Step |
Hyp |
Ref |
Expression |
1 |
|
itsclquadb.q |
|- Q = ( ( A ^ 2 ) + ( B ^ 2 ) ) |
2 |
|
itsclquadb.t |
|- T = -u ( 2 x. ( B x. C ) ) |
3 |
|
itsclquadb.u |
|- U = ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) |
4 |
|
simpl1 |
|- ( ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) /\ x e. RR ) -> ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) ) |
5 |
|
simp2 |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> R e. RR+ ) |
6 |
5
|
adantr |
|- ( ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) /\ x e. RR ) -> R e. RR+ ) |
7 |
|
simp3 |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> Y e. RR ) |
8 |
7
|
anim1ci |
|- ( ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) /\ x e. RR ) -> ( x e. RR /\ Y e. RR ) ) |
9 |
1 2 3
|
itscnhlc0yqe |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ ( x e. RR /\ Y e. RR ) ) -> ( ( ( ( x ^ 2 ) + ( Y ^ 2 ) ) = ( R ^ 2 ) /\ ( ( A x. x ) + ( B x. Y ) ) = C ) -> ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 ) ) |
10 |
4 6 8 9
|
syl3anc |
|- ( ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) /\ x e. RR ) -> ( ( ( ( x ^ 2 ) + ( Y ^ 2 ) ) = ( R ^ 2 ) /\ ( ( A x. x ) + ( B x. Y ) ) = C ) -> ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 ) ) |
11 |
10
|
rexlimdva |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( E. x e. RR ( ( ( x ^ 2 ) + ( Y ^ 2 ) ) = ( R ^ 2 ) /\ ( ( A x. x ) + ( B x. Y ) ) = C ) -> ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 ) ) |
12 |
|
simp3 |
|- ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) -> C e. RR ) |
13 |
12
|
3ad2ant1 |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> C e. RR ) |
14 |
|
simp2 |
|- ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) -> B e. RR ) |
15 |
14
|
3ad2ant1 |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> B e. RR ) |
16 |
15 7
|
remulcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( B x. Y ) e. RR ) |
17 |
13 16
|
resubcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( C - ( B x. Y ) ) e. RR ) |
18 |
|
simp11l |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> A e. RR ) |
19 |
|
simp11r |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> A =/= 0 ) |
20 |
17 18 19
|
redivcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C - ( B x. Y ) ) / A ) e. RR ) |
21 |
20
|
adantr |
|- ( ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) /\ ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 ) -> ( ( C - ( B x. Y ) ) / A ) e. RR ) |
22 |
|
oveq1 |
|- ( x = ( ( C - ( B x. Y ) ) / A ) -> ( x ^ 2 ) = ( ( ( C - ( B x. Y ) ) / A ) ^ 2 ) ) |
23 |
22
|
oveq1d |
|- ( x = ( ( C - ( B x. Y ) ) / A ) -> ( ( x ^ 2 ) + ( Y ^ 2 ) ) = ( ( ( ( C - ( B x. Y ) ) / A ) ^ 2 ) + ( Y ^ 2 ) ) ) |
24 |
23
|
eqeq1d |
|- ( x = ( ( C - ( B x. Y ) ) / A ) -> ( ( ( x ^ 2 ) + ( Y ^ 2 ) ) = ( R ^ 2 ) <-> ( ( ( ( C - ( B x. Y ) ) / A ) ^ 2 ) + ( Y ^ 2 ) ) = ( R ^ 2 ) ) ) |
25 |
|
oveq2 |
|- ( x = ( ( C - ( B x. Y ) ) / A ) -> ( A x. x ) = ( A x. ( ( C - ( B x. Y ) ) / A ) ) ) |
26 |
25
|
oveq1d |
|- ( x = ( ( C - ( B x. Y ) ) / A ) -> ( ( A x. x ) + ( B x. Y ) ) = ( ( A x. ( ( C - ( B x. Y ) ) / A ) ) + ( B x. Y ) ) ) |
27 |
26
|
eqeq1d |
|- ( x = ( ( C - ( B x. Y ) ) / A ) -> ( ( ( A x. x ) + ( B x. Y ) ) = C <-> ( ( A x. ( ( C - ( B x. Y ) ) / A ) ) + ( B x. Y ) ) = C ) ) |
28 |
24 27
|
anbi12d |
|- ( x = ( ( C - ( B x. Y ) ) / A ) -> ( ( ( ( x ^ 2 ) + ( Y ^ 2 ) ) = ( R ^ 2 ) /\ ( ( A x. x ) + ( B x. Y ) ) = C ) <-> ( ( ( ( ( C - ( B x. Y ) ) / A ) ^ 2 ) + ( Y ^ 2 ) ) = ( R ^ 2 ) /\ ( ( A x. ( ( C - ( B x. Y ) ) / A ) ) + ( B x. Y ) ) = C ) ) ) |
29 |
28
|
adantl |
|- ( ( ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) /\ ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 ) /\ x = ( ( C - ( B x. Y ) ) / A ) ) -> ( ( ( ( x ^ 2 ) + ( Y ^ 2 ) ) = ( R ^ 2 ) /\ ( ( A x. x ) + ( B x. Y ) ) = C ) <-> ( ( ( ( ( C - ( B x. Y ) ) / A ) ^ 2 ) + ( Y ^ 2 ) ) = ( R ^ 2 ) /\ ( ( A x. ( ( C - ( B x. Y ) ) / A ) ) + ( B x. Y ) ) = C ) ) ) |
30 |
17
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( C - ( B x. Y ) ) e. CC ) |
