Step |
Hyp |
Ref |
Expression |
1 |
|
cxpaddle.1 |
|- ( ph -> A e. RR ) |
2 |
|
cxpaddle.2 |
|- ( ph -> 0 <_ A ) |
3 |
|
cxpaddle.3 |
|- ( ph -> B e. RR ) |
4 |
|
cxpaddle.4 |
|- ( ph -> 0 <_ B ) |
5 |
|
cxpaddle.5 |
|- ( ph -> C e. RR+ ) |
6 |
|
cxpaddle.6 |
|- ( ph -> C <_ 1 ) |
7 |
1 3
|
readdcld |
|- ( ph -> ( A + B ) e. RR ) |
8 |
1 3 2 4
|
addge0d |
|- ( ph -> 0 <_ ( A + B ) ) |
9 |
5
|
rpred |
|- ( ph -> C e. RR ) |
10 |
7 8 9
|
recxpcld |
|- ( ph -> ( ( A + B ) ^c C ) e. RR ) |
11 |
10
|
adantr |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( A + B ) ^c C ) e. RR ) |
12 |
11
|
recnd |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( A + B ) ^c C ) e. CC ) |
13 |
12
|
mulid2d |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( 1 x. ( ( A + B ) ^c C ) ) = ( ( A + B ) ^c C ) ) |
14 |
1
|
adantr |
|- ( ( ph /\ 0 < ( A + B ) ) -> A e. RR ) |
15 |
7
|
anim1i |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( A + B ) e. RR /\ 0 < ( A + B ) ) ) |
16 |
|
elrp |
|- ( ( A + B ) e. RR+ <-> ( ( A + B ) e. RR /\ 0 < ( A + B ) ) ) |
17 |
15 16
|
sylibr |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( A + B ) e. RR+ ) |
18 |
14 17
|
rerpdivcld |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( A / ( A + B ) ) e. RR ) |
19 |
3
|
adantr |
|- ( ( ph /\ 0 < ( A + B ) ) -> B e. RR ) |
20 |
19 17
|
rerpdivcld |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( B / ( A + B ) ) e. RR ) |
21 |
2
|
adantr |
|- ( ( ph /\ 0 < ( A + B ) ) -> 0 <_ A ) |
22 |
7
|
adantr |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( A + B ) e. RR ) |
23 |
|
simpr |
|- ( ( ph /\ 0 < ( A + B ) ) -> 0 < ( A + B ) ) |
24 |
|
divge0 |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( ( A + B ) e. RR /\ 0 < ( A + B ) ) ) -> 0 <_ ( A / ( A + B ) ) ) |
25 |
14 21 22 23 24
|
syl22anc |
|- ( ( ph /\ 0 < ( A + B ) ) -> 0 <_ ( A / ( A + B ) ) ) |
26 |
9
|
adantr |
|- ( ( ph /\ 0 < ( A + B ) ) -> C e. RR ) |
27 |
18 25 26
|
recxpcld |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( A / ( A + B ) ) ^c C ) e. RR ) |
28 |
4
|
adantr |
|- ( ( ph /\ 0 < ( A + B ) ) -> 0 <_ B ) |
29 |
|
divge0 |
|- ( ( ( B e. RR /\ 0 <_ B ) /\ ( ( A + B ) e. RR /\ 0 < ( A + B ) ) ) -> 0 <_ ( B / ( A + B ) ) ) |
30 |
19 28 22 23 29
|
syl22anc |
|- ( ( ph /\ 0 < ( A + B ) ) -> 0 <_ ( B / ( A + B ) ) ) |
31 |
20 30 26
|
recxpcld |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( B / ( A + B ) ) ^c C ) e. RR ) |
32 |
1 3
|
addge01d |
|- ( ph -> ( 0 <_ B <-> A <_ ( A + B ) ) ) |
33 |
4 32
|
mpbid |
|- ( ph -> A <_ ( A + B ) ) |
34 |
33
|
adantr |
|- ( ( ph /\ 0 < ( A + B ) ) -> A <_ ( A + B ) ) |
35 |
22
|
recnd |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( A + B ) e. CC ) |
36 |
35
|
mulid1d |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( A + B ) x. 1 ) = ( A + B ) ) |
37 |
34 36
|
breqtrrd |
|- ( ( ph /\ 0 < ( A + B ) ) -> A <_ ( ( A + B ) x. 1 ) ) |
38 |
|
1red |
