Step |
Hyp |
Ref |
Expression |
1 |
|
fltltc.a |
|- ( ph -> A e. NN ) |
2 |
|
fltltc.b |
|- ( ph -> B e. NN ) |
3 |
|
fltltc.c |
|- ( ph -> C e. NN ) |
4 |
|
fltltc.n |
|- ( ph -> N e. ( ZZ>= ` 3 ) ) |
5 |
|
fltltc.1 |
|- ( ph -> ( ( A ^ N ) + ( B ^ N ) ) = ( C ^ N ) ) |
6 |
|
fltnlta.1 |
|- ( ph -> A < B ) |
7 |
|
eluzge3nn |
|- ( N e. ( ZZ>= ` 3 ) -> N e. NN ) |
8 |
4 7
|
syl |
|- ( ph -> N e. NN ) |
9 |
8
|
nnred |
|- ( ph -> N e. RR ) |
10 |
3
|
nnred |
|- ( ph -> C e. RR ) |
11 |
2
|
nnred |
|- ( ph -> B e. RR ) |
12 |
10 11
|
resubcld |
|- ( ph -> ( C - B ) e. RR ) |
13 |
|
uzuzle23 |
|- ( N e. ( ZZ>= ` 3 ) -> N e. ( ZZ>= ` 2 ) ) |
14 |
|
uz2m1nn |
|- ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) e. NN ) |
15 |
4 13 14
|
3syl |
|- ( ph -> ( N - 1 ) e. NN ) |
16 |
15
|
nnnn0d |
|- ( ph -> ( N - 1 ) e. NN0 ) |
17 |
10 16
|
reexpcld |
|- ( ph -> ( C ^ ( N - 1 ) ) e. RR ) |
18 |
15
|
nnred |
|- ( ph -> ( N - 1 ) e. RR ) |
19 |
11 16
|
reexpcld |
|- ( ph -> ( B ^ ( N - 1 ) ) e. RR ) |
20 |
18 19
|
remulcld |
|- ( ph -> ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) e. RR ) |
21 |
17 20
|
readdcld |
|- ( ph -> ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) e. RR ) |
22 |
12 21
|
remulcld |
|- ( ph -> ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) e. RR ) |
23 |
1
|
nnrpd |
|- ( ph -> A e. RR+ ) |
24 |
15
|
nnzd |
|- ( ph -> ( N - 1 ) e. ZZ ) |
25 |
23 24
|
rpexpcld |
|- ( ph -> ( A ^ ( N - 1 ) ) e. RR+ ) |
26 |
22 25
|
rerpdivcld |
|- ( ph -> ( ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) e. RR ) |
27 |
1
|
nnred |
|- ( ph -> A e. RR ) |
28 |
19 20
|
readdcld |
|- ( ph -> ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) e. RR ) |
29 |
12 28
|
remulcld |
|- ( ph -> ( ( C - B ) x. ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) e. RR ) |
30 |
29 25
|
rerpdivcld |
|- ( ph -> ( ( ( C - B ) x. ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) e. RR ) |
31 |
12 9
|
remulcld |
|- ( ph -> ( ( C - B ) x. N ) e. RR ) |
32 |
|
1cnd |
|- ( ph -> 1 e. CC ) |
33 |
15
|
nncnd |
|- ( ph -> ( N - 1 ) e. CC ) |
34 |
19
|
recnd |
|- ( ph -> ( B ^ ( N - 1 ) ) e. CC ) |
35 |
32 33 34
|
adddird |
|- ( ph -> ( ( 1 + ( N - 1 ) ) x. ( B ^ ( N - 1 ) ) ) = ( ( 1 x. ( B ^ ( N - 1 ) ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) |
36 |
8
|
nncnd |
|- ( ph -> N e. CC ) |
37 |
32 36
|
pncan3d |
|- ( ph -> ( 1 + ( N - 1 ) ) = N ) |
38 |
37
|
oveq1d |
|- ( ph -> ( ( 1 + ( N - 1 ) ) x. ( B ^ ( N - 1 ) ) ) = ( N x. ( B ^ ( N - 1 ) ) ) ) |
39 |
34
|
mulid2d |
|- ( ph -> ( 1 x. ( B ^ ( N - 1 ) ) ) = ( B ^ ( N - 1 ) ) ) |
40 |
39
|
oveq1d |
