Step |
Hyp |
Ref |
Expression |
1 |
|
1re |
|- 1 e. RR |
2 |
|
elicc01 |
|- ( T e. ( 0 [,] 1 ) <-> ( T e. RR /\ 0 <_ T /\ T <_ 1 ) ) |
3 |
2
|
simp1bi |
|- ( T e. ( 0 [,] 1 ) -> T e. RR ) |
4 |
3
|
ad2antrl |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> T e. RR ) |
5 |
|
resubcl |
|- ( ( 1 e. RR /\ T e. RR ) -> ( 1 - T ) e. RR ) |
6 |
1 4 5
|
sylancr |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 1 - T ) e. RR ) |
7 |
6
|
recnd |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 1 - T ) e. CC ) |
8 |
7
|
mul02d |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 0 x. ( 1 - T ) ) = 0 ) |
9 |
8
|
eqcomd |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> 0 = ( 0 x. ( 1 - T ) ) ) |
10 |
|
elicc01 |
|- ( S e. ( 0 [,] 1 ) <-> ( S e. RR /\ 0 <_ S /\ S <_ 1 ) ) |
11 |
10
|
simp1bi |
|- ( S e. ( 0 [,] 1 ) -> S e. RR ) |
12 |
11
|
ad2antll |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> S e. RR ) |
13 |
|
resubcl |
|- ( ( 1 e. RR /\ S e. RR ) -> ( 1 - S ) e. RR ) |
14 |
1 12 13
|
sylancr |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 1 - S ) e. RR ) |
15 |
14
|
recnd |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 1 - S ) e. CC ) |
16 |
15
|
mulid2d |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 1 x. ( 1 - S ) ) = ( 1 - S ) ) |
17 |
|
oveq2 |
|- ( S = 0 -> ( 1 - S ) = ( 1 - 0 ) ) |
18 |
17
|
adantr |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 1 - S ) = ( 1 - 0 ) ) |
19 |
|
1m0e1 |
|- ( 1 - 0 ) = 1 |
20 |
18 19
|
eqtrdi |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 1 - S ) = 1 ) |
21 |
16 20
|
eqtr2d |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> 1 = ( 1 x. ( 1 - S ) ) ) |
22 |
4
|
recnd |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> T e. CC ) |
23 |
22
|
mul02d |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 0 x. T ) = 0 ) |
24 |
|
oveq2 |
|- ( S = 0 -> ( 1 x. S ) = ( 1 x. 0 ) ) |
25 |
24
|
adantr |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 1 x. S ) = ( 1 x. 0 ) ) |
26 |
|
ax-1cn |
|- 1 e. CC |
27 |
26
|
mul01i |
|- ( 1 x. 0 ) = 0 |
28 |
25 27
|
eqtrdi |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 1 x. S ) = 0 ) |
29 |
23 28
|
eqtr4d |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 0 x. T ) = ( 1 x. S ) ) |
30 |
|
1elunit |
|- 1 e. ( 0 [,] 1 ) |
31 |
|
0elunit |
|- 0 e. ( 0 [,] 1 ) |
32 |
|
oveq2 |
|- ( r = 1 -> ( 1 - r ) = ( 1 - 1 ) ) |
33 |
|
1m1e0 |
|- ( 1 - 1 ) = 0 |
34 |
32 33
|
eqtrdi |
|- ( r = 1 -> ( 1 - r ) = 0 ) |
35 |
34
|
oveq1d |
|- ( r = 1 -> ( ( 1 - r ) x. ( 1 - T ) ) = ( 0 x. ( 1 - T ) ) ) |
36 |
35
|
eqeq2d |
|- ( r = 1 -> ( p = ( ( 1 - r ) x. ( 1 - T ) ) <-> p = ( 0 x. ( 1 - T ) ) ) ) |
37 |
|
eqeq1 |
|- ( r = 1 -> ( r = ( ( 1 - p ) x. ( 1 - S ) ) <-> 1 = ( ( 1 - p ) x. ( 1 - S ) ) ) ) |
38 |
34
|
oveq1d |
|- ( r = 1 -> ( ( 1 - r ) x. T ) = ( 0 x. T ) ) |
39 |
38
|
eqeq1d |
|- ( r = 1 -> ( ( ( 1 - r ) x. T ) = ( ( 1 - p ) x. S ) <-> ( 0 x. T ) = ( ( 1 - p ) x. S ) ) ) |
40 |
36 37 39
|
3anbi123d |
|- ( r = 1 -> ( ( p = ( ( 1 - r ) x. ( 1 - T ) ) /\ r = ( ( 1 - p ) x. ( 1 - S ) ) /\ ( ( 1 - r ) x. T ) = ( ( 1 - p ) x. S ) ) <-> ( p = ( 0 x. ( 1 - T ) ) /\ 1 = ( ( 1 - p ) x. ( 1 - S ) ) /\ ( 0 x. T ) = ( ( 1 - p ) x. S ) ) ) ) |
41 |
|
eqeq1 |
|- ( p = 0 -> ( p = ( 0 x. ( 1 - T ) ) <-> 0 = ( 0 x. ( 1 - T ) ) ) ) |
42 |
|
oveq2 |
|- ( p = 0 -> ( 1 - p ) = ( 1 - 0 ) ) |
43 |
42 19
|
eqtrdi |
|- ( p = 0 -> ( 1 - p ) = 1 ) |
44 |
43
|
oveq1d |
|- ( p = 0 -> ( ( 1 - p ) x. ( 1 - S ) ) = ( 1 x. ( 1 - S ) ) ) |
45 |
44
|