|- ( ( ph /\ 0 < ( A + B ) ) -> 1 e. RR ) |
39 |
|
ledivmul |
|- ( ( A e. RR /\ 1 e. RR /\ ( ( A + B ) e. RR /\ 0 < ( A + B ) ) ) -> ( ( A / ( A + B ) ) <_ 1 <-> A <_ ( ( A + B ) x. 1 ) ) ) |
40 |
14 38 22 23 39
|
syl112anc |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( A / ( A + B ) ) <_ 1 <-> A <_ ( ( A + B ) x. 1 ) ) ) |
41 |
37 40
|
mpbird |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( A / ( A + B ) ) <_ 1 ) |
42 |
5
|
adantr |
|- ( ( ph /\ 0 < ( A + B ) ) -> C e. RR+ ) |
43 |
6
|
adantr |
|- ( ( ph /\ 0 < ( A + B ) ) -> C <_ 1 ) |
44 |
18 25 41 42 43
|
cxpaddlelem |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( A / ( A + B ) ) <_ ( ( A / ( A + B ) ) ^c C ) ) |
45 |
3 1
|
addge02d |
|- ( ph -> ( 0 <_ A <-> B <_ ( A + B ) ) ) |
46 |
2 45
|
mpbid |
|- ( ph -> B <_ ( A + B ) ) |
47 |
46
|
adantr |
|- ( ( ph /\ 0 < ( A + B ) ) -> B <_ ( A + B ) ) |
48 |
47 36
|
breqtrrd |
|- ( ( ph /\ 0 < ( A + B ) ) -> B <_ ( ( A + B ) x. 1 ) ) |
49 |
|
ledivmul |
|- ( ( B e. RR /\ 1 e. RR /\ ( ( A + B ) e. RR /\ 0 < ( A + B ) ) ) -> ( ( B / ( A + B ) ) <_ 1 <-> B <_ ( ( A + B ) x. 1 ) ) ) |
50 |
19 38 22 23 49
|
syl112anc |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( B / ( A + B ) ) <_ 1 <-> B <_ ( ( A + B ) x. 1 ) ) ) |
51 |
48 50
|
mpbird |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( B / ( A + B ) ) <_ 1 ) |
52 |
20 30 51 42 43
|
cxpaddlelem |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( B / ( A + B ) ) <_ ( ( B / ( A + B ) ) ^c C ) ) |
53 |
18 20 27 31 44 52
|
le2addd |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( A / ( A + B ) ) + ( B / ( A + B ) ) ) <_ ( ( ( A / ( A + B ) ) ^c C ) + ( ( B / ( A + B ) ) ^c C ) ) ) |
54 |
14
|
recnd |
|- ( ( ph /\ 0 < ( A + B ) ) -> A e. CC ) |
55 |
19
|
recnd |
|- ( ( ph /\ 0 < ( A + B ) ) -> B e. CC ) |
56 |
17
|
rpne0d |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( A + B ) =/= 0 ) |
57 |
54 55 35 56
|
divdird |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( A + B ) / ( A + B ) ) = ( ( A / ( A + B ) ) + ( B / ( A + B ) ) ) ) |
58 |
35 56
|
dividd |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( A + B ) / ( A + B ) ) = 1 ) |
59 |
57 58
|
eqtr3d |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( A / ( A + B ) ) + ( B / ( A + B ) ) ) = 1 ) |
60 |
9
|
recnd |
|- ( ph -> C e. CC ) |
61 |
60
|
adantr |
|- ( ( ph /\ 0 < ( A + B ) ) -> C e. CC ) |
62 |
14 21 17 61
|
divcxpd |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( A / ( A + B ) ) ^c C ) = ( ( A ^c C ) / ( ( A + B ) ^c C ) ) ) |
63 |
19 28 17 61
|
divcxpd |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( B / ( A + B ) ) ^c C ) = ( ( B ^c C ) / ( ( A + B ) ^c C ) ) ) |
64 |
62 63
|
oveq12d |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( ( A / ( A + B ) ) ^c C ) + ( ( B / ( A + B ) ) ^c C ) ) = ( ( ( A ^c C ) / ( ( A + B ) ^c C ) ) + ( ( B ^c C ) / ( ( A + B ) ^c C ) ) ) ) |
65 |
1 2 9
|
recxpcld |
