Step |
Hyp |
Ref |
Expression |
1 |
|
itsclc0.i |
|- I = { 1 , 2 } |
2 |
|
itsclc0.e |
|- E = ( RR^ ` I ) |
3 |
|
itsclc0.p |
|- P = ( RR ^m I ) |
4 |
|
itsclc0.s |
|- S = ( Sphere ` E ) |
5 |
|
itsclc0.0 |
|- .0. = ( I X. { 0 } ) |
6 |
|
itsclc0.q |
|- Q = ( ( A ^ 2 ) + ( B ^ 2 ) ) |
7 |
|
itsclc0.d |
|- D = ( ( ( R ^ 2 ) x. Q ) - ( C ^ 2 ) ) |
8 |
|
itsclc0.l |
|- L = { p e. P | ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C } |
9 |
|
rprege0 |
|- ( R e. RR+ -> ( R e. RR /\ 0 <_ R ) ) |
10 |
|
elrege0 |
|- ( R e. ( 0 [,) +oo ) <-> ( R e. RR /\ 0 <_ R ) ) |
11 |
9 10
|
sylibr |
|- ( R e. RR+ -> R e. ( 0 [,) +oo ) ) |
12 |
11
|
adantr |
|- ( ( R e. RR+ /\ 0 <_ D ) -> R e. ( 0 [,) +oo ) ) |
13 |
12
|
3ad2ant3 |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) /\ ( R e. RR+ /\ 0 <_ D ) ) -> R e. ( 0 [,) +oo ) ) |
14 |
|
eqid |
|- { p e. P | ( ( ( p ` 1 ) ^ 2 ) + ( ( p ` 2 ) ^ 2 ) ) = ( R ^ 2 ) } = { p e. P | ( ( ( p ` 1 ) ^ 2 ) + ( ( p ` 2 ) ^ 2 ) ) = ( R ^ 2 ) } |
15 |
1 2 3 4 5 14
|
2sphere0 |
|- ( R e. ( 0 [,) +oo ) -> ( .0. S R ) = { p e. P | ( ( ( p ` 1 ) ^ 2 ) + ( ( p ` 2 ) ^ 2 ) ) = ( R ^ 2 ) } ) |
16 |
15
|
eleq2d |
|- ( R e. ( 0 [,) +oo ) -> ( X e. ( .0. S R ) <-> X e. { p e. P | ( ( ( p ` 1 ) ^ 2 ) + ( ( p ` 2 ) ^ 2 ) ) = ( R ^ 2 ) } ) ) |
17 |
13 16
|
syl |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) /\ ( R e. RR+ /\ 0 <_ D ) ) -> ( X e. ( .0. S R ) <-> X e. { p e. P | ( ( ( p ` 1 ) ^ 2 ) + ( ( p ` 2 ) ^ 2 ) ) = ( R ^ 2 ) } ) ) |
18 |
|
fveq1 |
|- ( p = X -> ( p ` 1 ) = ( X ` 1 ) ) |
19 |
18
|
oveq2d |
|- ( p = X -> ( A x. ( p ` 1 ) ) = ( A x. ( X ` 1 ) ) ) |
20 |
|
fveq1 |
|- ( p = X -> ( p ` 2 ) = ( X ` 2 ) ) |
21 |
20
|
oveq2d |
|- ( p = X -> ( B x. ( p ` 2 ) ) = ( B x. ( X ` 2 ) ) ) |
22 |
19 21
|
oveq12d |
|- ( p = X -> ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = ( ( A x. ( X ` 1 ) ) + ( B x. ( X ` 2 ) ) ) ) |
23 |
22
|
eqeq1d |
|- ( p = X -> ( ( ( A x. ( p ` 1 ) ) + ( B x. ( p ` 2 ) ) ) = C <-> ( ( A x. ( X ` 1 ) ) + ( B x. ( X ` 2 ) ) ) = C ) ) |
24 |
23 8
|
elrab2 |
|- ( X e. L <-> ( X e. P /\ ( ( A x. ( X ` 1 ) ) + ( B x. ( X ` 2 ) ) ) = C ) ) |
25 |
24
|
a1i |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) /\ ( R e. RR+ /\ 0 <_ D ) ) -> ( X e. L <-> ( X e. P /\ ( ( A x. ( X ` 1 ) ) + ( B x. ( X ` 2 ) ) ) = C ) ) ) |
26 |
17 25
|
anbi12d |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) /\ ( R e. RR+ /\ 0 <_ D ) ) -> ( ( X e. ( .0. S R ) /\ X e. L ) <-> ( X e. { p e. P | ( ( ( p ` 1 ) ^ 2 ) + ( ( p ` 2 ) ^ 2 ) ) = ( R ^ 2 ) } /\ ( X e. P /\ ( ( A x. ( X ` 1 ) ) + ( B x. ( X ` 2 ) ) ) = C ) ) ) ) |
27 |
18
|
oveq1d |
