Step |
Hyp |
Ref |
Expression |
1 |
|
normlem1.1 |
|- S e. CC |
2 |
|
normlem1.2 |
|- F e. ~H |
3 |
|
normlem1.3 |
|- G e. ~H |
4 |
|
normlem2.4 |
|- B = -u ( ( ( * ` S ) x. ( F .ih G ) ) + ( S x. ( G .ih F ) ) ) |
5 |
|
normlem3.5 |
|- A = ( G .ih G ) |
6 |
|
normlem3.6 |
|- C = ( F .ih F ) |
7 |
|
normlem6.7 |
|- ( abs ` S ) = 1 |
8 |
|
hiidrcl |
|- ( G e. ~H -> ( G .ih G ) e. RR ) |
9 |
3 8
|
ax-mp |
|- ( G .ih G ) e. RR |
10 |
5 9
|
eqeltri |
|- A e. RR |
11 |
10
|
a1i |
|- ( T. -> A e. RR ) |
12 |
1 2 3 4
|
normlem2 |
|- B e. RR |
13 |
12
|
a1i |
|- ( T. -> B e. RR ) |
14 |
|
hiidrcl |
|- ( F e. ~H -> ( F .ih F ) e. RR ) |
15 |
2 14
|
ax-mp |
|- ( F .ih F ) e. RR |
16 |
6 15
|
eqeltri |
|- C e. RR |
17 |
16
|
a1i |
|- ( T. -> C e. RR ) |
18 |
|
oveq1 |
|- ( x = if ( x e. RR , x , 0 ) -> ( x ^ 2 ) = ( if ( x e. RR , x , 0 ) ^ 2 ) ) |
19 |
18
|
oveq2d |
|- ( x = if ( x e. RR , x , 0 ) -> ( A x. ( x ^ 2 ) ) = ( A x. ( if ( x e. RR , x , 0 ) ^ 2 ) ) ) |
20 |
|
oveq2 |
|- ( x = if ( x e. RR , x , 0 ) -> ( B x. x ) = ( B x. if ( x e. RR , x , 0 ) ) ) |
21 |
19 20
|
oveq12d |
|- ( x = if ( x e. RR , x , 0 ) -> ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) = ( ( A x. ( if ( x e. RR , x , 0 ) ^ 2 ) ) + ( B x. if ( x e. RR , x , 0 ) ) ) ) |
22 |
21
|
oveq1d |
|- ( x = if ( x e. RR , x , 0 ) -> ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) = ( ( ( A x. ( if ( x e. RR , x , 0 ) ^ 2 ) ) + ( B x. if ( x e. RR , x , 0 ) ) ) + C ) ) |
23 |
22
|
breq2d |
|- ( x = if ( x e. RR , x , 0 ) -> ( 0 <_ ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) <-> 0 <_ ( ( ( A x. ( if ( x e. RR , x , 0 ) ^ 2 ) ) + ( B x. if ( x e. RR , x , 0 ) ) ) + C ) ) ) |
24 |
|
0re |
|- 0 e. RR |
25 |
24
|
elimel |
|- if ( x e. RR , x , 0 ) e. RR |
26 |
1 2 3 4 5 6 25 7
|
normlem5 |
|- 0 <_ ( ( ( A x. ( if ( x e. RR , x , 0 ) ^ 2 ) ) + ( B x. if ( x e. RR , x , 0 ) ) ) + C ) |
27 |
23 26
|
dedth |
|- ( x e. RR -> 0 <_ ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) ) |
28 |
27
|
adantl |
|- ( ( T. /\ x e. RR ) -> 0 <_ ( ( ( A x. ( x ^ 2 ) ) + ( B x. x ) ) + C ) ) |
29 |
11 13 17 28
|
discr |
|- ( T. -> ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) <_ 0 ) |
30 |
29
|
mptru |
|- ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) <_ 0 |
31 |
12
|
resqcli |
|- ( B ^ 2 ) e. RR |
32 |
|
4re |
|- 4 e. RR |
33 |
10 16
|
remulcli |
|- ( A x. C ) e. RR |
34 |
32 33
|
remulcli |
|- ( 4 x. ( A x. C ) ) e. RR |
35 |
31 34 24
|
lesubadd2i |
|- ( ( ( B ^ 2 ) - ( 4 x. ( A x. C ) ) ) <_ 0 <-> ( B ^ 2 ) <_ ( ( 4 x. ( A x. C ) ) + 0 ) ) |
36 |
30 35
|
mpbi |
|- ( B ^ 2 ) <_ ( ( 4 x. ( A x. C ) ) + 0 ) |
37 |
34
|
recni |
|- ( 4 x. ( A x. C ) ) e. CC |
38 |
37
|
addid1i |
|- ( ( 4 x. ( A x. C ) ) + 0 ) = ( 4 x. ( A x. C ) ) |
39 |
36 38
|
breqtri |
|- ( B ^ 2 ) <_ ( 4 x. ( A x. C ) ) |
40 |
12
|
sqge0i |
|- 0 <_ ( B ^ 2 ) |
41 |
|
4pos |
|- 0 < 4 |
42 |
24 32 41
|
ltleii |
|- 0 <_ 4 |
43 |
|
hiidge0 |
|- ( G e. ~H -> 0 <_ ( G .ih G ) ) |
44 |
3 43
|
ax-mp |
|- 0 <_ ( G .ih G ) |
45 |
44 5
|
breqtrri |
|- 0 <_ A |
46 |
|
hiidge0 |
|- ( F e. ~H -> 0 <_ ( F .ih F ) ) |
47 |
2 46
|
ax-mp |
|- 0 <_ ( F .ih F ) |
48 |
47 6
|
breqtrri |
|- 0 <_ C |
49 |
10 16
|
mulge0i |
|- ( ( 0 <_ A /\ 0 <_ C ) -> 0 <_ ( A x. C ) ) |
50 |
45 48 49
|
mp2an |
|- 0 <_ ( A x. C ) |
51 |
32 33
|
mulge0i |
|- ( ( 0 <_ 4 /\ 0 <_ ( A x. C ) ) -> 0 <_ ( 4 x. ( A x. C ) ) ) |
52 |
42 50 51
|
mp2an |
|- 0 <_ ( 4 x. ( A x. C ) ) |
53 |
31 34
|
sqrtlei |
|- ( ( 0 <_ ( B ^ 2 ) /\ 0 <_ ( 4 x. ( A x. C ) ) ) -> ( ( B ^ 2 ) <_ ( 4 x. ( A x. C ) ) <-> ( sqrt ` ( B ^ 2 ) ) <_ ( sqrt ` ( 4 x. ( A x. C ) ) ) ) ) |
54 |
40 52 53
|
mp2an |
|- ( ( B ^ 2 ) <_ ( 4 x. ( A x. C ) ) <-> ( sqrt ` ( B ^ 2 ) ) <_ ( sqrt ` ( 4 x. ( A x. C ) ) ) ) |
55 |
39 54
|
mpbi |
|- ( sqrt ` ( B ^ 2 ) ) <_ ( sqrt ` ( 4 x. ( A x. C ) ) ) |
56 |
12
|
absrei |
|- ( abs ` B ) = ( sqrt ` ( B ^ 2 ) ) |
57 |
32 33 42 50
|
sqrtmulii |
|- ( sqrt ` ( 4 x. ( A x. C ) ) ) = ( ( sqrt ` 4 ) x. ( sqrt ` ( A x. C ) ) ) |
58 |
|
sqrt4 |
|- ( sqrt ` 4 ) = 2 |
59 |
10 16 45 48
|
sqrtmulii |
|- ( sqrt ` ( A x. C ) ) = ( ( sqrt ` A ) x. ( sqrt ` C ) ) |
60 |
58 59
|
oveq12i |
|- ( ( sqrt ` 4 ) x. ( sqrt ` ( A x. C ) ) ) = ( 2 x. ( ( sqrt ` A ) x. ( sqrt ` C ) ) ) |
61 |
57 60
|
eqtr2i |
|- ( 2 x. ( ( sqrt ` A ) x. ( sqrt ` C ) ) ) = ( sqrt ` ( 4 x. ( A x. C ) ) ) |
62 |
55 56 61
|
3brtr4i |
|- ( abs ` B ) <_ ( 2 x. ( ( sqrt ` A ) x. ( sqrt ` C ) ) ) |