉβπ)) = ((((πΉβπ)β2) / (πΉβπ)) β (π΄ / (πΉβπ)))) |
77 | 73 | sqvald 10650 |
. . . . . . . . . . . . . . . 16
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β ((πΉβπ)β2) = ((πΉβπ) Β· (πΉβπ))) |
78 | 77 | oveq1d 5889 |
. . . . . . . . . . . . . . 15
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β (((πΉβπ)β2) / (πΉβπ)) = (((πΉβπ) Β· (πΉβπ)) / (πΉβπ))) |
79 | 73, 73, 75 | divcanap3d 8751 |
. . . . . . . . . . . . . . 15
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β (((πΉβπ) Β· (πΉβπ)) / (πΉβπ)) = (πΉβπ)) |
80 | 78, 79 | eqtrd 2210 |
. . . . . . . . . . . . . 14
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β (((πΉβπ)β2) / (πΉβπ)) = (πΉβπ)) |
81 | 80 | oveq1d 5889 |
. . . . . . . . . . . . 13
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β ((((πΉβπ)β2) / (πΉβπ)) β (π΄ / (πΉβπ))) = ((πΉβπ) β (π΄ / (πΉβπ)))) |
82 | 76, 81 | eqtrd 2210 |
. . . . . . . . . . . 12
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β ((((πΉβπ)β2) β π΄) / (πΉβπ)) = ((πΉβπ) β (π΄ / (πΉβπ)))) |
83 | 67, 82 | breqtrd 4029 |
. . . . . . . . . . 11
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β 0 < ((πΉβπ) β (π΄ / (πΉβπ)))) |
84 | 55, 83 | gt0ap0d 8585 |
. . . . . . . . . 10
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β ((πΉβπ) β (π΄ / (πΉβπ))) # 0) |
85 | 55, 84 | sqgt0apd 10681 |
. . . . . . . . 9
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β 0 < (((πΉβπ) β (π΄ / (πΉβπ)))β2)) |
86 | | 4pos 9015 |
. . . . . . . . . 10
β’ 0 <
4 |
87 | 86 | a1i 9 |
. . . . . . . . 9
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β 0 < 4) |
88 | 56, 58, 85, 87 | divgt0d 8891 |
. . . . . . . 8
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β 0 < ((((πΉβπ) β (π΄ / (πΉβπ)))β2) / 4)) |
89 | 57, 86 | gt0ap0ii 8584 |
. . . . . . . . . . 11
β’ 4 #
0 |
90 | 89 | a1i 9 |
. . . . . . . . . 10
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β 4 # 0) |
91 | 56, 58, 90 | redivclapd 8791 |
. . . . . . . . 9
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β ((((πΉβπ) β (π΄ / (πΉβπ)))β2) / 4) β
β) |
92 | 52 | adantr 276 |
. . . . . . . . 9
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β π΄ β β) |
93 | 91, 92 | ltaddpos2d 8486 |
. . . . . . . 8
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β (0 < ((((πΉβπ) β (π΄ / (πΉβπ)))β2) / 4) β π΄ < (((((πΉβπ) β (π΄ / (πΉβπ)))β2) / 4) + π΄))) |
94 | 88, 93 | mpbid 147 |
. . . . . . 7
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β π΄ < (((((πΉβπ) β (π΄ / (πΉβπ)))β2) / 4) + π΄)) |
95 | 39, 17, 27 | resqrexlemfp1 11017 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β β) β (πΉβ(π + 1)) = (((πΉβπ) + (π΄ / (πΉβπ))) / 2)) |
96 | 95 | oveq1d 5889 |
. . . . . . . . . . . . 13
β’ ((π β§ π β β) β ((πΉβ(π + 1))β2) = ((((πΉβπ) + (π΄ / (πΉβπ))) / 2)β2)) |
97 | 51, 53 | readdcld 7986 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β β) β ((πΉβπ) + (π΄ / (πΉβπ))) β β) |