eqeq2d |
|- ( p = 0 -> ( 1 = ( ( 1 - p ) x. ( 1 - S ) ) <-> 1 = ( 1 x. ( 1 - S ) ) ) ) |
46 |
43
|
oveq1d |
|- ( p = 0 -> ( ( 1 - p ) x. S ) = ( 1 x. S ) ) |
47 |
46
|
eqeq2d |
|- ( p = 0 -> ( ( 0 x. T ) = ( ( 1 - p ) x. S ) <-> ( 0 x. T ) = ( 1 x. S ) ) ) |
48 |
41 45 47
|
3anbi123d |
|- ( p = 0 -> ( ( p = ( 0 x. ( 1 - T ) ) /\ 1 = ( ( 1 - p ) x. ( 1 - S ) ) /\ ( 0 x. T ) = ( ( 1 - p ) x. S ) ) <-> ( 0 = ( 0 x. ( 1 - T ) ) /\ 1 = ( 1 x. ( 1 - S ) ) /\ ( 0 x. T ) = ( 1 x. S ) ) ) ) |
49 |
40 48
|
rspc2ev |
|- ( ( 1 e. ( 0 [,] 1 ) /\ 0 e. ( 0 [,] 1 ) /\ ( 0 = ( 0 x. ( 1 - T ) ) /\ 1 = ( 1 x. ( 1 - S ) ) /\ ( 0 x. T ) = ( 1 x. S ) ) ) -> E. r e. ( 0 [,] 1 ) E. p e. ( 0 [,] 1 ) ( p = ( ( 1 - r ) x. ( 1 - T ) ) /\ r = ( ( 1 - p ) x. ( 1 - S ) ) /\ ( ( 1 - r ) x. T ) = ( ( 1 - p ) x. S ) ) ) |
50 |
30 31 49
|
mp3an12 |
|- ( ( 0 = ( 0 x. ( 1 - T ) ) /\ 1 = ( 1 x. ( 1 - S ) ) /\ ( 0 x. T ) = ( 1 x. S ) ) -> E. r e. ( 0 [,] 1 ) E. p e. ( 0 [,] 1 ) ( p = ( ( 1 - r ) x. ( 1 - T ) ) /\ r = ( ( 1 - p ) x. ( 1 - S ) ) /\ ( ( 1 - r ) x. T ) = ( ( 1 - p ) x. S ) ) ) |
51 |
9 21 29 50
|
syl3anc |
|- ( ( S = 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> E. r e. ( 0 [,] 1 ) E. p e. ( 0 [,] 1 ) ( p = ( ( 1 - r ) x. ( 1 - T ) ) /\ r = ( ( 1 - p ) x. ( 1 - S ) ) /\ ( ( 1 - r ) x. T ) = ( ( 1 - p ) x. S ) ) ) |
52 |
51
|
ex |
|- ( S = 0 -> ( ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) -> E. r e. ( 0 [,] 1 ) E. p e. ( 0 [,] 1 ) ( p = ( ( 1 - r ) x. ( 1 - T ) ) /\ r = ( ( 1 - p ) x. ( 1 - S ) ) /\ ( ( 1 - r ) x. T ) = ( ( 1 - p ) x. S ) ) ) ) |
53 |
3
|
ad2antrl |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> T e. RR ) |
54 |
11
|
ad2antll |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> S e. RR ) |
55 |
54 53
|
remulcld |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S x. T ) e. RR ) |
56 |
53 55
|
resubcld |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( T - ( S x. T ) ) e. RR ) |
57 |
54 53
|
readdcld |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S + T ) e. RR ) |
58 |
57 55
|
resubcld |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S + T ) - ( S x. T ) ) e. RR ) |
59 |
|
1red |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> 1 e. RR ) |
60 |
2
|
simp2bi |
|- ( T e. ( 0 [,] 1 ) -> 0 <_ T ) |
61 |
60
|
ad2antrl |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> 0 <_ T ) |
62 |
10
|
simp3bi |
|- ( S e. ( 0 [,] 1 ) -> S <_ 1 ) |
63 |
62
|
ad2antll |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> S <_ 1 ) |
64 |
54 59 53 61 63
|
lemul1ad |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S x. T ) <_ ( 1 x. T ) ) |
65 |
53
|
recnd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> T e. CC ) |
66 |
65
|
mulid2d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 1 x. T ) = T ) |
67 |
64 66
|
breqtrd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S x. T ) <_ T ) |
68 |
10
|
simp2bi |
|- ( S e. ( 0 [,] 1 ) -> 0 <_ S ) |
69 |
68
|
ad2antll |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> 0 <_ S ) |
70 |
|
simpl |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> S =/= 0 ) |
71 |
54 69 70
|
ne0gt0d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> 0 < S ) |
72 |
54 53
|
ltaddpos2d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 0 < S <-> T < ( S + T ) ) ) |
73 |
71 72
|
mpbid |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> T < ( S + T ) ) |
74 |
55 53 57 67 73
|
lelttrd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S x. T ) < ( S + T ) ) |
75 |
55 57
|
