Step |
Hyp |
Ref |
Expression |
1 |
|
siii.1 |
|- X = ( BaseSet ` U ) |
2 |
|
siii.6 |
|- N = ( normCV ` U ) |
3 |
|
siii.7 |
|- P = ( .iOLD ` U ) |
4 |
|
siii.9 |
|- U e. CPreHilOLD |
5 |
|
siii.a |
|- A e. X |
6 |
|
siii.b |
|- B e. X |
7 |
|
sii1.3 |
|- M = ( -v ` U ) |
8 |
|
sii1.4 |
|- S = ( .sOLD ` U ) |
9 |
|
sii1.c |
|- C e. CC |
10 |
|
sii1.r |
|- ( C x. ( A P B ) ) e. RR |
11 |
|
sii1.z |
|- 0 <_ ( C x. ( A P B ) ) |
12 |
4
|
phnvi |
|- U e. NrmCVec |
13 |
9
|
cjcli |
|- ( * ` C ) e. CC |
14 |
1 8
|
nvscl |
|- ( ( U e. NrmCVec /\ ( * ` C ) e. CC /\ B e. X ) -> ( ( * ` C ) S B ) e. X ) |
15 |
12 13 6 14
|
mp3an |
|- ( ( * ` C ) S B ) e. X |
16 |
1 7
|
nvmcl |
|- ( ( U e. NrmCVec /\ A e. X /\ ( ( * ` C ) S B ) e. X ) -> ( A M ( ( * ` C ) S B ) ) e. X ) |
17 |
12 5 15 16
|
mp3an |
|- ( A M ( ( * ` C ) S B ) ) e. X |
18 |
1 2 12 17
|
nvcli |
|- ( N ` ( A M ( ( * ` C ) S B ) ) ) e. RR |
19 |
18
|
sqge0i |
|- 0 <_ ( ( N ` ( A M ( ( * ` C ) S B ) ) ) ^ 2 ) |
20 |
17 5 15
|
3pm3.2i |
|- ( ( A M ( ( * ` C ) S B ) ) e. X /\ A e. X /\ ( ( * ` C ) S B ) e. X ) |
21 |
1 7 3
|
dipsubdi |
|- ( ( U e. CPreHilOLD /\ ( ( A M ( ( * ` C ) S B ) ) e. X /\ A e. X /\ ( ( * ` C ) S B ) e. X ) ) -> ( ( A M ( ( * ` C ) S B ) ) P ( A M ( ( * ` C ) S B ) ) ) = ( ( ( A M ( ( * ` C ) S B ) ) P A ) - ( ( A M ( ( * ` C ) S B ) ) P ( ( * ` C ) S B ) ) ) ) |
22 |
4 20 21
|
mp2an |
|- ( ( A M ( ( * ` C ) S B ) ) P ( A M ( ( * ` C ) S B ) ) ) = ( ( ( A M ( ( * ` C ) S B ) ) P A ) - ( ( A M ( ( * ` C ) S B ) ) P ( ( * ` C ) S B ) ) ) |
23 |
1 2 3
|
ipidsq |
|- ( ( U e. NrmCVec /\ ( A M ( ( * ` C ) S B ) ) e. X ) -> ( ( A M ( ( * ` C ) S B ) ) P ( A M ( ( * ` C ) S B ) ) ) = ( ( N ` ( A M ( ( * ` C ) S B ) ) ) ^ 2 ) ) |
24 |
12 17 23
|
mp2an |
|- ( ( A M ( ( * ` C ) S B ) ) P ( A M ( ( * ` C ) S B ) ) ) = ( ( N ` ( A M ( ( * ` C ) S B ) ) ) ^ 2 ) |
25 |
13 6 15
|
3pm3.2i |
|- ( ( * ` C ) e. CC /\ B e. X /\ ( ( * ` C ) S B ) e. X ) |
26 |
1 8 3
|
dipass |
|- ( ( U e. CPreHilOLD /\ ( ( * ` C ) e. CC /\ B e. X /\ ( ( * ` C ) S B ) e. X ) ) -> ( ( ( * ` C ) S B ) P ( ( * ` C ) S B ) ) = ( ( * ` C ) x. ( B P ( ( * ` C ) S B ) ) ) ) |
27 |
4 25 26
|
mp2an |
|- ( ( ( * ` C ) S B ) P ( ( * ` C ) S B ) ) = ( ( * ` C ) x. ( B P ( ( * ` C ) S B ) ) ) |
28 |
6 9 6
|
3pm3.2i |
|- ( B e. X /\ C e. CC /\ B e. X ) |
29 |
1 8 3
|
dipassr2 |
|- ( ( U e. CPreHilOLD /\ ( B e. X /\ C e. CC /\ B e. X ) ) -> ( B P ( ( * ` C ) S B ) ) = ( C x. ( B P B ) ) ) |
30 |
4 28 29
|
mp2an |