98 | 97 | recnd 7985 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β β) β ((πΉβπ) + (π΄ / (πΉβπ))) β β) |
99 | | 2cnd 8991 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β β) β 2 β
β) |
100 | | 2ap0 9011 |
. . . . . . . . . . . . . . 15
β’ 2 #
0 |
101 | 100 | a1i 9 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β β) β 2 #
0) |
102 | 98, 99, 101 | sqdivapd 10666 |
. . . . . . . . . . . . 13
β’ ((π β§ π β β) β ((((πΉβπ) + (π΄ / (πΉβπ))) / 2)β2) = ((((πΉβπ) + (π΄ / (πΉβπ)))β2) / (2β2))) |
103 | 96, 102 | eqtrd 2210 |
. . . . . . . . . . . 12
β’ ((π β§ π β β) β ((πΉβ(π + 1))β2) = ((((πΉβπ) + (π΄ / (πΉβπ)))β2) / (2β2))) |
104 | | sq2 10615 |
. . . . . . . . . . . . 13
β’
(2β2) = 4 |
105 | 104 | oveq2i 5885 |
. . . . . . . . . . . 12
β’ ((((πΉβπ) + (π΄ / (πΉβπ)))β2) / (2β2)) = ((((πΉβπ) + (π΄ / (πΉβπ)))β2) / 4) |
106 | 103, 105 | eqtrdi 2226 |
. . . . . . . . . . 11
β’ ((π β§ π β β) β ((πΉβ(π + 1))β2) = ((((πΉβπ) + (π΄ / (πΉβπ)))β2) / 4)) |
107 | 71, 68, 74 | divcanap2d 8748 |
. . . . . . . . . . . . . . . . 17
β’ ((π β§ π β β) β ((πΉβπ) Β· (π΄ / (πΉβπ))) = π΄) |
108 | 107 | oveq2d 5890 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ π β β) β (2 Β· ((πΉβπ) Β· (π΄ / (πΉβπ)))) = (2 Β· π΄)) |
109 | 108 | oveq2d 5890 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β β) β (((πΉβπ)β2) β (2 Β· ((πΉβπ) Β· (π΄ / (πΉβπ))))) = (((πΉβπ)β2) β (2 Β· π΄))) |
110 | 109 | oveq1d 5889 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β β) β ((((πΉβπ)β2) β (2 Β· ((πΉβπ) Β· (π΄ / (πΉβπ))))) + ((π΄ / (πΉβπ))β2)) = ((((πΉβπ)β2) β (2 Β· π΄)) + ((π΄ / (πΉβπ))β2))) |
111 | 110 | oveq1d 5889 |
. . . . . . . . . . . . 13
β’ ((π β§ π β β) β (((((πΉβπ)β2) β (2 Β· ((πΉβπ) Β· (π΄ / (πΉβπ))))) + ((π΄ / (πΉβπ))β2)) + (4 Β· π΄)) = (((((πΉβπ)β2) β (2 Β· π΄)) + ((π΄ / (πΉβπ))β2)) + (4 Β· π΄))) |
112 | 53 | recnd 7985 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β β) β (π΄ / (πΉβπ)) β β) |
113 | | binom2sub 10633 |
. . . . . . . . . . . . . . 15
β’ (((πΉβπ) β β β§ (π΄ / (πΉβπ)) β β) β (((πΉβπ) β (π΄ / (πΉβπ)))β2) = ((((πΉβπ)β2) β (2 Β· ((πΉβπ) Β· (π΄ / (πΉβπ))))) + ((π΄ / (πΉβπ))β2))) |
114 | 68, 112, 113 | syl2anc 411 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β β) β (((πΉβπ) β (π΄ / (πΉβπ)))β2) = ((((πΉβπ)β2) β (2 Β· ((πΉβπ) Β· (π΄ / (πΉβπ))))) + ((π΄ / (πΉβπ))β2))) |
115 | 114 | oveq1d 5889 |
. . . . . . . . . . . . 13
β’ ((π β§ π β β) β ((((πΉβπ) β (π΄ / (πΉβπ)))β2) + (4 Β· π΄)) = (((((πΉβπ)β2) β (2 Β· ((πΉβπ) Β· (π΄ / (πΉβπ))))) + ((π΄ / (πΉβπ))β2)) + (4 Β· π΄))) |
116 | | binom2 10631 |
. . . . . . . . . . . . . . . 16
β’ (((πΉβπ) β β β§ (π΄ / (πΉβπ)) β β) β (((πΉβπ) + (π΄ / (πΉβπ)))β2) = ((((πΉβπ)β2) + (2 Β· ((πΉβπ) Β· (π΄ / (πΉβπ))))) + ((π΄ / (πΉβπ))β2))) |
