Step |
Hyp |
Ref |
Expression |
1 |
|
jm3.1.a |
|- ( ph -> A e. ( ZZ>= ` 2 ) ) |
2 |
|
jm3.1.b |
|- ( ph -> K e. ( ZZ>= ` 2 ) ) |
3 |
|
jm3.1.c |
|- ( ph -> N e. NN ) |
4 |
|
jm3.1.d |
|- ( ph -> ( K rmY ( N + 1 ) ) <_ A ) |
5 |
|
eluzelre |
|- ( K e. ( ZZ>= ` 2 ) -> K e. RR ) |
6 |
2 5
|
syl |
|- ( ph -> K e. RR ) |
7 |
3
|
nnnn0d |
|- ( ph -> N e. NN0 ) |
8 |
6 7
|
reexpcld |
|- ( ph -> ( K ^ N ) e. RR ) |
9 |
|
eluzelre |
|- ( A e. ( ZZ>= ` 2 ) -> A e. RR ) |
10 |
1 9
|
syl |
|- ( ph -> A e. RR ) |
11 |
|
2re |
|- 2 e. RR |
12 |
|
remulcl |
|- ( ( 2 e. RR /\ A e. RR ) -> ( 2 x. A ) e. RR ) |
13 |
11 10 12
|
sylancr |
|- ( ph -> ( 2 x. A ) e. RR ) |
14 |
13 6
|
remulcld |
|- ( ph -> ( ( 2 x. A ) x. K ) e. RR ) |
15 |
6
|
resqcld |
|- ( ph -> ( K ^ 2 ) e. RR ) |
16 |
14 15
|
resubcld |
|- ( ph -> ( ( ( 2 x. A ) x. K ) - ( K ^ 2 ) ) e. RR ) |
17 |
|
1re |
|- 1 e. RR |
18 |
|
resubcl |
|- ( ( ( ( ( 2 x. A ) x. K ) - ( K ^ 2 ) ) e. RR /\ 1 e. RR ) -> ( ( ( ( 2 x. A ) x. K ) - ( K ^ 2 ) ) - 1 ) e. RR ) |
19 |
16 17 18
|
sylancl |
|- ( ph -> ( ( ( ( 2 x. A ) x. K ) - ( K ^ 2 ) ) - 1 ) e. RR ) |
20 |
1 2 3 4
|
jm3.1lem1 |
|- ( ph -> ( K ^ N ) < A ) |
21 |
10 6
|
remulcld |
|- ( ph -> ( A x. K ) e. RR ) |
22 |
|
resubcl |
|- ( ( K e. RR /\ 1 e. RR ) -> ( K - 1 ) e. RR ) |
23 |
6 17 22
|
sylancl |
|- ( ph -> ( K - 1 ) e. RR ) |
24 |
21 23
|
readdcld |
|- ( ph -> ( ( A x. K ) + ( K - 1 ) ) e. RR ) |
25 |
|
eluz2b1 |
|- ( K e. ( ZZ>= ` 2 ) <-> ( K e. ZZ /\ 1 < K ) ) |
26 |
25
|
simprbi |
|- ( K e. ( ZZ>= ` 2 ) -> 1 < K ) |
27 |
2 26
|
syl |
|- ( ph -> 1 < K ) |
28 |
|
eluz2nn |
|- ( A e. ( ZZ>= ` 2 ) -> A e. NN ) |
29 |
1 28
|
syl |
|- ( ph -> A e. NN ) |
30 |
29
|
nngt0d |
|- ( ph -> 0 < A ) |
31 |
|
ltmulgt11 |
|- ( ( A e. RR /\ K e. RR /\ 0 < A ) -> ( 1 < K <-> A < ( A x. K ) ) ) |
32 |
10 6 30 31
|
syl3anc |
|- ( ph -> ( 1 < K <-> A < ( A x. K ) ) ) |
33 |
27 32
|
mpbid |
|- ( ph -> A < ( A x. K ) ) |