31 |
18
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> A e. CC ) |
32 |
30 31 19
|
sqdivd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( C - ( B x. Y ) ) / A ) ^ 2 ) = ( ( ( C - ( B x. Y ) ) ^ 2 ) / ( A ^ 2 ) ) ) |
33 |
13
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> C e. CC ) |
34 |
16
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( B x. Y ) e. CC ) |
35 |
|
binom2sub |
|- ( ( C e. CC /\ ( B x. Y ) e. CC ) -> ( ( C - ( B x. Y ) ) ^ 2 ) = ( ( ( C ^ 2 ) - ( 2 x. ( C x. ( B x. Y ) ) ) ) + ( ( B x. Y ) ^ 2 ) ) ) |
36 |
33 34 35
|
syl2anc |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C - ( B x. Y ) ) ^ 2 ) = ( ( ( C ^ 2 ) - ( 2 x. ( C x. ( B x. Y ) ) ) ) + ( ( B x. Y ) ^ 2 ) ) ) |
37 |
13
|
resqcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( C ^ 2 ) e. RR ) |
38 |
37
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( C ^ 2 ) e. CC ) |
39 |
|
2re |
|- 2 e. RR |
40 |
39
|
a1i |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> 2 e. RR ) |
41 |
13 16
|
remulcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( C x. ( B x. Y ) ) e. RR ) |
42 |
40 41
|
remulcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( 2 x. ( C x. ( B x. Y ) ) ) e. RR ) |
43 |
42
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( 2 x. ( C x. ( B x. Y ) ) ) e. CC ) |
44 |
38 43
|
negsubd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C ^ 2 ) + -u ( 2 x. ( C x. ( B x. Y ) ) ) ) = ( ( C ^ 2 ) - ( 2 x. ( C x. ( B x. Y ) ) ) ) ) |
45 |
15
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> B e. CC ) |
46 |
7
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> Y e. CC ) |
47 |
33 45 46
|
mulassd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C x. B ) x. Y ) = ( C x. ( B x. Y ) ) ) |
48 |
47
|
eqcomd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( C x. ( B x. Y ) ) = ( ( C x. B ) x. Y ) ) |
49 |
48
|
oveq2d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( 2 x. ( C x. ( B x. Y ) ) ) = ( 2 x. ( ( C x. B ) x. Y ) ) ) |
50 |
|
2cnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> 2 e. CC ) |
51 |
13 15
|
remulcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( C x. B ) e. RR ) |
52 |
51
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( C x. B ) e. CC ) |
53 |
50 52 46
|
mulassd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( 2 x. ( C x. B ) ) x. Y ) = ( 2 x. ( ( C x. B ) x. Y ) ) ) |
54 |
53
|
eqcomd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( 2 x. ( ( C x. B ) x. Y ) ) = ( ( 2 x. ( C x. B ) ) x. Y ) ) |
55 |
33 45
|
mulcomd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( C x. B ) = ( B x. C ) ) |
56 |
55
|
oveq2d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( 2 x. ( C x. B ) ) = ( 2 x. ( B x. C ) ) ) |
57 |
56
|
oveq1d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( 2 x. ( C x. B ) ) x. Y ) = ( ( 2 x. ( B x. C ) ) x. Y ) ) |
58 |
49 54 57
|
3eqtrd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( 2 x. ( C x. ( B x. Y ) ) ) = ( ( 2 x. ( B x. C ) ) x. Y ) ) |
59 |
58
|
negeqd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> -u ( 2 x. ( C x. ( B x. Y ) ) ) = -u ( ( 2 x. ( B x. C ) ) x. Y ) ) |
60 |
59
|
oveq2d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C ^ 2 ) + -u ( 2 x. ( C x. ( B x. Y ) ) ) ) = ( ( C ^ 2 ) + -u ( ( 2 x. ( B x. C ) ) x. Y ) ) ) |
61 |
44 60
|
eqtr3d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C ^ 2 ) - ( 2 x. ( C x. ( B x. Y ) ) ) ) = ( ( C ^ 2 ) + -u ( ( 2 x. ( B x. C ) ) x. Y ) ) ) |
62 |
45 46
|
sqmuld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( B x. Y ) ^ 2 ) = ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) |
63 |
61 62
|
oveq12d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( C ^ 2 ) - ( 2 x. ( C x. ( B x. Y ) ) ) ) + ( ( B x. Y ) ^ 2 ) ) = ( ( ( C ^ 2 ) + -u ( ( 2 x. ( B x. C ) ) x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) ) |
64 |
15 13
|
remulcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( B x. C ) e. RR ) |
65 |
40 64
|
remulcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( 2 x. ( B x. C ) ) e. RR ) |
66 |