117 | 68, 112, 116 | syl2anc 411 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β β) β (((πΉβπ) + (π΄ / (πΉβπ)))β2) = ((((πΉβπ)β2) + (2 Β· ((πΉβπ) Β· (π΄ / (πΉβπ))))) + ((π΄ / (πΉβπ))β2))) |
118 | 108 | oveq2d 5890 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ π β β) β (((πΉβπ)β2) + (2 Β· ((πΉβπ) Β· (π΄ / (πΉβπ))))) = (((πΉβπ)β2) + (2 Β· π΄))) |
119 | 118 | oveq1d 5889 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β β) β ((((πΉβπ)β2) + (2 Β· ((πΉβπ) Β· (π΄ / (πΉβπ))))) + ((π΄ / (πΉβπ))β2)) = ((((πΉβπ)β2) + (2 Β· π΄)) + ((π΄ / (πΉβπ))β2))) |
120 | 117, 119 | eqtrd 2210 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β β) β (((πΉβπ) + (π΄ / (πΉβπ)))β2) = ((((πΉβπ)β2) + (2 Β· π΄)) + ((π΄ / (πΉβπ))β2))) |
121 | 99, 71 | mulcld 7977 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β§ π β β) β (2 Β· π΄) β
β) |
122 | 121 | negcld 8254 |
. . . . . . . . . . . . . . . . 17
β’ ((π β§ π β β) β -(2 Β· π΄) β
β) |
123 | | 4cn 8996 |
. . . . . . . . . . . . . . . . . . 19
β’ 4 β
β |
124 | 123 | a1i 9 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β§ π β β) β 4 β
β) |
125 | 124, 71 | mulcld 7977 |
. . . . . . . . . . . . . . . . 17
β’ ((π β§ π β β) β (4 Β· π΄) β
β) |
126 | 69, 122, 125 | addassd 7979 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ π β β) β ((((πΉβπ)β2) + -(2 Β· π΄)) + (4 Β· π΄)) = (((πΉβπ)β2) + (-(2 Β· π΄) + (4 Β· π΄)))) |
127 | 69, 121 | negsubd 8273 |
. . . . . . . . . . . . . . . . 17
β’ ((π β§ π β β) β (((πΉβπ)β2) + -(2 Β· π΄)) = (((πΉβπ)β2) β (2 Β· π΄))) |
128 | 127 | oveq1d 5889 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ π β β) β ((((πΉβπ)β2) + -(2 Β· π΄)) + (4 Β· π΄)) = ((((πΉβπ)β2) β (2 Β· π΄)) + (4 Β· π΄))) |
129 | | 2cn 8989 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ 2 β
β |
130 | 129 | negcli 8224 |
. . . . . . . . . . . . . . . . . . . . 21
β’ -2 β
β |
131 | 130 | a1i 9 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π β§ π β β) β -2 β
β) |
132 | 131, 124,
71 | adddird 7982 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π β§ π β β) β ((-2 + 4) Β·
π΄) = ((-2 Β· π΄) + (4 Β· π΄))) |
133 | 99, 71 | mulneg1d 8367 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π β§ π β β) β (-2 Β· π΄) = -(2 Β· π΄)) |
134 | 133 | oveq1d 5889 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π β§ π β β) β ((-2 Β· π΄) + (4 Β· π΄)) = (-(2 Β· π΄) + (4 Β· π΄))) |
135 | 132, 134 | eqtrd 2210 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β§ π β β) β ((-2 + 4) Β·
π΄) = (-(2 Β· π΄) + (4 Β· π΄))) |
136 | 130, 129,
129 | addassi 7964 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((-2 + 2)
+ 2) = (-2 + (2 + 2)) |
137 | 129 | subidi 8227 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (2
β 2) = 0 |
138 | 137 | negeqi 8150 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ -(2
β 2) = -0 |
139 | 129, 129 | negsubdii 8241 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ -(2
