Step |
Hyp |
Ref |
Expression |
1 |
|
oexpreposd.n |
|- ( ph -> N e. RR ) |
2 |
|
oexpreposd.m |
|- ( ph -> M e. NN ) |
3 |
|
oexpreposd.1 |
|- ( ph -> -. ( M / 2 ) e. NN ) |
4 |
1
|
adantr |
|- ( ( ph /\ 0 < N ) -> N e. RR ) |
5 |
2
|
nnzd |
|- ( ph -> M e. ZZ ) |
6 |
5
|
adantr |
|- ( ( ph /\ 0 < N ) -> M e. ZZ ) |
7 |
|
simpr |
|- ( ( ph /\ 0 < N ) -> 0 < N ) |
8 |
|
expgt0 |
|- ( ( N e. RR /\ M e. ZZ /\ 0 < N ) -> 0 < ( N ^ M ) ) |
9 |
4 6 7 8
|
syl3anc |
|- ( ( ph /\ 0 < N ) -> 0 < ( N ^ M ) ) |
10 |
9
|
ex |
|- ( ph -> ( 0 < N -> 0 < ( N ^ M ) ) ) |
11 |
|
0red |
|- ( ph -> 0 e. RR ) |
12 |
11 1
|
lttrid |
|- ( ph -> ( 0 < N <-> -. ( 0 = N \/ N < 0 ) ) ) |
13 |
12
|
notbid |
|- ( ph -> ( -. 0 < N <-> -. -. ( 0 = N \/ N < 0 ) ) ) |
14 |
|
notnotr |
|- ( -. -. ( 0 = N \/ N < 0 ) -> ( 0 = N \/ N < 0 ) ) |
15 |
|
0re |
|- 0 e. RR |
16 |
15
|
ltnri |
|- -. 0 < 0 |
17 |
2
|
0expd |
|- ( ph -> ( 0 ^ M ) = 0 ) |
18 |
17
|
breq2d |
|- ( ph -> ( 0 < ( 0 ^ M ) <-> 0 < 0 ) ) |
19 |
16 18
|
mtbiri |
|- ( ph -> -. 0 < ( 0 ^ M ) ) |
20 |
19
|
adantr |
|- ( ( ph /\ 0 = N ) -> -. 0 < ( 0 ^ M ) ) |
21 |
|
simpr |
|- ( ( ph /\ 0 = N ) -> 0 = N ) |
22 |
21
|
eqcomd |
|- ( ( ph /\ 0 = N ) -> N = 0 ) |
23 |
22
|
oveq1d |
|- ( ( ph /\ 0 = N ) -> ( N ^ M ) = ( 0 ^ M ) ) |
24 |
23
|
breq2d |
|- ( ( ph /\ 0 = N ) -> ( 0 < ( N ^ M ) <-> 0 < ( 0 ^ M ) ) ) |
25 |
20 24
|
mtbird |
|- ( ( ph /\ 0 = N ) -> -. 0 < ( N ^ M ) ) |
26 |
25
|
ex |
|- ( ph -> ( 0 = N -> -. 0 < ( N ^ M ) ) ) |
27 |
1
|
renegcld |
|- ( ph -> -u N e. RR ) |
28 |
27
|
adantr |
|- ( ( ph /\ 0 < -u N ) -> -u N e. RR ) |
29 |
5
|
adantr |
|- ( ( ph /\ 0 < -u N ) -> M e. ZZ ) |
30 |
|
simpr |
|- ( ( ph /\ 0 < -u N ) -> 0 < -u N ) |
31 |
|
expgt0 |
|- ( ( -u N e. RR /\ M e. ZZ /\ 0 < -u N ) -> 0 < ( -u N ^ M ) ) |
32 |
28 29 30 31
|
syl3anc |
|- ( ( ph /\ 0 < -u N ) -> 0 < ( -u N ^ M ) ) |
33 |
32
|
ex |
|- ( ph -> ( 0 < -u N -> 0 < ( -u N ^ M ) ) ) |
34 |
1
|
recnd |
|- ( ph -> N e. CC ) |
35 |
|
simpr |
|- ( ( ph /\ ( M / 2 ) e. ZZ ) -> ( M / 2 ) e. ZZ ) |
36 |
|
zq |
|- ( ( M / 2 ) e. ZZ -> ( M / 2 ) e. QQ ) |
37 |
36
|
adantl |
|- ( ( ph /\ ( M / 2 ) e. ZZ ) -> ( M / 2 ) e. QQ ) |
38 |
|
qden1elz |
|- ( ( M / 2 ) e. QQ -> ( ( denom ` ( M / 2 ) ) = 1 <-> ( M / 2 ) e. ZZ ) ) |
39 |
37 38
|
syl |
|- ( ( ph /\ ( M / 2 ) e. ZZ ) -> ( ( denom ` ( M / 2 ) ) = 1 <-> ( M / 2 ) e. ZZ ) ) |
40 |
35 39
|
mpbird |
|- ( ( ph /\ ( M / 2 ) e. ZZ ) -> ( denom ` ( M / 2 ) ) = 1 ) |
41 |
40
|
oveq2d |
|- ( ( ph /\ ( M / 2 ) e. ZZ ) -> ( ( M / 2 ) x. ( denom ` ( M / 2 ) ) ) = ( ( M / 2 ) x. 1 ) ) |
42 |
|
qmuldeneqnum |