34 |
|
uz2m1nn |
|- ( K e. ( ZZ>= ` 2 ) -> ( K - 1 ) e. NN ) |
35 |
2 34
|
syl |
|- ( ph -> ( K - 1 ) e. NN ) |
36 |
35
|
nnrpd |
|- ( ph -> ( K - 1 ) e. RR+ ) |
37 |
21 36
|
ltaddrpd |
|- ( ph -> ( A x. K ) < ( ( A x. K ) + ( K - 1 ) ) ) |
38 |
10 21 24 33 37
|
lttrd |
|- ( ph -> A < ( ( A x. K ) + ( K - 1 ) ) ) |
39 |
|
peano2re |
|- ( K e. RR -> ( K + 1 ) e. RR ) |
40 |
6 39
|
syl |
|- ( ph -> ( K + 1 ) e. RR ) |
41 |
40 6
|
remulcld |
|- ( ph -> ( ( K + 1 ) x. K ) e. RR ) |
42 |
|
resubcl |
|- ( ( ( A x. K ) e. RR /\ 1 e. RR ) -> ( ( A x. K ) - 1 ) e. RR ) |
43 |
21 17 42
|
sylancl |
|- ( ph -> ( ( A x. K ) - 1 ) e. RR ) |
44 |
43 15
|
resubcld |
|- ( ph -> ( ( ( A x. K ) - 1 ) - ( K ^ 2 ) ) e. RR ) |
45 |
6
|
recnd |
|- ( ph -> K e. CC ) |
46 |
45
|
exp1d |
|- ( ph -> ( K ^ 1 ) = K ) |
47 |
|
eluz2nn |
|- ( K e. ( ZZ>= ` 2 ) -> K e. NN ) |
48 |
2 47
|
syl |
|- ( ph -> K e. NN ) |
49 |
48
|
nnge1d |
|- ( ph -> 1 <_ K ) |
50 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
51 |
3 50
|
eleqtrdi |
|- ( ph -> N e. ( ZZ>= ` 1 ) ) |
52 |
6 49 51
|
leexp2ad |
|- ( ph -> ( K ^ 1 ) <_ ( K ^ N ) ) |
53 |
46 52
|
eqbrtrrd |
|- ( ph -> K <_ ( K ^ N ) ) |
54 |
6 8 10 53 20
|
lelttrd |
|- ( ph -> K < A ) |
55 |
|
eluzelz |
|- ( K e. ( ZZ>= ` 2 ) -> K e. ZZ ) |
56 |
2 55
|
syl |
|- ( ph -> K e. ZZ ) |
57 |
|
eluzelz |
|- ( A e. ( ZZ>= ` 2 ) -> A e. ZZ ) |
58 |
1 57
|
syl |
|- ( ph -> A e. ZZ ) |
59 |
|
zltp1le |
|- ( ( K e. ZZ /\ A e. ZZ ) -> ( K < A <-> ( K + 1 ) <_ A ) ) |
60 |
56 58 59
|
syl2anc |
|- ( ph -> ( K < A <-> ( K + 1 ) <_ A ) ) |
61 |
54 60
|
mpbid |
|- ( ph -> ( K + 1 ) <_ A ) |
62 |
48
|
nngt0d |
|- ( ph -> 0 < K ) |
63 |
|
lemul1 |
|- ( ( ( K + 1 ) e. RR /\ A e. RR /\ ( K e. RR /\ 0 < K ) ) -> ( ( K + 1 ) <_ A <-> ( ( K + 1 ) x. K ) <_ ( A x. K ) ) ) |
64 |
40 10 6 62 63
|
syl112anc |
|- ( ph -> ( ( K + 1 ) <_ A <-> ( ( K + 1 ) x. K ) <_ ( A x. K ) ) ) |
65 |
61 64
|
mpbid |