|- ( ph -> ( ( 1 x. ( B ^ ( N - 1 ) ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) = ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) |
41 |
35 38 40
|
3eqtr3rd |
|- ( ph -> ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) = ( N x. ( B ^ ( N - 1 ) ) ) ) |
42 |
41
|
oveq2d |
|- ( ph -> ( ( C - B ) x. ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) = ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) ) |
43 |
42
|
oveq1d |
|- ( ph -> ( ( ( C - B ) x. ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) = ( ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) ) |
44 |
43 30
|
eqeltrrd |
|- ( ph -> ( ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) e. RR ) |
45 |
8
|
nnnn0d |
|- ( ph -> N e. NN0 ) |
46 |
45
|
nn0ge0d |
|- ( ph -> 0 <_ N ) |
47 |
|
1red |
|- ( ph -> 1 e. RR ) |
48 |
1 2 3 4 5
|
fltltc |
|- ( ph -> B < C ) |
49 |
|
nnltp1le |
|- ( ( B e. NN /\ C e. NN ) -> ( B < C <-> ( B + 1 ) <_ C ) ) |
50 |
2 3 49
|
syl2anc |
|- ( ph -> ( B < C <-> ( B + 1 ) <_ C ) ) |
51 |
48 50
|
mpbid |
|- ( ph -> ( B + 1 ) <_ C ) |
52 |
11
|
leidd |
|- ( ph -> B <_ B ) |
53 |
10 11 47 11 51 52
|
lesub3d |
|- ( ph -> 1 <_ ( C - B ) ) |
54 |
9 12 46 53
|
lemulge12d |
|- ( ph -> N <_ ( ( C - B ) x. N ) ) |
55 |
12
|
recnd |
|- ( ph -> ( C - B ) e. CC ) |
56 |
25
|
rpred |
|- ( ph -> ( A ^ ( N - 1 ) ) e. RR ) |
57 |
56
|
recnd |
|- ( ph -> ( A ^ ( N - 1 ) ) e. CC ) |
58 |
55 36 57
|
mulassd |
|- ( ph -> ( ( ( C - B ) x. N ) x. ( A ^ ( N - 1 ) ) ) = ( ( C - B ) x. ( N x. ( A ^ ( N - 1 ) ) ) ) ) |
59 |
58
|
oveq1d |
|- ( ph -> ( ( ( ( C - B ) x. N ) x. ( A ^ ( N - 1 ) ) ) / ( A ^ ( N - 1 ) ) ) = ( ( ( C - B ) x. ( N x. ( A ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) ) |
60 |
55 36
|
mulcld |
|- ( ph -> ( ( C - B ) x. N ) e. CC ) |
61 |
1
|
nncnd |
|- ( ph -> A e. CC ) |
62 |
1
|
nnne0d |
|- ( ph -> A =/= 0 ) |
63 |
61 62 24
|
expne0d |
|- ( ph -> ( A ^ ( N - 1 ) ) =/= 0 ) |
64 |
60 57 63
|
divcan4d |
|- ( ph -> ( ( ( ( C - B ) x. N ) x. ( A ^ ( N - 1 ) ) ) / ( A ^ ( N - 1 ) ) ) = ( ( C - B ) x. N ) ) |
65 |
59 64
|
eqtr3d |
|- ( ph -> ( ( ( C - B ) x. ( N x. ( A ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) = ( ( C - B ) x. N ) ) |
66 |
9 56
|
remulcld |
|- ( ph -> ( N x. ( A ^ ( N - 1 ) ) ) e. RR ) |
67 |
12 66
|
remulcld |
|- ( ph -> ( ( C - B ) x. ( N x. ( A ^ ( N - 1 ) ) ) ) e. RR ) |
68 |
42 29
|
eqeltrrd |
|- ( ph -> ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) e. RR ) |
69 |
41 28
|
eqeltrrd |
|- ( ph -> ( N x. ( B ^ ( N - 1 ) ) ) e. RR ) |
70 |
|
difrp |
|- ( ( B e. RR /\ C e. RR ) -> ( B < C <-> ( C - B ) e. RR+ ) ) |
71 |
11 10 70
|