posdifd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S x. T ) < ( S + T ) <-> 0 < ( ( S + T ) - ( S x. T ) ) ) ) |
76 |
74 75
|
mpbid |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> 0 < ( ( S + T ) - ( S x. T ) ) ) |
77 |
76
|
gt0ne0d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S + T ) - ( S x. T ) ) =/= 0 ) |
78 |
56 58 77
|
redivcld |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) e. RR ) |
79 |
53 55
|
subge0d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 0 <_ ( T - ( S x. T ) ) <-> ( S x. T ) <_ T ) ) |
80 |
67 79
|
mpbird |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> 0 <_ ( T - ( S x. T ) ) ) |
81 |
|
divge0 |
|- ( ( ( ( T - ( S x. T ) ) e. RR /\ 0 <_ ( T - ( S x. T ) ) ) /\ ( ( ( S + T ) - ( S x. T ) ) e. RR /\ 0 < ( ( S + T ) - ( S x. T ) ) ) ) -> 0 <_ ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) |
82 |
56 80 58 76 81
|
syl22anc |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> 0 <_ ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) |
83 |
53 57 73
|
ltled |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> T <_ ( S + T ) ) |
84 |
53 57 55 83
|
lesub1dd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( T - ( S x. T ) ) <_ ( ( S + T ) - ( S x. T ) ) ) |
85 |
58
|
recnd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S + T ) - ( S x. T ) ) e. CC ) |
86 |
85
|
mulid2d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 1 x. ( ( S + T ) - ( S x. T ) ) ) = ( ( S + T ) - ( S x. T ) ) ) |
87 |
84 86
|
breqtrrd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( T - ( S x. T ) ) <_ ( 1 x. ( ( S + T ) - ( S x. T ) ) ) ) |
88 |
|
ledivmul2 |
|- ( ( ( T - ( S x. T ) ) e. RR /\ 1 e. RR /\ ( ( ( S + T ) - ( S x. T ) ) e. RR /\ 0 < ( ( S + T ) - ( S x. T ) ) ) ) -> ( ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) <_ 1 <-> ( T - ( S x. T ) ) <_ ( 1 x. ( ( S + T ) - ( S x. T ) ) ) ) ) |
89 |
56 59 58 76 88
|
syl112anc |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) <_ 1 <-> ( T - ( S x. T ) ) <_ ( 1 x. ( ( S + T ) - ( S x. T ) ) ) ) ) |
90 |
87 89
|
mpbird |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) <_ 1 ) |
91 |
|
elicc01 |
|- ( ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) e. ( 0 [,] 1 ) <-> ( ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) e. RR /\ 0 <_ ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) /\ ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) <_ 1 ) ) |
92 |
78 82 90 91
|
syl3anbrc |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) e. ( 0 [,] 1 ) ) |
93 |
54 55
|
resubcld |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S - ( S x. T ) ) e. RR ) |
94 |
93 58 77
|
redivcld |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) e. RR ) |
95 |
2
|
simp3bi |
|- ( T e. ( 0 [,] 1 ) -> T <_ 1 ) |
96 |
95
|
ad2antrl |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> T <_ 1 ) |
97 |
53 59 54 69 96
|
lemul2ad |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S x. T ) <_ ( S x. 1 ) ) |
98 |
54
|
recnd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> S e. CC ) |
99 |
98
|
mulid1d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S x. 1 ) = S ) |
100 |
97 99
|
breqtrd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S x. T ) <_ S ) |
101 |
54 55
|
subge0d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 0 <_ ( S - ( S x. T ) ) <-> ( S x. T ) <_ S ) ) |
102 |
100 101
|
mpbird |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> 0 <_ ( S - ( S x. T ) ) ) |
103 |
|
divge0 |