|- ( B P ( ( * ` C ) S B ) ) = ( C x. ( B P B ) ) |
31 |
1 2 3
|
ipidsq |
|- ( ( U e. NrmCVec /\ B e. X ) -> ( B P B ) = ( ( N ` B ) ^ 2 ) ) |
32 |
12 6 31
|
mp2an |
|- ( B P B ) = ( ( N ` B ) ^ 2 ) |
33 |
32
|
oveq2i |
|- ( C x. ( B P B ) ) = ( C x. ( ( N ` B ) ^ 2 ) ) |
34 |
30 33
|
eqtri |
|- ( B P ( ( * ` C ) S B ) ) = ( C x. ( ( N ` B ) ^ 2 ) ) |
35 |
34
|
oveq2i |
|- ( ( * ` C ) x. ( B P ( ( * ` C ) S B ) ) ) = ( ( * ` C ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) |
36 |
27 35
|
eqtri |
|- ( ( ( * ` C ) S B ) P ( ( * ` C ) S B ) ) = ( ( * ` C ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) |
37 |
36
|
oveq2i |
|- ( ( C x. ( A P B ) ) - ( ( ( * ` C ) S B ) P ( ( * ` C ) S B ) ) ) = ( ( C x. ( A P B ) ) - ( ( * ` C ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) ) |
38 |
37
|
oveq2i |
|- ( ( ( ( N ` A ) ^ 2 ) - ( ( * ` C ) x. ( B P A ) ) ) - ( ( C x. ( A P B ) ) - ( ( ( * ` C ) S B ) P ( ( * ` C ) S B ) ) ) ) = ( ( ( ( N ` A ) ^ 2 ) - ( ( * ` C ) x. ( B P A ) ) ) - ( ( C x. ( A P B ) ) - ( ( * ` C ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) ) ) |
39 |
1 2 12 5
|
nvcli |
|- ( N ` A ) e. RR |
40 |
39
|
recni |
|- ( N ` A ) e. CC |
41 |
40
|
sqcli |
|- ( ( N ` A ) ^ 2 ) e. CC |
42 |
1 3
|
dipcl |
|- ( ( U e. NrmCVec /\ B e. X /\ A e. X ) -> ( B P A ) e. CC ) |
43 |
12 6 5 42
|
mp3an |
|- ( B P A ) e. CC |
44 |
13 43
|
mulcli |
|- ( ( * ` C ) x. ( B P A ) ) e. CC |
45 |
10
|
recni |
|- ( C x. ( A P B ) ) e. CC |
46 |
1 2 12 6
|
nvcli |
|- ( N ` B ) e. RR |
47 |
46
|
recni |
|- ( N ` B ) e. CC |
48 |
47
|
sqcli |
|- ( ( N ` B ) ^ 2 ) e. CC |
49 |
9 48
|
mulcli |
|- ( C x. ( ( N ` B ) ^ 2 ) ) e. CC |
50 |
13 49
|
mulcli |
|- ( ( * ` C ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) e. CC |
51 |
|
sub4 |
|- ( ( ( ( ( N ` A ) ^ 2 ) e. CC /\ ( ( * ` C ) x. ( B P A ) ) e. CC ) /\ ( ( C x. ( A P B ) ) e. CC /\ ( ( * ` C ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) e. CC ) ) -> ( ( ( ( N ` A ) ^ 2 ) - ( ( * ` C ) x. ( B P A ) ) ) - ( ( C x. ( A P B ) ) - ( ( * ` C ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) ) ) = ( ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) - ( ( ( * ` C ) x. ( B P A ) ) - ( ( * ` C ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) ) ) ) |
52 |
41 44 45 50 51
|
mp4an |
|- ( ( ( ( N ` A ) ^ 2 ) - ( ( * ` C ) x. ( B P A ) ) ) - ( ( C x. ( A P B ) ) - ( ( * ` C ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) ) ) = ( ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) - ( ( ( * ` C ) x. ( B P A ) ) - ( ( * ` C ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) ) ) |