|- ( ( M / 2 ) e. QQ -> ( ( M / 2 ) x. ( denom ` ( M / 2 ) ) ) = ( numer ` ( M / 2 ) ) ) |
43 |
37 42
|
syl |
|- ( ( ph /\ ( M / 2 ) e. ZZ ) -> ( ( M / 2 ) x. ( denom ` ( M / 2 ) ) ) = ( numer ` ( M / 2 ) ) ) |
44 |
35
|
zcnd |
|- ( ( ph /\ ( M / 2 ) e. ZZ ) -> ( M / 2 ) e. CC ) |
45 |
44
|
mulid1d |
|- ( ( ph /\ ( M / 2 ) e. ZZ ) -> ( ( M / 2 ) x. 1 ) = ( M / 2 ) ) |
46 |
41 43 45
|
3eqtr3rd |
|- ( ( ph /\ ( M / 2 ) e. ZZ ) -> ( M / 2 ) = ( numer ` ( M / 2 ) ) ) |
47 |
2
|
nnred |
|- ( ph -> M e. RR ) |
48 |
|
2re |
|- 2 e. RR |
49 |
48
|
a1i |
|- ( ph -> 2 e. RR ) |
50 |
2
|
nngt0d |
|- ( ph -> 0 < M ) |
51 |
|
2pos |
|- 0 < 2 |
52 |
51
|
a1i |
|- ( ph -> 0 < 2 ) |
53 |
47 49 50 52
|
divgt0d |
|- ( ph -> 0 < ( M / 2 ) ) |
54 |
|
qgt0numnn |
|- ( ( ( M / 2 ) e. QQ /\ 0 < ( M / 2 ) ) -> ( numer ` ( M / 2 ) ) e. NN ) |
55 |
36 53 54
|
syl2anr |
|- ( ( ph /\ ( M / 2 ) e. ZZ ) -> ( numer ` ( M / 2 ) ) e. NN ) |
56 |
46 55
|
eqeltrd |
|- ( ( ph /\ ( M / 2 ) e. ZZ ) -> ( M / 2 ) e. NN ) |
57 |
3 56
|
mtand |
|- ( ph -> -. ( M / 2 ) e. ZZ ) |
58 |
|
evend2 |
|- ( M e. ZZ -> ( 2 || M <-> ( M / 2 ) e. ZZ ) ) |
59 |
5 58
|
syl |
|- ( ph -> ( 2 || M <-> ( M / 2 ) e. ZZ ) ) |
60 |
57 59
|
mtbird |
|- ( ph -> -. 2 || M ) |
61 |
|
oexpneg |
|- ( ( N e. CC /\ M e. NN /\ -. 2 || M ) -> ( -u N ^ M ) = -u ( N ^ M ) ) |
62 |
34 2 60 61
|
syl3anc |
|- ( ph -> ( -u N ^ M ) = -u ( N ^ M ) ) |
63 |
62
|
breq2d |
|- ( ph -> ( 0 < ( -u N ^ M ) <-> 0 < -u ( N ^ M ) ) ) |
64 |
63
|
biimpd |
|- ( ph -> ( 0 < ( -u N ^ M ) -> 0 < -u ( N ^ M ) ) ) |
65 |
2
|
nnnn0d |
|- ( ph -> M e. NN0 ) |
66 |
1 65
|
reexpcld |
|- ( ph -> ( N ^ M ) e. RR ) |
67 |
66
|
renegcld |
|- ( ph -> -u ( N ^ M ) e. RR ) |
68 |
11 67
|
lttrid |
|- ( ph -> ( 0 < -u ( N ^ M ) <-> -. ( 0 = -u ( N ^ M ) \/ -u ( N ^ M ) < 0 ) ) ) |
69 |
|
pm2.46 |
|- ( -. ( 0 = -u ( N ^ M ) \/ -u ( N ^ M ) < 0 ) -> -. -u ( N ^ M ) < 0 ) |
70 |
68 69
|
syl6bi |
|- ( ph -> ( 0 < -u ( N ^ M ) -> -. -u ( N ^ M ) < 0 ) ) |
71 |
33 64 70
|
3syld |
|- ( ph -> ( 0 < -u N -> -. -u ( N ^ M ) < 0 ) ) |
72 |
1
|
lt0neg1d |
|- ( ph -> ( N < 0 <-> 0 < -u N ) ) |
73 |
66
|
lt0neg2d |
|- ( ph -> ( 0 < ( N ^ M ) <-> -u ( N ^ M ) < 0 ) ) |
74 |
73
|
notbid |
|- ( ph -> ( -. 0 < ( N ^ M ) <-> -. -u ( N ^ M ) < 0 ) ) |
75 |
71 72 74
|
3imtr4d |
|- ( ph -> ( N < 0 -> -. 0 < ( N ^ M ) ) ) |
76 |
26 75
|
jaod |
|- ( ph -> ( ( 0 = N \/ N < 0 ) -> -. 0 < ( N ^ M ) ) ) |
77 |
14 76
|
syl5 |
|- ( ph -> ( -. -. ( 0 = N \/ N < 0 ) -> -. 0 < ( N ^ M ) ) ) |
78 |
13 77
|
sylbid |
|- ( ph -> ( -. 0 < N -> -. 0 < ( N ^ M ) ) ) |
79 |
10 78
|
impcon4bid |
|- ( ph -> ( 0 < N <-> 0 < ( N ^ M ) ) ) |