|- ( ph -> ( A ^c C ) e. RR ) |
66 |
65
|
recnd |
|- ( ph -> ( A ^c C ) e. CC ) |
67 |
66
|
adantr |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( A ^c C ) e. CC ) |
68 |
3 4 9
|
recxpcld |
|- ( ph -> ( B ^c C ) e. RR ) |
69 |
68
|
recnd |
|- ( ph -> ( B ^c C ) e. CC ) |
70 |
69
|
adantr |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( B ^c C ) e. CC ) |
71 |
17 26
|
rpcxpcld |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( A + B ) ^c C ) e. RR+ ) |
72 |
71
|
rpne0d |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( A + B ) ^c C ) =/= 0 ) |
73 |
67 70 12 72
|
divdird |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( ( A ^c C ) + ( B ^c C ) ) / ( ( A + B ) ^c C ) ) = ( ( ( A ^c C ) / ( ( A + B ) ^c C ) ) + ( ( B ^c C ) / ( ( A + B ) ^c C ) ) ) ) |
74 |
64 73
|
eqtr4d |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( ( A / ( A + B ) ) ^c C ) + ( ( B / ( A + B ) ) ^c C ) ) = ( ( ( A ^c C ) + ( B ^c C ) ) / ( ( A + B ) ^c C ) ) ) |
75 |
53 59 74
|
3brtr3d |
|- ( ( ph /\ 0 < ( A + B ) ) -> 1 <_ ( ( ( A ^c C ) + ( B ^c C ) ) / ( ( A + B ) ^c C ) ) ) |
76 |
65 68
|
readdcld |
|- ( ph -> ( ( A ^c C ) + ( B ^c C ) ) e. RR ) |
77 |
76
|
adantr |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( A ^c C ) + ( B ^c C ) ) e. RR ) |
78 |
38 77 71
|
lemuldivd |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( 1 x. ( ( A + B ) ^c C ) ) <_ ( ( A ^c C ) + ( B ^c C ) ) <-> 1 <_ ( ( ( A ^c C ) + ( B ^c C ) ) / ( ( A + B ) ^c C ) ) ) ) |
79 |
75 78
|
mpbird |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( 1 x. ( ( A + B ) ^c C ) ) <_ ( ( A ^c C ) + ( B ^c C ) ) ) |
80 |
13 79
|
eqbrtrrd |
|- ( ( ph /\ 0 < ( A + B ) ) -> ( ( A + B ) ^c C ) <_ ( ( A ^c C ) + ( B ^c C ) ) ) |
81 |
5
|
rpne0d |
|- ( ph -> C =/= 0 ) |
82 |
60 81
|
0cxpd |
|- ( ph -> ( 0 ^c C ) = 0 ) |
83 |
1 2 9
|
cxpge0d |
|- ( ph -> 0 <_ ( A ^c C ) ) |
84 |
3 4 9
|
cxpge0d |
|- ( ph -> 0 <_ ( B ^c C ) ) |
85 |
65 68 83 84
|
addge0d |
|- ( ph -> 0 <_ ( ( A ^c C ) + ( B ^c C ) ) ) |
86 |
82 85
|
eqbrtrd |
|- ( ph -> ( 0 ^c C ) <_ ( ( A ^c C ) + ( B ^c C ) ) ) |
87 |
|
oveq1 |
|- ( 0 = ( A + B ) -> ( 0 ^c C ) = ( ( A + B ) ^c C ) ) |
88 |
87
|
breq1d |
|- ( 0 = ( A + B ) -> ( ( 0 ^c C ) <_ ( ( A ^c C ) + ( B ^c C ) ) <-> ( ( A + B ) ^c C ) <_ ( ( A ^c C ) + ( B ^c C ) ) ) ) |
89 |
86 88
|
syl5ibcom |
|- ( ph -> ( 0 = ( A + B ) -> ( ( A + B ) ^c C ) <_ ( ( A ^c C ) + ( B ^c C ) ) ) ) |
90 |
89
|
imp |
|- ( ( ph /\ 0 = ( A + B ) ) -> ( ( A + B ) ^c C ) <_ ( ( A ^c C ) + ( B ^c C ) ) ) |
91 |
|
0re |
|- 0 e. RR |
92 |
|
leloe |
|- ( ( 0 e. RR /\ ( A + B ) e. RR ) -> ( 0 <_ ( A + B ) <-> ( 0 < ( A + B ) \/ 0 = ( A + B ) ) ) ) |
93 |
91 7 92
|
sylancr |
|- ( ph -> ( 0 <_ ( A + B ) <-> ( 0 < ( A + B ) \/ 0 = ( A + B ) ) ) ) |
94 |
8 93
|
mpbid |
|- ( ph -> ( 0 < ( A + B ) \/ 0 = ( A + B ) ) ) |
95 |
80 90 94
|
mpjaodan |
|- ( ph -> ( ( A + B ) ^c C ) <_ ( ( A ^c C ) + ( B ^c C ) ) ) |