65
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( 2 x. ( B x. C ) ) e. CC ) |
67 |
66 46
|
mulneg1d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( -u ( 2 x. ( B x. C ) ) x. Y ) = -u ( ( 2 x. ( B x. C ) ) x. Y ) ) |
68 |
2
|
eqcomi |
|- -u ( 2 x. ( B x. C ) ) = T |
69 |
68
|
oveq1i |
|- ( -u ( 2 x. ( B x. C ) ) x. Y ) = ( T x. Y ) |
70 |
69
|
a1i |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( -u ( 2 x. ( B x. C ) ) x. Y ) = ( T x. Y ) ) |
71 |
67 70
|
eqtr3d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> -u ( ( 2 x. ( B x. C ) ) x. Y ) = ( T x. Y ) ) |
72 |
71
|
oveq2d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C ^ 2 ) + -u ( ( 2 x. ( B x. C ) ) x. Y ) ) = ( ( C ^ 2 ) + ( T x. Y ) ) ) |
73 |
72
|
oveq1d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( C ^ 2 ) + -u ( ( 2 x. ( B x. C ) ) x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) = ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) ) |
74 |
36 63 73
|
3eqtrd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C - ( B x. Y ) ) ^ 2 ) = ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) ) |
75 |
74
|
oveq1d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( C - ( B x. Y ) ) ^ 2 ) / ( A ^ 2 ) ) = ( ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) / ( A ^ 2 ) ) ) |
76 |
32 75
|
eqtrd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( C - ( B x. Y ) ) / A ) ^ 2 ) = ( ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) / ( A ^ 2 ) ) ) |
77 |
|
resqcl |
|- ( Y e. RR -> ( Y ^ 2 ) e. RR ) |
78 |
77
|
recnd |
|- ( Y e. RR -> ( Y ^ 2 ) e. CC ) |
79 |
78
|
3ad2ant3 |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( Y ^ 2 ) e. CC ) |
80 |
18
|
resqcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( A ^ 2 ) e. RR ) |
81 |
80
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( A ^ 2 ) e. CC ) |
82 |
|
recn |
|- ( A e. RR -> A e. CC ) |
83 |
|
sqne0 |
|- ( A e. CC -> ( ( A ^ 2 ) =/= 0 <-> A =/= 0 ) ) |
84 |
82 83
|
syl |
|- ( A e. RR -> ( ( A ^ 2 ) =/= 0 <-> A =/= 0 ) ) |
85 |
84
|
biimpar |
|- ( ( A e. RR /\ A =/= 0 ) -> ( A ^ 2 ) =/= 0 ) |
86 |
85
|
3ad2ant1 |
|- ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) -> ( A ^ 2 ) =/= 0 ) |
87 |
86
|
3ad2ant1 |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( A ^ 2 ) =/= 0 ) |
88 |
79 81 87
|
divcan2d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( A ^ 2 ) x. ( ( Y ^ 2 ) / ( A ^ 2 ) ) ) = ( Y ^ 2 ) ) |
89 |
88
|
eqcomd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( Y ^ 2 ) = ( ( A ^ 2 ) x. ( ( Y ^ 2 ) / ( A ^ 2 ) ) ) ) |
90 |
76 89
|
oveq12d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( ( C - ( B x. Y ) ) / A ) ^ 2 ) + ( Y ^ 2 ) ) = ( ( ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) / ( A ^ 2 ) ) + ( ( A ^ 2 ) x. ( ( Y ^ 2 ) / ( A ^ 2 ) ) ) ) ) |
91 |
81 79 81 87
|
divassd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( A ^ 2 ) x. ( Y ^ 2 ) ) / ( A ^ 2 ) ) = ( ( A ^ 2 ) x. ( ( Y ^ 2 ) / ( A ^ 2 ) ) ) ) |
92 |
91
|
eqcomd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( A ^ 2 ) x. ( ( Y ^ 2 ) / ( A ^ 2 ) ) ) = ( ( ( A ^ 2 ) x. ( Y ^ 2 ) ) / ( A ^ 2 ) ) ) |
93 |
92
|
oveq2d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) / ( A ^ 2 ) ) + ( ( A ^ 2 ) x. ( ( Y ^ 2 ) / ( A ^ 2 ) ) ) ) = ( ( ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) / ( A ^ 2 ) ) + ( ( ( A ^ 2 ) x. ( Y ^ 2 ) ) / ( A ^ 2 ) ) ) ) |
94 |
65
|
renegcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> -u ( 2 x. ( B x. C ) ) e. RR ) |
95 |
2 94
|
eqeltrid |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> T e. RR ) |
96 |
95 7
|
remulcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( T x. Y ) e. RR ) |
97 |
37 96
|
readdcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C ^ 2 ) + ( T x. Y ) ) e. RR ) |
98 |
15
|
resqcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( B ^ 2 ) e. RR ) |
99 |
7
|
resqcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( Y ^ 2 ) e. RR ) |
100 |
98 99
|
remulcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( B ^ 2 ) x. ( Y ^ 2 ) ) e. RR ) |
101 |