|- ( p = X -> ( ( p ` 1 ) ^ 2 ) = ( ( X ` 1 ) ^ 2 ) ) |
28 |
20
|
oveq1d |
|- ( p = X -> ( ( p ` 2 ) ^ 2 ) = ( ( X ` 2 ) ^ 2 ) ) |
29 |
27 28
|
oveq12d |
|- ( p = X -> ( ( ( p ` 1 ) ^ 2 ) + ( ( p ` 2 ) ^ 2 ) ) = ( ( ( X ` 1 ) ^ 2 ) + ( ( X ` 2 ) ^ 2 ) ) ) |
30 |
29
|
eqeq1d |
|- ( p = X -> ( ( ( ( p ` 1 ) ^ 2 ) + ( ( p ` 2 ) ^ 2 ) ) = ( R ^ 2 ) <-> ( ( ( X ` 1 ) ^ 2 ) + ( ( X ` 2 ) ^ 2 ) ) = ( R ^ 2 ) ) ) |
31 |
30
|
elrab |
|- ( X e. { p e. P | ( ( ( p ` 1 ) ^ 2 ) + ( ( p ` 2 ) ^ 2 ) ) = ( R ^ 2 ) } <-> ( X e. P /\ ( ( ( X ` 1 ) ^ 2 ) + ( ( X ` 2 ) ^ 2 ) ) = ( R ^ 2 ) ) ) |
32 |
31
|
anbi1i |
|- ( ( X e. { p e. P | ( ( ( p ` 1 ) ^ 2 ) + ( ( p ` 2 ) ^ 2 ) ) = ( R ^ 2 ) } /\ ( X e. P /\ ( ( A x. ( X ` 1 ) ) + ( B x. ( X ` 2 ) ) ) = C ) ) <-> ( ( X e. P /\ ( ( ( X ` 1 ) ^ 2 ) + ( ( X ` 2 ) ^ 2 ) ) = ( R ^ 2 ) ) /\ ( X e. P /\ ( ( A x. ( X ` 1 ) ) + ( B x. ( X ` 2 ) ) ) = C ) ) ) |
33 |
|
anandi |
|- ( ( X e. P /\ ( ( ( ( X ` 1 ) ^ 2 ) + ( ( X ` 2 ) ^ 2 ) ) = ( R ^ 2 ) /\ ( ( A x. ( X ` 1 ) ) + ( B x. ( X ` 2 ) ) ) = C ) ) <-> ( ( X e. P /\ ( ( ( X ` 1 ) ^ 2 ) + ( ( X ` 2 ) ^ 2 ) ) = ( R ^ 2 ) ) /\ ( X e. P /\ ( ( A x. ( X ` 1 ) ) + ( B x. ( X ` 2 ) ) ) = C ) ) ) |
34 |
|
simpl1 |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) /\ ( R e. RR+ /\ 0 <_ D ) ) /\ X e. P ) -> ( A e. RR /\ B e. RR /\ C e. RR ) ) |
35 |
|
simpl2 |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) /\ ( R e. RR+ /\ 0 <_ D ) ) /\ X e. P ) -> ( A =/= 0 \/ B =/= 0 ) ) |
36 |
|
simpl3l |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) /\ ( R e. RR+ /\ 0 <_ D ) ) /\ X e. P ) -> R e. RR+ ) |
37 |
|
simpl3r |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) /\ ( R e. RR+ /\ 0 <_ D ) ) /\ X e. P ) -> 0 <_ D ) |
38 |
1 3
|
rrx2pxel |
|- ( X e. P -> ( X ` 1 ) e. RR ) |
39 |
1 3
|
rrx2pyel |
|- ( X e. P -> ( X ` 2 ) e. RR ) |
40 |
38 39
|
jca |
|- ( X e. P -> ( ( X ` 1 ) e. RR /\ ( X ` 2 ) e. RR ) ) |
41 |
40
|
adantl |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) /\ ( R e. RR+ /\ 0 <_ D ) ) /\ X e. P ) -> ( ( X ` 1 ) e. RR /\ ( X ` 2 ) e. RR ) ) |
42 |
36 37 41
|
jca31 |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) /\ ( R e. RR+ /\ 0 <_ D ) ) /\ X e. P ) -> ( ( R e. RR+ /\ 0 <_ D ) /\ ( ( X ` 1 ) e. RR /\ ( X ` 2 ) e. RR ) ) ) |
43 |
6 7
|
itsclc0xyqsolb |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) ) /\ ( ( R e. RR+ /\ 0 <_ D ) /\ ( ( X ` 1 ) e. RR /\ ( X ` 2 ) e. RR ) ) ) -> ( ( ( ( ( X ` 1 ) ^ 2 ) + ( ( X ` 2 ) ^ 2 ) ) = ( R ^ 2 ) /\ ( ( A x. ( X ` 1 ) ) + ( B x. ( X ` 2 ) ) ) = C ) <-> ( ( ( X ` 1 ) = ( ( ( A x. C ) + ( B x. ( sqrt ` D ) ) ) / Q ) /\ ( X ` 2 ) = ( ( ( B x. C ) - ( A x. ( sqrt ` D ) ) ) / Q ) ) \/ ( ( X ` 1 ) = ( ( ( A x. C ) - ( B x. ( sqrt ` D ) ) ) / Q ) /\ ( X ` 2 ) = ( ( ( B x. C ) + ( A x. ( sqrt ` D ) ) ) / Q ) ) ) ) ) |