53 |
38 52
|
eqtri |
|- ( ( ( ( N ` A ) ^ 2 ) - ( ( * ` C ) x. ( B P A ) ) ) - ( ( C x. ( A P B ) ) - ( ( ( * ` C ) S B ) P ( ( * ` C ) S B ) ) ) ) = ( ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) - ( ( ( * ` C ) x. ( B P A ) ) - ( ( * ` C ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) ) ) |
54 |
5 15 5
|
3pm3.2i |
|- ( A e. X /\ ( ( * ` C ) S B ) e. X /\ A e. X ) |
55 |
1 7 3
|
dipsubdir |
|- ( ( U e. CPreHilOLD /\ ( A e. X /\ ( ( * ` C ) S B ) e. X /\ A e. X ) ) -> ( ( A M ( ( * ` C ) S B ) ) P A ) = ( ( A P A ) - ( ( ( * ` C ) S B ) P A ) ) ) |
56 |
4 54 55
|
mp2an |
|- ( ( A M ( ( * ` C ) S B ) ) P A ) = ( ( A P A ) - ( ( ( * ` C ) S B ) P A ) ) |
57 |
1 2 3
|
ipidsq |
|- ( ( U e. NrmCVec /\ A e. X ) -> ( A P A ) = ( ( N ` A ) ^ 2 ) ) |
58 |
12 5 57
|
mp2an |
|- ( A P A ) = ( ( N ` A ) ^ 2 ) |
59 |
13 6 5
|
3pm3.2i |
|- ( ( * ` C ) e. CC /\ B e. X /\ A e. X ) |
60 |
1 8 3
|
dipass |
|- ( ( U e. CPreHilOLD /\ ( ( * ` C ) e. CC /\ B e. X /\ A e. X ) ) -> ( ( ( * ` C ) S B ) P A ) = ( ( * ` C ) x. ( B P A ) ) ) |
61 |
4 59 60
|
mp2an |
|- ( ( ( * ` C ) S B ) P A ) = ( ( * ` C ) x. ( B P A ) ) |
62 |
58 61
|
oveq12i |
|- ( ( A P A ) - ( ( ( * ` C ) S B ) P A ) ) = ( ( ( N ` A ) ^ 2 ) - ( ( * ` C ) x. ( B P A ) ) ) |
63 |
56 62
|
eqtri |
|- ( ( A M ( ( * ` C ) S B ) ) P A ) = ( ( ( N ` A ) ^ 2 ) - ( ( * ` C ) x. ( B P A ) ) ) |
64 |
5 15 15
|
3pm3.2i |
|- ( A e. X /\ ( ( * ` C ) S B ) e. X /\ ( ( * ` C ) S B ) e. X ) |
65 |
1 7 3
|
dipsubdir |
|- ( ( U e. CPreHilOLD /\ ( A e. X /\ ( ( * ` C ) S B ) e. X /\ ( ( * ` C ) S B ) e. X ) ) -> ( ( A M ( ( * ` C ) S B ) ) P ( ( * ` C ) S B ) ) = ( ( A P ( ( * ` C ) S B ) ) - ( ( ( * ` C ) S B ) P ( ( * ` C ) S B ) ) ) ) |
66 |
4 64 65
|
mp2an |
|- ( ( A M ( ( * ` C ) S B ) ) P ( ( * ` C ) S B ) ) = ( ( A P ( ( * ` C ) S B ) ) - ( ( ( * ` C ) S B ) P ( ( * ` C ) S B ) ) ) |
67 |
5 9 6
|
3pm3.2i |
|- ( A e. X /\ C e. CC /\ B e. X ) |
68 |
1 8 3
|
dipassr2 |
|- ( ( U e. CPreHilOLD /\ ( A e. X /\ C e. CC /\ B e. X ) ) -> ( A P ( ( * ` C ) S B ) ) = ( C x. ( A P B ) ) ) |
69 |
4 67 68
|
mp2an |
|- ( A P ( ( * ` C ) S B ) ) = ( C x. ( A P B ) ) |
70 |
69
|
oveq1i |
|- ( ( A P ( ( * ` C ) S B ) ) - ( ( ( * ` C ) S B ) P ( ( * ` C ) S B ) ) ) = ( ( C x. ( A P B ) ) - ( ( ( * ` C ) S B ) P ( ( * ` C ) S B ) ) ) |
71 |
66 70
|
eqtri |