β 2) = (-2 + 2) |
140 | | neg0 8202 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ -0 =
0 |
141 | 138, 139,
140 | 3eqtr3i 2206 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (-2 + 2)
= 0 |
142 | 141 | oveq1i 5884 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((-2 + 2)
+ 2) = (0 + 2) |
143 | 129 | addid2i 8099 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (0 + 2) =
2 |
144 | 142, 143 | eqtri 2198 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((-2 + 2)
+ 2) = 2 |
145 | | 2p2e4 9045 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (2 + 2) =
4 |
146 | 145 | oveq2i 5885 |
. . . . . . . . . . . . . . . . . . . 20
β’ (-2 + (2
+ 2)) = (-2 + 4) |
147 | 136, 144,
146 | 3eqtr3ri 2207 |
. . . . . . . . . . . . . . . . . . 19
β’ (-2 + 4)
= 2 |
148 | 147 | oveq1i 5884 |
. . . . . . . . . . . . . . . . . 18
β’ ((-2 + 4)
Β· π΄) = (2 Β·
π΄) |
149 | 135, 148 | eqtr3di 2225 |
. . . . . . . . . . . . . . . . 17
β’ ((π β§ π β β) β (-(2 Β· π΄) + (4 Β· π΄)) = (2 Β· π΄)) |
150 | 149 | oveq2d 5890 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ π β β) β (((πΉβπ)β2) + (-(2 Β· π΄) + (4 Β· π΄))) = (((πΉβπ)β2) + (2 Β· π΄))) |
151 | 126, 128,
150 | 3eqtr3rd 2219 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β β) β (((πΉβπ)β2) + (2 Β· π΄)) = ((((πΉβπ)β2) β (2 Β· π΄)) + (4 Β· π΄))) |
152 | 151 | oveq1d 5889 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β β) β ((((πΉβπ)β2) + (2 Β· π΄)) + ((π΄ / (πΉβπ))β2)) = (((((πΉβπ)β2) β (2 Β· π΄)) + (4 Β· π΄)) + ((π΄ / (πΉβπ))β2))) |
153 | 19 | a1i 9 |
. . . . . . . . . . . . . . . . 17
β’ ((π β§ π β β) β 2 β
β) |
154 | 153, 52 | remulcld 7987 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ π β β) β (2 Β· π΄) β
β) |
155 | 59, 154 | resubcld 8337 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β β) β (((πΉβπ)β2) β (2 Β· π΄)) β
β) |
156 | 57 | a1i 9 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ π β β) β 4 β
β) |
157 | 156, 52 | remulcld 7987 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β β) β (4 Β· π΄) β
β) |
158 | 53 | resqcld 10679 |
. . . . . . . . . . . . . . 15
β’ ((π β§ π β β) β ((π΄ / (πΉβπ))β2) β β) |
159 | | recn 7943 |
. . . . . . . . . . . . . . . . 17
β’ (π β β β π β
β) |
160 | | recn 7943 |
. . . . . . . . . . . . . . . . 17
β’ (π β β β π β
β) |
161 | | addcom 8093 |
. . . . . . . . . . . . . . . . 17
β’ ((π β β β§ π β β) β (π + π) = (π + π)) |
162 | 159, 160,
161 | syl2an 289 |
. . . . . . . . . . . . . . . 16
β’ ((π β β β§ π β β) β (π + π) = (π + π)) |
163 | 162 | adantl 277 |
. . . . . . . . . . . . . . 15
β’ (((π β§ π β β) β§ (π β β β§ π β β)) β (π + π) = (π + π)) |
164 | | recn 7943 |
. . . . . . . . . . . . . . . . 17
β’ (β β β β β β β) |
165 | | addass 7940 |
. . . . . . . . . . . . . . . . 17
β’ ((π β β β§ π β β β§ β β β) β ((π + π) + β) = (π + (π + β))) |
166 | 159, 160,