|- ( ph -> ( ( K + 1 ) x. K ) <_ ( A x. K ) ) |
66 |
41 21 44 65
|
leadd1dd |
|- ( ph -> ( ( ( K + 1 ) x. K ) + ( ( ( A x. K ) - 1 ) - ( K ^ 2 ) ) ) <_ ( ( A x. K ) + ( ( ( A x. K ) - 1 ) - ( K ^ 2 ) ) ) ) |
67 |
21
|
recnd |
|- ( ph -> ( A x. K ) e. CC ) |
68 |
41 15
|
resubcld |
|- ( ph -> ( ( ( K + 1 ) x. K ) - ( K ^ 2 ) ) e. RR ) |
69 |
68
|
recnd |
|- ( ph -> ( ( ( K + 1 ) x. K ) - ( K ^ 2 ) ) e. CC ) |
70 |
|
1cnd |
|- ( ph -> 1 e. CC ) |
71 |
67 69 70
|
addsub12d |
|- ( ph -> ( ( A x. K ) + ( ( ( ( K + 1 ) x. K ) - ( K ^ 2 ) ) - 1 ) ) = ( ( ( ( K + 1 ) x. K ) - ( K ^ 2 ) ) + ( ( A x. K ) - 1 ) ) ) |
72 |
45 70 45
|
adddird |
|- ( ph -> ( ( K + 1 ) x. K ) = ( ( K x. K ) + ( 1 x. K ) ) ) |
73 |
45
|
sqvald |
|- ( ph -> ( K ^ 2 ) = ( K x. K ) ) |
74 |
72 73
|
oveq12d |
|- ( ph -> ( ( ( K + 1 ) x. K ) - ( K ^ 2 ) ) = ( ( ( K x. K ) + ( 1 x. K ) ) - ( K x. K ) ) ) |
75 |
45 45
|
mulcld |
|- ( ph -> ( K x. K ) e. CC ) |
76 |
|
ax-1cn |
|- 1 e. CC |
77 |
|
mulcl |
|- ( ( 1 e. CC /\ K e. CC ) -> ( 1 x. K ) e. CC ) |
78 |
76 45 77
|
sylancr |
|- ( ph -> ( 1 x. K ) e. CC ) |
79 |
75 78
|
pncan2d |
|- ( ph -> ( ( ( K x. K ) + ( 1 x. K ) ) - ( K x. K ) ) = ( 1 x. K ) ) |
80 |
45
|
mulid2d |
|- ( ph -> ( 1 x. K ) = K ) |
81 |
74 79 80
|
3eqtrd |
|- ( ph -> ( ( ( K + 1 ) x. K ) - ( K ^ 2 ) ) = K ) |
82 |
81
|
oveq1d |
|- ( ph -> ( ( ( ( K + 1 ) x. K ) - ( K ^ 2 ) ) - 1 ) = ( K - 1 ) ) |
83 |
82
|
oveq2d |
|- ( ph -> ( ( A x. K ) + ( ( ( ( K + 1 ) x. K ) - ( K ^ 2 ) ) - 1 ) ) = ( ( A x. K ) + ( K - 1 ) ) ) |
84 |
41
|
recnd |
|- ( ph -> ( ( K + 1 ) x. K ) e. CC ) |
85 |
15
|
recnd |
|- ( ph -> ( K ^ 2 ) e. CC ) |
86 |
43
|
recnd |
|- ( ph -> ( ( A x. K ) - 1 ) e. CC ) |
87 |
84 85 86
|
subadd23d |
|- ( ph -> ( ( ( ( K + 1 ) x. K ) - ( K ^ 2 ) ) + ( ( A x. K ) - 1 ) ) = ( ( ( K + 1 ) x. K ) + ( ( ( A x. K ) - 1 ) - ( K ^ 2 ) ) ) ) |
88 |
71 83 87
|
3eqtr3d |
|- ( ph -> ( ( A x. K ) + ( K - 1 ) ) = ( ( ( K + 1 ) x. K ) + ( ( ( A x. K ) - 1 ) - ( K ^ 2 ) ) ) ) |