44 |
34 35 42 43
|
syl21anc |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) /\ ( R e. RR+ /\ 0 <_ D ) ) /\ X e. P ) -> ( ( ( ( ( X ` 1 ) ^ 2 ) + ( ( X ` 2 ) ^ 2 ) ) = ( R ^ 2 ) /\ ( ( A x. ( X ` 1 ) ) + ( B x. ( X ` 2 ) ) ) = C ) <-> ( ( ( X ` 1 ) = ( ( ( A x. C ) + ( B x. ( sqrt ` D ) ) ) / Q ) /\ ( X ` 2 ) = ( ( ( B x. C ) - ( A x. ( sqrt ` D ) ) ) / Q ) ) \/ ( ( X ` 1 ) = ( ( ( A x. C ) - ( B x. ( sqrt ` D ) ) ) / Q ) /\ ( X ` 2 ) = ( ( ( B x. C ) + ( A x. ( sqrt ` D ) ) ) / Q ) ) ) ) ) |
45 |
44
|
pm5.32da |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) /\ ( R e. RR+ /\ 0 <_ D ) ) -> ( ( X e. P /\ ( ( ( ( X ` 1 ) ^ 2 ) + ( ( X ` 2 ) ^ 2 ) ) = ( R ^ 2 ) /\ ( ( A x. ( X ` 1 ) ) + ( B x. ( X ` 2 ) ) ) = C ) ) <-> ( X e. P /\ ( ( ( X ` 1 ) = ( ( ( A x. C ) + ( B x. ( sqrt ` D ) ) ) / Q ) /\ ( X ` 2 ) = ( ( ( B x. C ) - ( A x. ( sqrt ` D ) ) ) / Q ) ) \/ ( ( X ` 1 ) = ( ( ( A x. C ) - ( B x. ( sqrt ` D ) ) ) / Q ) /\ ( X ` 2 ) = ( ( ( B x. C ) + ( A x. ( sqrt ` D ) ) ) / Q ) ) ) ) ) ) |
46 |
33 45
|
bitr3id |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) /\ ( R e. RR+ /\ 0 <_ D ) ) -> ( ( ( X e. P /\ ( ( ( X ` 1 ) ^ 2 ) + ( ( X ` 2 ) ^ 2 ) ) = ( R ^ 2 ) ) /\ ( X e. P /\ ( ( A x. ( X ` 1 ) ) + ( B x. ( X ` 2 ) ) ) = C ) ) <-> ( X e. P /\ ( ( ( X ` 1 ) = ( ( ( A x. C ) + ( B x. ( sqrt ` D ) ) ) / Q ) /\ ( X ` 2 ) = ( ( ( B x. C ) - ( A x. ( sqrt ` D ) ) ) / Q ) ) \/ ( ( X ` 1 ) = ( ( ( A x. C ) - ( B x. ( sqrt ` D ) ) ) / Q ) /\ ( X ` 2 ) = ( ( ( B x. C ) + ( A x. ( sqrt ` D ) ) ) / Q ) ) ) ) ) ) |
47 |
32 46
|
syl5bb |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) /\ ( R e. RR+ /\ 0 <_ D ) ) -> ( ( X e. { p e. P | ( ( ( p ` 1 ) ^ 2 ) + ( ( p ` 2 ) ^ 2 ) ) = ( R ^ 2 ) } /\ ( X e. P /\ ( ( A x. ( X ` 1 ) ) + ( B x. ( X ` 2 ) ) ) = C ) ) <-> ( X e. P /\ ( ( ( X ` 1 ) = ( ( ( A x. C ) + ( B x. ( sqrt ` D ) ) ) / Q ) /\ ( X ` 2 ) = ( ( ( B x. C ) - ( A x. ( sqrt ` D ) ) ) / Q ) ) \/ ( ( X ` 1 ) = ( ( ( A x. C ) - ( B x. ( sqrt ` D ) ) ) / Q ) /\ ( X ` 2 ) = ( ( ( B x. C ) + ( A x. ( sqrt ` D ) ) ) / Q ) ) ) ) ) ) |
48 |
26 47
|
bitrd |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A =/= 0 \/ B =/= 0 ) /\ ( R e. RR+ /\ 0 <_ D ) ) -> ( ( X e. ( .0. S R ) /\ X e. L ) <-> ( X e. P /\ ( ( ( X ` 1 ) = ( ( ( A x. C ) + ( B x. ( sqrt ` D ) ) ) / Q ) /\ ( X ` 2 ) = ( ( ( B x. C ) - ( A x. ( sqrt ` D ) ) ) / Q ) ) \/ ( ( X ` 1 ) = ( ( ( A x. C ) - ( B x. ( sqrt ` D ) ) ) / Q ) /\ ( X ` 2 ) = ( ( ( B x. C ) + ( A x. ( sqrt ` D ) ) ) / Q ) ) ) ) ) ) |