|- ( ( ( ( S - ( S x. T ) ) e. RR /\ 0 <_ ( S - ( S x. T ) ) ) /\ ( ( ( S + T ) - ( S x. T ) ) e. RR /\ 0 < ( ( S + T ) - ( S x. T ) ) ) ) -> 0 <_ ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) |
104 |
93 102 58 76 103
|
syl22anc |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> 0 <_ ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) |
105 |
54 53
|
addge01d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 0 <_ T <-> S <_ ( S + T ) ) ) |
106 |
61 105
|
mpbid |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> S <_ ( S + T ) ) |
107 |
54 57 55 106
|
lesub1dd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S - ( S x. T ) ) <_ ( ( S + T ) - ( S x. T ) ) ) |
108 |
107 86
|
breqtrrd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S - ( S x. T ) ) <_ ( 1 x. ( ( S + T ) - ( S x. T ) ) ) ) |
109 |
|
ledivmul2 |
|- ( ( ( S - ( S x. T ) ) e. RR /\ 1 e. RR /\ ( ( ( S + T ) - ( S x. T ) ) e. RR /\ 0 < ( ( S + T ) - ( S x. T ) ) ) ) -> ( ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) <_ 1 <-> ( S - ( S x. T ) ) <_ ( 1 x. ( ( S + T ) - ( S x. T ) ) ) ) ) |
110 |
93 59 58 76 109
|
syl112anc |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) <_ 1 <-> ( S - ( S x. T ) ) <_ ( 1 x. ( ( S + T ) - ( S x. T ) ) ) ) ) |
111 |
108 110
|
mpbird |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) <_ 1 ) |
112 |
|
elicc01 |
|- ( ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) e. ( 0 [,] 1 ) <-> ( ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) e. RR /\ 0 <_ ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) /\ ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) <_ 1 ) ) |
113 |
94 104 111 112
|
syl3anbrc |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) e. ( 0 [,] 1 ) ) |
114 |
1 53 5
|
sylancr |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 1 - T ) e. RR ) |
115 |
114
|
recnd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 1 - T ) e. CC ) |
116 |
98 115 85 77
|
div23d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S x. ( 1 - T ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( S / ( ( S + T ) - ( S x. T ) ) ) x. ( 1 - T ) ) ) |
117 |
|
subdi |
|- ( ( S e. CC /\ 1 e. CC /\ T e. CC ) -> ( S x. ( 1 - T ) ) = ( ( S x. 1 ) - ( S x. T ) ) ) |
118 |
26 117
|
mp3an2 |
|- ( ( S e. CC /\ T e. CC ) -> ( S x. ( 1 - T ) ) = ( ( S x. 1 ) - ( S x. T ) ) ) |
119 |
98 65 118
|
syl2anc |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S x. ( 1 - T ) ) = ( ( S x. 1 ) - ( S x. T ) ) ) |
120 |
99
|
oveq1d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S x. 1 ) - ( S x. T ) ) = ( S - ( S x. T ) ) ) |
121 |
119 120
|
eqtrd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S x. ( 1 - T ) ) = ( S - ( S x. T ) ) ) |
122 |
121
|
oveq1d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S x. ( 1 - T ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) |
123 |
56
|
recnd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( T - ( S x. T ) ) e. CC ) |
124 |
85 123 85 77
|
divsubdird |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( ( ( S + T ) - ( S x. T ) ) - ( T - ( S x. T ) ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( ( ( S + T ) - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) ) |
125 |
57
|
recnd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S + T ) e. CC ) |
126 |
55
|
recnd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S x. T ) e. CC ) |
127 |
125 65 126
|
nnncan2d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( ( S + T ) - ( S x. T ) ) - ( T - ( S x. T ) ) ) = ( ( S + T ) - T ) ) |