164, 165 | syl3an 1280 |
. . . . . . . . . . . . . . . 16
β’ ((π β β β§ π β β β§ β β β) β ((π + π) + β) = (π + (π + β))) |
167 | 166 | adantl 277 |
. . . . . . . . . . . . . . 15
β’ (((π β§ π β β) β§ (π β β β§ π β β β§ β β β)) β ((π + π) + β) = (π + (π + β))) |
168 | 155, 157,
158, 163, 167 | caov32d 6054 |
. . . . . . . . . . . . . 14
β’ ((π β§ π β β) β (((((πΉβπ)β2) β (2 Β· π΄)) + (4 Β· π΄)) + ((π΄ / (πΉβπ))β2)) = (((((πΉβπ)β2) β (2 Β· π΄)) + ((π΄ / (πΉβπ))β2)) + (4 Β· π΄))) |
169 | 120, 152,
168 | 3eqtrd 2214 |
. . . . . . . . . . . . 13
β’ ((π β§ π β β) β (((πΉβπ) + (π΄ / (πΉβπ)))β2) = (((((πΉβπ)β2) β (2 Β· π΄)) + ((π΄ / (πΉβπ))β2)) + (4 Β· π΄))) |
170 | 111, 115,
169 | 3eqtr4rd 2221 |
. . . . . . . . . . . 12
β’ ((π β§ π β β) β (((πΉβπ) + (π΄ / (πΉβπ)))β2) = ((((πΉβπ) β (π΄ / (πΉβπ)))β2) + (4 Β· π΄))) |
171 | 170 | oveq1d 5889 |
. . . . . . . . . . 11
β’ ((π β§ π β β) β ((((πΉβπ) + (π΄ / (πΉβπ)))β2) / 4) = (((((πΉβπ) β (π΄ / (πΉβπ)))β2) + (4 Β· π΄)) / 4)) |
172 | 106, 171 | eqtrd 2210 |
. . . . . . . . . 10
β’ ((π β§ π β β) β ((πΉβ(π + 1))β2) = (((((πΉβπ) β (π΄ / (πΉβπ)))β2) + (4 Β· π΄)) / 4)) |
173 | 68, 112 | subcld 8267 |
. . . . . . . . . . . 12
β’ ((π β§ π β β) β ((πΉβπ) β (π΄ / (πΉβπ))) β β) |
174 | 173 | sqcld 10651 |
. . . . . . . . . . 11
β’ ((π β§ π β β) β (((πΉβπ) β (π΄ / (πΉβπ)))β2) β β) |
175 | 89 | a1i 9 |
. . . . . . . . . . 11
β’ ((π β§ π β β) β 4 #
0) |
176 | 174, 125,
124, 175 | divdirapd 8785 |
. . . . . . . . . 10
β’ ((π β§ π β β) β (((((πΉβπ) β (π΄ / (πΉβπ)))β2) + (4 Β· π΄)) / 4) = (((((πΉβπ) β (π΄ / (πΉβπ)))β2) / 4) + ((4 Β· π΄) / 4))) |
177 | 71, 124, 175 | divcanap3d 8751 |
. . . . . . . . . . 11
β’ ((π β§ π β β) β ((4 Β· π΄) / 4) = π΄) |
178 | 177 | oveq2d 5890 |
. . . . . . . . . 10
β’ ((π β§ π β β) β (((((πΉβπ) β (π΄ / (πΉβπ)))β2) / 4) + ((4 Β· π΄) / 4)) = (((((πΉβπ) β (π΄ / (πΉβπ)))β2) / 4) + π΄)) |
179 | 172, 176,
178 | 3eqtrd 2214 |
. . . . . . . . 9
β’ ((π β§ π β β) β ((πΉβ(π + 1))β2) = (((((πΉβπ) β (π΄ / (πΉβπ)))β2) / 4) + π΄)) |
180 | 179 | breq2d 4015 |
. . . . . . . 8
β’ ((π β§ π β β) β (π΄ < ((πΉβ(π + 1))β2) β π΄ < (((((πΉβπ) β (π΄ / (πΉβπ)))β2) / 4) + π΄))) |
181 | 180 | adantr 276 |
. . . . . . 7
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β (π΄ < ((πΉβ(π + 1))β2) β π΄ < (((((πΉβπ) β (π΄ / (πΉβπ)))β2) / 4) + π΄))) |
182 | 94, 181 | mpbird 167 |
. . . . . 6
β’ (((π β§ π β β) β§ π΄ < ((πΉβπ)β2)) β π΄ < ((πΉβ(π + 1))β2)) |
183 | 182 | ex 115 |
. . . . 5
β’ ((π β§ π β β) β (π΄ < ((πΉβπ)β2) β π΄ < ((πΉβ(π + 1))β2))) |
184 | 183 | expcom 116 |
. . . 4
β’ (π β β β (π β (π΄ < ((πΉβπ)β2) β π΄ < ((πΉβ(π + 1))β2)))) |
185 | 184 | a2d 26 |
. . 3
β’ (π β β β ((π β π΄ < ((πΉβπ)β2)) β (π β π΄ < ((πΉβ(π + 1))β2)))) |
186 | 4, 8, 12, 16, 48, 185 | nnind 8934 |
. 2
β’ (π β β β (π β π΄ < ((πΉβπ)β2))) |
187 | 186 | impcom 125 |
1
β’ ((π β§ π β β) β π΄ < ((πΉβπ)β2)) |