syl2anc |
|- ( ph -> ( B < C <-> ( C - B ) e. RR+ ) ) |
72 |
48 71
|
mpbid |
|- ( ph -> ( C - B ) e. RR+ ) |
73 |
8
|
nnrpd |
|- ( ph -> N e. RR+ ) |
74 |
2
|
nnrpd |
|- ( ph -> B e. RR+ ) |
75 |
23 74 15 6
|
ltexp1dd |
|- ( ph -> ( A ^ ( N - 1 ) ) < ( B ^ ( N - 1 ) ) ) |
76 |
56 19 73 75
|
ltmul2dd |
|- ( ph -> ( N x. ( A ^ ( N - 1 ) ) ) < ( N x. ( B ^ ( N - 1 ) ) ) ) |
77 |
66 69 72 76
|
ltmul2dd |
|- ( ph -> ( ( C - B ) x. ( N x. ( A ^ ( N - 1 ) ) ) ) < ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) ) |
78 |
67 68 25 77
|
ltdiv1dd |
|- ( ph -> ( ( ( C - B ) x. ( N x. ( A ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) < ( ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) ) |
79 |
65 78
|
eqbrtrrd |
|- ( ph -> ( ( C - B ) x. N ) < ( ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) ) |
80 |
9 31 44 54 79
|
lelttrd |
|- ( ph -> N < ( ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) ) |
81 |
80 43
|
breqtrrd |
|- ( ph -> N < ( ( ( C - B ) x. ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) ) |
82 |
3
|
nnrpd |
|- ( ph -> C e. RR+ ) |
83 |
74 82 15 48
|
ltexp1dd |
|- ( ph -> ( B ^ ( N - 1 ) ) < ( C ^ ( N - 1 ) ) ) |
84 |
19 17 20 83
|
ltadd1dd |
|- ( ph -> ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) < ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) |
85 |
28 21 72 84
|
ltmul2dd |
|- ( ph -> ( ( C - B ) x. ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) < ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) ) |
86 |
29 22 25 85
|
ltdiv1dd |
|- ( ph -> ( ( ( C - B ) x. ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) < ( ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) ) |
87 |
9 30 26 81 86
|
lttrd |
|- ( ph -> N < ( ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) ) |
88 |
27 45
|
reexpcld |
|- ( ph -> ( A ^ N ) e. RR ) |
89 |
1 2 3 4 5
|
fltnltalem |
|- ( ph -> ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) < ( A ^ N ) ) |
90 |
22 88 25 89
|
ltdiv1dd |
|- ( ph -> ( ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) < ( ( A ^ N ) / ( A ^ ( N - 1 ) ) ) ) |
91 |
36 32
|
nncand |
|- ( ph -> ( N - ( N - 1 ) ) = 1 ) |
92 |
91
|
oveq2d |
|- ( ph -> ( A ^ ( N - ( N - 1 ) ) ) = ( A ^ 1 ) ) |
93 |
8
|
nnzd |
|- ( ph -> N e. ZZ ) |
94 |
61 62 24 93
|
expsubd |
|- ( ph -> ( A ^ ( N - ( N - 1 ) ) ) = ( ( A ^ N ) / ( A ^ ( N - 1 ) ) ) ) |
95 |
61
|
exp1d |
|- ( ph -> ( A ^ 1 ) = A ) |
96 |
92 94 95
|
3eqtr3d |
|- ( ph -> ( ( A ^ N ) / ( A ^ ( N - 1 ) ) ) = A ) |
97 |
90 96
|
breqtrd |
|- ( ph -> ( ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) < A ) |
98 |
9 26 27 87 97
|
lttrd |
|- ( ph -> N < A ) |