128 |
98 65
|
pncand |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S + T ) - T ) = S ) |
129 |
127 128
|
eqtrd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( ( S + T ) - ( S x. T ) ) - ( T - ( S x. T ) ) ) = S ) |
130 |
129
|
oveq1d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( ( ( S + T ) - ( S x. T ) ) - ( T - ( S x. T ) ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( S / ( ( S + T ) - ( S x. T ) ) ) ) |
131 |
85 77
|
dividd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( ( S + T ) - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) = 1 ) |
132 |
131
|
oveq1d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( ( ( S + T ) - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) = ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) ) |
133 |
124 130 132
|
3eqtr3d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S / ( ( S + T ) - ( S x. T ) ) ) = ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) ) |
134 |
133
|
oveq1d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S / ( ( S + T ) - ( S x. T ) ) ) x. ( 1 - T ) ) = ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - T ) ) ) |
135 |
116 122 134
|
3eqtr3d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - T ) ) ) |
136 |
1 54 13
|
sylancr |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 1 - S ) e. RR ) |
137 |
136
|
recnd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( 1 - S ) e. CC ) |
138 |
65 137 85 77
|
div23d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( T x. ( 1 - S ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( T / ( ( S + T ) - ( S x. T ) ) ) x. ( 1 - S ) ) ) |
139 |
|
subdi |
|- ( ( T e. CC /\ 1 e. CC /\ S e. CC ) -> ( T x. ( 1 - S ) ) = ( ( T x. 1 ) - ( T x. S ) ) ) |
140 |
26 139
|
mp3an2 |
|- ( ( T e. CC /\ S e. CC ) -> ( T x. ( 1 - S ) ) = ( ( T x. 1 ) - ( T x. S ) ) ) |
141 |
65 98 140
|
syl2anc |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( T x. ( 1 - S ) ) = ( ( T x. 1 ) - ( T x. S ) ) ) |
142 |
65
|
mulid1d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( T x. 1 ) = T ) |
143 |
65 98
|
mulcomd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( T x. S ) = ( S x. T ) ) |
144 |
142 143
|
oveq12d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( T x. 1 ) - ( T x. S ) ) = ( T - ( S x. T ) ) ) |
145 |
141 144
|
eqtrd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( T x. ( 1 - S ) ) = ( T - ( S x. T ) ) ) |
146 |
145
|
oveq1d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( T x. ( 1 - S ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) |
147 |
93
|
recnd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S - ( S x. T ) ) e. CC ) |
148 |
85 147 85 77
|
divsubdird |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( ( ( S + T ) - ( S x. T ) ) - ( S - ( S x. T ) ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( ( ( S + T ) - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) ) |
149 |
125 98 126
|
nnncan2d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( ( S + T ) - ( S x. T ) ) - ( S - ( S x. T ) ) ) = ( ( S + T ) - S ) ) |
150 |
98 65
|
pncan2d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S + T ) - S ) = T ) |
151 |
149 150
|
eqtrd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( ( S + T ) - ( S x. T ) ) - ( S - ( S x. T ) ) ) = T ) |
152 |
151
|
oveq1d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( ( ( S + T ) - ( S x. T ) ) - ( S - ( S x. T ) ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( T / ( ( S + T ) - ( S x. T ) ) ) ) |