|- ( ( A M ( ( * ` C ) S B ) ) P ( ( * ` C ) S B ) ) = ( ( C x. ( A P B ) ) - ( ( ( * ` C ) S B ) P ( ( * ` C ) S B ) ) ) |
72 |
63 71
|
oveq12i |
|- ( ( ( A M ( ( * ` C ) S B ) ) P A ) - ( ( A M ( ( * ` C ) S B ) ) P ( ( * ` C ) S B ) ) ) = ( ( ( ( N ` A ) ^ 2 ) - ( ( * ` C ) x. ( B P A ) ) ) - ( ( C x. ( A P B ) ) - ( ( ( * ` C ) S B ) P ( ( * ` C ) S B ) ) ) ) |
73 |
13 43 49
|
subdii |
|- ( ( * ` C ) x. ( ( B P A ) - ( C x. ( ( N ` B ) ^ 2 ) ) ) ) = ( ( ( * ` C ) x. ( B P A ) ) - ( ( * ` C ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) ) |
74 |
73
|
oveq2i |
|- ( ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) - ( ( * ` C ) x. ( ( B P A ) - ( C x. ( ( N ` B ) ^ 2 ) ) ) ) ) = ( ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) - ( ( ( * ` C ) x. ( B P A ) ) - ( ( * ` C ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) ) ) |
75 |
53 72 74
|
3eqtr4i |
|- ( ( ( A M ( ( * ` C ) S B ) ) P A ) - ( ( A M ( ( * ` C ) S B ) ) P ( ( * ` C ) S B ) ) ) = ( ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) - ( ( * ` C ) x. ( ( B P A ) - ( C x. ( ( N ` B ) ^ 2 ) ) ) ) ) |
76 |
22 24 75
|
3eqtr3i |
|- ( ( N ` ( A M ( ( * ` C ) S B ) ) ) ^ 2 ) = ( ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) - ( ( * ` C ) x. ( ( B P A ) - ( C x. ( ( N ` B ) ^ 2 ) ) ) ) ) |
77 |
19 76
|
breqtri |
|- 0 <_ ( ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) - ( ( * ` C ) x. ( ( B P A ) - ( C x. ( ( N ` B ) ^ 2 ) ) ) ) ) |
78 |
43 49
|
subeq0i |
|- ( ( ( B P A ) - ( C x. ( ( N ` B ) ^ 2 ) ) ) = 0 <-> ( B P A ) = ( C x. ( ( N ` B ) ^ 2 ) ) ) |
79 |
|
oveq2 |
|- ( ( ( B P A ) - ( C x. ( ( N ` B ) ^ 2 ) ) ) = 0 -> ( ( * ` C ) x. ( ( B P A ) - ( C x. ( ( N ` B ) ^ 2 ) ) ) ) = ( ( * ` C ) x. 0 ) ) |
80 |
13
|
mul01i |
|- ( ( * ` C ) x. 0 ) = 0 |
81 |
79 80
|
eqtrdi |
|- ( ( ( B P A ) - ( C x. ( ( N ` B ) ^ 2 ) ) ) = 0 -> ( ( * ` C ) x. ( ( B P A ) - ( C x. ( ( N ` B ) ^ 2 ) ) ) ) = 0 ) |
82 |
78 81
|
sylbir |
|- ( ( B P A ) = ( C x. ( ( N ` B ) ^ 2 ) ) -> ( ( * ` C ) x. ( ( B P A ) - ( C x. ( ( N ` B ) ^ 2 ) ) ) ) = 0 ) |
83 |
82
|
oveq2d |
|- ( ( B P A ) = ( C x. ( ( N ` B ) ^ 2 ) ) -> ( ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) - ( ( * ` C ) x. ( ( B P A ) - ( C x. ( ( N ` B ) ^ 2 ) ) ) ) ) = ( ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) - 0 ) ) |
84 |
39
|
resqcli |
|- ( ( N ` A ) ^ 2 ) e. RR |
85 |
84
|
recni |
|- ( ( N ` A ) ^ 2 ) e. CC |
86 |
85 45
|
subcli |
|- ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) e. CC |
87 |
86
|
