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