153 |
131
|
oveq1d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( ( ( S + T ) - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) = ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) ) |
154 |
148 152 153
|
3eqtr3d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( T / ( ( S + T ) - ( S x. T ) ) ) = ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) ) |
155 |
154
|
oveq1d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( T / ( ( S + T ) - ( S x. T ) ) ) x. ( 1 - S ) ) = ( ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - S ) ) ) |
156 |
138 146 155
|
3eqtr3d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - S ) ) ) |
157 |
98 65
|
mulcomd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( S x. T ) = ( T x. S ) ) |
158 |
157
|
oveq1d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S x. T ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( T x. S ) / ( ( S + T ) - ( S x. T ) ) ) ) |
159 |
98 65 85 77
|
div23d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S x. T ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( S / ( ( S + T ) - ( S x. T ) ) ) x. T ) ) |
160 |
133
|
oveq1d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S / ( ( S + T ) - ( S x. T ) ) ) x. T ) = ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. T ) ) |
161 |
159 160
|
eqtrd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( S x. T ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. T ) ) |
162 |
65 98 85 77
|
div23d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( T x. S ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( T / ( ( S + T ) - ( S x. T ) ) ) x. S ) ) |
163 |
154
|
oveq1d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( T / ( ( S + T ) - ( S x. T ) ) ) x. S ) = ( ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. S ) ) |
164 |
162 163
|
eqtrd |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( T x. S ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. S ) ) |
165 |
158 161 164
|
3eqtr3d |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. T ) = ( ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. S ) ) |
166 |
|
oveq2 |
|- ( r = ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) -> ( 1 - r ) = ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) ) |
167 |
166
|
oveq1d |
|- ( r = ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) -> ( ( 1 - r ) x. ( 1 - T ) ) = ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - T ) ) ) |
168 |
167
|
eqeq2d |
|- ( r = ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) -> ( p = ( ( 1 - r ) x. ( 1 - T ) ) <-> p = ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - T ) ) ) ) |
169 |
|
eqeq1 |
|- ( r = ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) -> ( r = ( ( 1 - p ) x. ( 1 - S ) ) <-> ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( 1 - p ) x. ( 1 - S ) ) ) ) |
170 |
166
|
oveq1d |
|- ( r = ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) -> ( ( 1 - r ) x. T ) = ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. T ) ) |
171 |
170
|
eqeq1d |
|- ( r = ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) -> ( ( ( 1 - r ) x. T ) = ( ( 1 - p ) x. S ) <-> ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. T ) = ( ( 1 - p ) x. S ) ) ) |
172 |
168 169 171
|
3anbi123d |
|- ( r = ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) -> ( ( p = ( ( 1 - r ) x. ( 1 - T ) ) /\ r = ( ( 1 - p ) x. ( 1 - S ) ) /\ ( ( 1 - r ) x. T ) = ( ( 1 - p ) x. S ) ) <-> ( p = ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - T ) ) /\ ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( 1 - p ) x. ( 1 - S ) ) /\ ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. T ) = ( ( 1 - p ) x. S ) ) ) ) |