97 100
|
readdcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) e. RR ) |
102 |
101
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) e. CC ) |
103 |
80 99
|
remulcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( A ^ 2 ) x. ( Y ^ 2 ) ) e. RR ) |
104 |
103
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( A ^ 2 ) x. ( Y ^ 2 ) ) e. CC ) |
105 |
102 104 81 87
|
divdird |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) + ( ( A ^ 2 ) x. ( Y ^ 2 ) ) ) / ( A ^ 2 ) ) = ( ( ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) / ( A ^ 2 ) ) + ( ( ( A ^ 2 ) x. ( Y ^ 2 ) ) / ( A ^ 2 ) ) ) ) |
106 |
105
|
eqcomd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) / ( A ^ 2 ) ) + ( ( ( A ^ 2 ) x. ( Y ^ 2 ) ) / ( A ^ 2 ) ) ) = ( ( ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) + ( ( A ^ 2 ) x. ( Y ^ 2 ) ) ) / ( A ^ 2 ) ) ) |
107 |
90 93 106
|
3eqtrd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( ( C - ( B x. Y ) ) / A ) ^ 2 ) + ( Y ^ 2 ) ) = ( ( ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) + ( ( A ^ 2 ) x. ( Y ^ 2 ) ) ) / ( A ^ 2 ) ) ) |
108 |
107
|
adantr |
|- ( ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) /\ ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 ) -> ( ( ( ( C - ( B x. Y ) ) / A ) ^ 2 ) + ( Y ^ 2 ) ) = ( ( ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) + ( ( A ^ 2 ) x. ( Y ^ 2 ) ) ) / ( A ^ 2 ) ) ) |
109 |
97
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C ^ 2 ) + ( T x. Y ) ) e. CC ) |
110 |
100
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( B ^ 2 ) x. ( Y ^ 2 ) ) e. CC ) |
111 |
109 110 104
|
addassd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) + ( ( A ^ 2 ) x. ( Y ^ 2 ) ) ) = ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( ( B ^ 2 ) x. ( Y ^ 2 ) ) + ( ( A ^ 2 ) x. ( Y ^ 2 ) ) ) ) ) |
112 |
98
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( B ^ 2 ) e. CC ) |
113 |
99
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( Y ^ 2 ) e. CC ) |
114 |
112 81 113
|
adddird |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( B ^ 2 ) + ( A ^ 2 ) ) x. ( Y ^ 2 ) ) = ( ( ( B ^ 2 ) x. ( Y ^ 2 ) ) + ( ( A ^ 2 ) x. ( Y ^ 2 ) ) ) ) |
115 |
112 81
|
addcomd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( B ^ 2 ) + ( A ^ 2 ) ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) ) |
116 |
115
|
oveq1d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( B ^ 2 ) + ( A ^ 2 ) ) x. ( Y ^ 2 ) ) = ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) ) |
117 |
114 116
|
eqtr3d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( B ^ 2 ) x. ( Y ^ 2 ) ) + ( ( A ^ 2 ) x. ( Y ^ 2 ) ) ) = ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) ) |
118 |
117
|
oveq2d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( ( B ^ 2 ) x. ( Y ^ 2 ) ) + ( ( A ^ 2 ) x. ( Y ^ 2 ) ) ) ) = ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) ) ) |
119 |
96
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( T x. Y ) e. CC ) |
120 |
80 98
|
readdcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( A ^ 2 ) + ( B ^ 2 ) ) e. RR ) |
121 |
120 99
|
remulcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) e. RR ) |
122 |
121
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) e. CC ) |
123 |
38 119 122
|
addassd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) ) = ( ( C ^ 2 ) + ( ( T x. Y ) + ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) ) ) ) |
124 |
119 122
|
addcomd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( T x. Y ) + ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) ) = ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) ) |
125 |
124
|
oveq2d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C ^ 2 ) + ( ( T x. Y ) + ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) ) ) = ( ( C ^ 2 ) + ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) ) ) |
126 |
123 125
|
eqtrd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) ) = ( ( C ^ 2 ) + ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) ) ) |
127 |
111 118 126
|