89 |
|
2cnd |
|- ( ph -> 2 e. CC ) |
90 |
10
|
recnd |
|- ( ph -> A e. CC ) |
91 |
89 90 45
|
mulassd |
|- ( ph -> ( ( 2 x. A ) x. K ) = ( 2 x. ( A x. K ) ) ) |
92 |
67
|
2timesd |
|- ( ph -> ( 2 x. ( A x. K ) ) = ( ( A x. K ) + ( A x. K ) ) ) |
93 |
91 92
|
eqtrd |
|- ( ph -> ( ( 2 x. A ) x. K ) = ( ( A x. K ) + ( A x. K ) ) ) |
94 |
93
|
oveq1d |
|- ( ph -> ( ( ( 2 x. A ) x. K ) - ( K ^ 2 ) ) = ( ( ( A x. K ) + ( A x. K ) ) - ( K ^ 2 ) ) ) |
95 |
94
|
oveq1d |
|- ( ph -> ( ( ( ( 2 x. A ) x. K ) - ( K ^ 2 ) ) - 1 ) = ( ( ( ( A x. K ) + ( A x. K ) ) - ( K ^ 2 ) ) - 1 ) ) |
96 |
21 21
|
readdcld |
|- ( ph -> ( ( A x. K ) + ( A x. K ) ) e. RR ) |
97 |
96
|
recnd |
|- ( ph -> ( ( A x. K ) + ( A x. K ) ) e. CC ) |
98 |
97 85 70
|
sub32d |
|- ( ph -> ( ( ( ( A x. K ) + ( A x. K ) ) - ( K ^ 2 ) ) - 1 ) = ( ( ( ( A x. K ) + ( A x. K ) ) - 1 ) - ( K ^ 2 ) ) ) |
99 |
67 67 70
|
addsubassd |
|- ( ph -> ( ( ( A x. K ) + ( A x. K ) ) - 1 ) = ( ( A x. K ) + ( ( A x. K ) - 1 ) ) ) |
100 |
99
|
oveq1d |
|- ( ph -> ( ( ( ( A x. K ) + ( A x. K ) ) - 1 ) - ( K ^ 2 ) ) = ( ( ( A x. K ) + ( ( A x. K ) - 1 ) ) - ( K ^ 2 ) ) ) |
101 |
67 86 85
|
addsubassd |
|- ( ph -> ( ( ( A x. K ) + ( ( A x. K ) - 1 ) ) - ( K ^ 2 ) ) = ( ( A x. K ) + ( ( ( A x. K ) - 1 ) - ( K ^ 2 ) ) ) ) |
102 |
100 101
|
eqtrd |
|- ( ph -> ( ( ( ( A x. K ) + ( A x. K ) ) - 1 ) - ( K ^ 2 ) ) = ( ( A x. K ) + ( ( ( A x. K ) - 1 ) - ( K ^ 2 ) ) ) ) |
103 |
95 98 102
|
3eqtrd |
|- ( ph -> ( ( ( ( 2 x. A ) x. K ) - ( K ^ 2 ) ) - 1 ) = ( ( A x. K ) + ( ( ( A x. K ) - 1 ) - ( K ^ 2 ) ) ) ) |
104 |
66 88 103
|
3brtr4d |
|- ( ph -> ( ( A x. K ) + ( K - 1 ) ) <_ ( ( ( ( 2 x. A ) x. K ) - ( K ^ 2 ) ) - 1 ) ) |
105 |
10 24 19 38 104
|
ltletrd |
|- ( ph -> A < ( ( ( ( 2 x. A ) x. K ) - ( K ^ 2 ) ) - 1 ) ) |
106 |
8 10 19 20 105
|
lttrd |
|- ( ph -> ( K ^ N ) < ( ( ( ( 2 x. A ) x. K ) - ( K ^ 2 ) ) - 1 ) ) |