173 |
|
eqeq1 |
|- ( p = ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) -> ( p = ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - T ) ) <-> ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - T ) ) ) ) |
174 |
|
oveq2 |
|- ( p = ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) -> ( 1 - p ) = ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) ) |
175 |
174
|
oveq1d |
|- ( p = ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) -> ( ( 1 - p ) x. ( 1 - S ) ) = ( ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - S ) ) ) |
176 |
175
|
eqeq2d |
|- ( p = ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) -> ( ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( 1 - p ) x. ( 1 - S ) ) <-> ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - S ) ) ) ) |
177 |
174
|
oveq1d |
|- ( p = ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) -> ( ( 1 - p ) x. S ) = ( ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. S ) ) |
178 |
177
|
eqeq2d |
|- ( p = ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) -> ( ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. T ) = ( ( 1 - p ) x. S ) <-> ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. T ) = ( ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. S ) ) ) |
179 |
173 176 178
|
3anbi123d |
|- ( p = ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) -> ( ( p = ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - T ) ) /\ ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( 1 - p ) x. ( 1 - S ) ) /\ ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. T ) = ( ( 1 - p ) x. S ) ) <-> ( ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - T ) ) /\ ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - S ) ) /\ ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. T ) = ( ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. S ) ) ) ) |
180 |
172 179
|
rspc2ev |
|- ( ( ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) e. ( 0 [,] 1 ) /\ ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) e. ( 0 [,] 1 ) /\ ( ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - T ) ) /\ ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) = ( ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. ( 1 - S ) ) /\ ( ( 1 - ( ( T - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. T ) = ( ( 1 - ( ( S - ( S x. T ) ) / ( ( S + T ) - ( S x. T ) ) ) ) x. S ) ) ) -> E. r e. ( 0 [,] 1 ) E. p e. ( 0 [,] 1 ) ( p = ( ( 1 - r ) x. ( 1 - T ) ) /\ r = ( ( 1 - p ) x. ( 1 - S ) ) /\ ( ( 1 - r ) x. T ) = ( ( 1 - p ) x. S ) ) ) |
181 |
92 113 135 156 165 180
|
syl113anc |
|- ( ( S =/= 0 /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) ) -> E. r e. ( 0 [,] 1 ) E. p e. ( 0 [,] 1 ) ( p = ( ( 1 - r ) x. ( 1 - T ) ) /\ r = ( ( 1 - p ) x. ( 1 - S ) ) /\ ( ( 1 - r ) x. T ) = ( ( 1 - p ) x. S ) ) ) |
182 |
181
|
ex |
|- ( S =/= 0 -> ( ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) -> E. r e. ( 0 [,] 1 ) E. p e. ( 0 [,] 1 ) ( p = ( ( 1 - r ) x. ( 1 - T ) ) /\ r = ( ( 1 - p ) x. ( 1 - S ) ) /\ ( ( 1 - r ) x. T ) = ( ( 1 - p ) x. S ) ) ) ) |
183 |
52 182
|
pm2.61ine |
|- ( ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) -> E. r e. ( 0 [,] 1 ) E. p e. ( 0 [,] 1 ) ( p = ( ( 1 - r ) x. ( 1 - T ) ) /\ r = ( ( 1 - p ) x. ( 1 - S ) ) /\ ( ( 1 - r ) x. T ) = ( ( 1 - p ) x. S ) ) ) |