subid1i |
|- ( ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) - 0 ) = ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) |
88 |
83 87
|
eqtrdi |
|- ( ( B P A ) = ( C x. ( ( N ` B ) ^ 2 ) ) -> ( ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) - ( ( * ` C ) x. ( ( B P A ) - ( C x. ( ( N ` B ) ^ 2 ) ) ) ) ) = ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) ) |
89 |
77 88
|
breqtrid |
|- ( ( B P A ) = ( C x. ( ( N ` B ) ^ 2 ) ) -> 0 <_ ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) ) |
90 |
84 10
|
subge0i |
|- ( 0 <_ ( ( ( N ` A ) ^ 2 ) - ( C x. ( A P B ) ) ) <-> ( C x. ( A P B ) ) <_ ( ( N ` A ) ^ 2 ) ) |
91 |
89 90
|
sylib |
|- ( ( B P A ) = ( C x. ( ( N ` B ) ^ 2 ) ) -> ( C x. ( A P B ) ) <_ ( ( N ` A ) ^ 2 ) ) |
92 |
46
|
resqcli |
|- ( ( N ` B ) ^ 2 ) e. RR |
93 |
46
|
sqge0i |
|- 0 <_ ( ( N ` B ) ^ 2 ) |
94 |
92 93
|
pm3.2i |
|- ( ( ( N ` B ) ^ 2 ) e. RR /\ 0 <_ ( ( N ` B ) ^ 2 ) ) |
95 |
10 84 94
|
3pm3.2i |
|- ( ( C x. ( A P B ) ) e. RR /\ ( ( N ` A ) ^ 2 ) e. RR /\ ( ( ( N ` B ) ^ 2 ) e. RR /\ 0 <_ ( ( N ` B ) ^ 2 ) ) ) |
96 |
|
lemul1a |
|- ( ( ( ( C x. ( A P B ) ) e. RR /\ ( ( N ` A ) ^ 2 ) e. RR /\ ( ( ( N ` B ) ^ 2 ) e. RR /\ 0 <_ ( ( N ` B ) ^ 2 ) ) ) /\ ( C x. ( A P B ) ) <_ ( ( N ` A ) ^ 2 ) ) -> ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) <_ ( ( ( N ` A ) ^ 2 ) x. ( ( N ` B ) ^ 2 ) ) ) |
97 |
95 96
|
mpan |
|- ( ( C x. ( A P B ) ) <_ ( ( N ` A ) ^ 2 ) -> ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) <_ ( ( ( N ` A ) ^ 2 ) x. ( ( N ` B ) ^ 2 ) ) ) |
98 |
91 97
|
syl |
|- ( ( B P A ) = ( C x. ( ( N ` B ) ^ 2 ) ) -> ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) <_ ( ( ( N ` A ) ^ 2 ) x. ( ( N ` B ) ^ 2 ) ) ) |
99 |
40 47
|
sqmuli |
|- ( ( ( N ` A ) x. ( N ` B ) ) ^ 2 ) = ( ( ( N ` A ) ^ 2 ) x. ( ( N ` B ) ^ 2 ) ) |
100 |
98 99
|
breqtrrdi |
|- ( ( B P A ) = ( C x. ( ( N ` B ) ^ 2 ) ) -> ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) <_ ( ( ( N ` A ) x. ( N ` B ) ) ^ 2 ) ) |
101 |
10 92
|
mulge0i |
|- ( ( 0 <_ ( C x. ( A P B ) ) /\ 0 <_ ( ( N ` B ) ^ 2 ) ) -> 0 <_ ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) ) |
102 |
11 93 101
|
mp2an |
|- 0 <_ ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) |
103 |
39 46
|
remulcli |
|- ( ( N ` A ) x. ( N ` B ) ) e. RR |
104 |
103
|
sqge0i |
|- 0 <_ ( ( ( N ` A ) x. ( N ` B ) ) ^ 2 ) |
105 |
10 92
|
remulcli |
|- ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) e. RR |
106 |
103
|
resqcli |