3eqtrd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) + ( ( A ^ 2 ) x. ( Y ^ 2 ) ) ) = ( ( C ^ 2 ) + ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) ) ) |
128 |
127
|
adantr |
|- ( ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) /\ ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 ) -> ( ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) + ( ( A ^ 2 ) x. ( Y ^ 2 ) ) ) = ( ( C ^ 2 ) + ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) ) ) |
129 |
128
|
oveq1d |
|- ( ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) /\ ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 ) -> ( ( ( ( ( C ^ 2 ) + ( T x. Y ) ) + ( ( B ^ 2 ) x. ( Y ^ 2 ) ) ) + ( ( A ^ 2 ) x. ( Y ^ 2 ) ) ) / ( A ^ 2 ) ) = ( ( ( C ^ 2 ) + ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) ) / ( A ^ 2 ) ) ) |
130 |
1
|
oveq1i |
|- ( Q x. ( Y ^ 2 ) ) = ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) |
131 |
3
|
oveq2i |
|- ( ( T x. Y ) + U ) = ( ( T x. Y ) + ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) |
132 |
130 131
|
oveq12i |
|- ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( ( T x. Y ) + ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) ) |
133 |
|
rpre |
|- ( R e. RR+ -> R e. RR ) |
134 |
133
|
resqcld |
|- ( R e. RR+ -> ( R ^ 2 ) e. RR ) |
135 |
134
|
3ad2ant2 |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( R ^ 2 ) e. RR ) |
136 |
80 135
|
remulcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( A ^ 2 ) x. ( R ^ 2 ) ) e. RR ) |
137 |
37 136
|
resubcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) e. RR ) |
138 |
137
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) e. CC ) |
139 |
122 119 138
|
addassd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) + ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) = ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( ( T x. Y ) + ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) ) ) |
140 |
132 139
|
eqtr4id |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = ( ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) + ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) ) |
141 |
140
|
eqeq1d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 <-> ( ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) + ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) = 0 ) ) |
142 |
121 96
|
readdcld |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) e. RR ) |
143 |
142
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) e. CC ) |
144 |
|
addeq0 |
|- ( ( ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) e. CC /\ ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) e. CC ) -> ( ( ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) + ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) = 0 <-> ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) = -u ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) ) |
145 |
143 138 144
|
syl2anc |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) + ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) = 0 <-> ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) = -u ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) ) |
146 |
141 145
|
bitrd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 <-> ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) = -u ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) ) |
147 |
|
oveq2 |
|- ( ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) = -u ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) -> ( ( C ^ 2 ) + ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) ) = ( ( C ^ 2 ) + -u ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) ) |
148 |
147
|
oveq1d |
|- ( ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) = -u ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) -> ( ( ( C ^ 2 ) + ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) ) / ( A ^ 2 ) ) = ( ( ( C ^ 2 ) + -u ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) / ( A ^ 2 ) ) ) |
149 |
38 138
|
negsubd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C ^ 2 ) + -u ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) = ( ( C ^ 2 ) - ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) ) |
150 |
136
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( A ^ 2 ) x. ( R ^ 2 ) ) e. CC ) |
151 |
38 150
|
nncand |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C ^ 2 ) - ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) = ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) |
152 |
149 151
|
eqtrd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C ^ 2 ) + -u ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) = ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) |
153 |
152
|
oveq1d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( C ^ 2 ) + -u ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) / ( A ^ 2 ) ) = ( ( ( A ^ 2 ) x. ( R ^ 2 ) ) / ( A ^ 2 ) ) ) |
154 |
135
|
recnd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( R ^ 2 ) e. CC ) |
155 |
154 81 87
|
divcan3d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( A ^ 2 ) x. ( R ^ 2 ) ) / ( A ^ 2 ) ) = ( R ^ 2 ) ) |
156 |
153 155
|
eqtrd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( C ^ 2 ) + -u ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) / ( A ^ 2 ) ) = ( R ^ 2 ) ) |
157 |
148 156
|
sylan9eqr |
|- ( ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) /\ ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) = -u ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) ) -> ( ( ( C ^ 2 ) + ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) ) / ( A ^ 2 ) ) = ( R ^ 2 ) ) |
158 |
157
|
ex |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) = -u ( ( C ^ 2 ) - ( ( A ^ 2 ) x. ( R ^ 2 ) ) ) -> ( ( ( C ^ 2 ) + ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) ) / ( A ^ 2 ) ) = ( R ^ 2 ) ) ) |
159 |
146 158
|
sylbid |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 -> ( ( ( C ^ 2 ) + ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) ) / ( A ^ 2 ) ) = ( R ^ 2 ) ) ) |
160 |
159
|
imp |
|- ( ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) /\ ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 ) -> ( ( ( C ^ 2 ) + ( ( ( ( A ^ 2 ) + ( B ^ 2 ) ) x. ( Y ^ 2 ) ) + ( T x. Y ) ) ) / ( A ^ 2 ) ) = ( R ^ 2 ) ) |
161 |
108 129 160
|
3eqtrd |
|- ( ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) /\ ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 ) -> ( ( ( ( C - ( B x. Y ) ) / A ) ^ 2 ) + ( Y ^ 2 ) ) = ( R ^ 2 ) ) |
162 |
30 31 19
|
divcan2d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( A x. ( ( C - ( B x. Y ) ) / A ) ) = ( C - ( B x. Y ) ) ) |
163 |
162
|
oveq1d |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( A x. ( ( C - ( B x. Y ) ) / A ) ) + ( B x. Y ) ) = ( ( C - ( B x. Y ) ) + ( B x. Y ) ) ) |
164 |
33 34
|
npcand |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( C - ( B x. Y ) ) + ( B x. Y ) ) = C ) |
165 |
163 164
|
eqtrd |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( A x. ( ( C - ( B x. Y ) ) / A ) ) + ( B x. Y ) ) = C ) |
166 |
165
|
adantr |
|- ( ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) /\ ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 ) -> ( ( A x. ( ( C - ( B x. Y ) ) / A ) ) + ( B x. Y ) ) = C ) |
167 |
161 166
|
jca |
|- ( ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) /\ ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 ) -> ( ( ( ( ( C - ( B x. Y ) ) / A ) ^ 2 ) + ( Y ^ 2 ) ) = ( R ^ 2 ) /\ ( ( A x. ( ( C - ( B x. Y ) ) / A ) ) + ( B x. Y ) ) = C ) ) |
168 |
21 29 167
|
rspcedvd |
|- ( ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) /\ ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 ) -> E. x e. RR ( ( ( x ^ 2 ) + ( Y ^ 2 ) ) = ( R ^ 2 ) /\ ( ( A x. x ) + ( B x. Y ) ) = C ) ) |
169 |
168
|
ex |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 -> E. x e. RR ( ( ( x ^ 2 ) + ( Y ^ 2 ) ) = ( R ^ 2 ) /\ ( ( A x. x ) + ( B x. Y ) ) = C ) ) ) |
170 |
11 169
|
impbid |
|- ( ( ( ( A e. RR /\ A =/= 0 ) /\ B e. RR /\ C e. RR ) /\ R e. RR+ /\ Y e. RR ) -> ( E. x e. RR ( ( ( x ^ 2 ) + ( Y ^ 2 ) ) = ( R ^ 2 ) /\ ( ( A x. x ) + ( B x. Y ) ) = C ) <-> ( ( Q x. ( Y ^ 2 ) ) + ( ( T x. Y ) + U ) ) = 0 ) ) |