|- ( ( ( N ` A ) x. ( N ` B ) ) ^ 2 ) e. RR |
107 |
105 106
|
sqrtlei |
|- ( ( 0 <_ ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) /\ 0 <_ ( ( ( N ` A ) x. ( N ` B ) ) ^ 2 ) ) -> ( ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) <_ ( ( ( N ` A ) x. ( N ` B ) ) ^ 2 ) <-> ( sqrt ` ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) ) <_ ( sqrt ` ( ( ( N ` A ) x. ( N ` B ) ) ^ 2 ) ) ) ) |
108 |
102 104 107
|
mp2an |
|- ( ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) <_ ( ( ( N ` A ) x. ( N ` B ) ) ^ 2 ) <-> ( sqrt ` ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) ) <_ ( sqrt ` ( ( ( N ` A ) x. ( N ` B ) ) ^ 2 ) ) ) |
109 |
100 108
|
sylib |
|- ( ( B P A ) = ( C x. ( ( N ` B ) ^ 2 ) ) -> ( sqrt ` ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) ) <_ ( sqrt ` ( ( ( N ` A ) x. ( N ` B ) ) ^ 2 ) ) ) |
110 |
1 3
|
dipcl |
|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) e. CC ) |
111 |
12 5 6 110
|
mp3an |
|- ( A P B ) e. CC |
112 |
9 111
|
mulcomi |
|- ( C x. ( A P B ) ) = ( ( A P B ) x. C ) |
113 |
112
|
oveq1i |
|- ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) = ( ( ( A P B ) x. C ) x. ( ( N ` B ) ^ 2 ) ) |
114 |
92
|
recni |
|- ( ( N ` B ) ^ 2 ) e. CC |
115 |
111 9 114
|
mulassi |
|- ( ( ( A P B ) x. C ) x. ( ( N ` B ) ^ 2 ) ) = ( ( A P B ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) |
116 |
113 115
|
eqtri |
|- ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) = ( ( A P B ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) |
117 |
116
|
fveq2i |
|- ( sqrt ` ( ( C x. ( A P B ) ) x. ( ( N ` B ) ^ 2 ) ) ) = ( sqrt ` ( ( A P B ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) ) |
118 |
1 2
|
nvge0 |
|- ( ( U e. NrmCVec /\ A e. X ) -> 0 <_ ( N ` A ) ) |
119 |
12 5 118
|
mp2an |
|- 0 <_ ( N ` A ) |
120 |
1 2
|
nvge0 |
|- ( ( U e. NrmCVec /\ B e. X ) -> 0 <_ ( N ` B ) ) |
121 |
12 6 120
|
mp2an |
|- 0 <_ ( N ` B ) |
122 |
39 46
|
mulge0i |
|- ( ( 0 <_ ( N ` A ) /\ 0 <_ ( N ` B ) ) -> 0 <_ ( ( N ` A ) x. ( N ` B ) ) ) |
123 |
119 121 122
|
mp2an |
|- 0 <_ ( ( N ` A ) x. ( N ` B ) ) |
124 |
103
|
sqrtsqi |
|- ( 0 <_ ( ( N ` A ) x. ( N ` B ) ) -> ( sqrt ` ( ( ( N ` A ) x. ( N ` B ) ) ^ 2 ) ) = ( ( N ` A ) x. ( N ` B ) ) ) |
125 |
123 124
|
ax-mp |
|- ( sqrt ` ( ( ( N ` A ) x. ( N ` B ) ) ^ 2 ) ) = ( ( N ` A ) x. ( N ` B ) ) |
126 |
109 117 125
|
3brtr3g |
|- ( ( B P A ) = ( C x. ( ( N ` B ) ^ 2 ) ) -> ( sqrt ` ( ( A P B ) x. ( C x. ( ( N ` B ) ^ 2 ) ) ) ) <_ ( ( N ` A ) x. ( N ` B ) ) ) |