Step |
Hyp |
Ref |
Expression |
1 |
|
hmopre |
⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ∈ ℝ ) |
2 |
1
|
adantlr |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ∈ ℝ ) |
3 |
1
|
recnd |
⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ∈ ℂ ) |
4 |
3
|
abscld |
⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( abs ‘ ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ) ∈ ℝ ) |
5 |
4
|
adantlr |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( abs ‘ ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ) ∈ ℝ ) |
6 |
|
idhmop |
⊢ Iop ∈ HrmOp |
7 |
|
hmopm |
⊢ ( ( ( normop ‘ 𝑇 ) ∈ ℝ ∧ Iop ∈ HrmOp ) → ( ( normop ‘ 𝑇 ) ·op Iop ) ∈ HrmOp ) |
8 |
6 7
|
mpan2 |
⊢ ( ( normop ‘ 𝑇 ) ∈ ℝ → ( ( normop ‘ 𝑇 ) ·op Iop ) ∈ HrmOp ) |
9 |
|
hmopre |
⊢ ( ( ( ( normop ‘ 𝑇 ) ·op Iop ) ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( ( ( ( normop ‘ 𝑇 ) ·op Iop ) ‘ 𝑥 ) ·ih 𝑥 ) ∈ ℝ ) |
10 |
8 9
|
sylan |
⊢ ( ( ( normop ‘ 𝑇 ) ∈ ℝ ∧ 𝑥 ∈ ℋ ) → ( ( ( ( normop ‘ 𝑇 ) ·op Iop ) ‘ 𝑥 ) ·ih 𝑥 ) ∈ ℝ ) |
11 |
10
|
adantll |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( ( ( normop ‘ 𝑇 ) ·op Iop ) ‘ 𝑥 ) ·ih 𝑥 ) ∈ ℝ ) |
12 |
1
|
leabsd |
⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ≤ ( abs ‘ ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ) ) |
13 |
12
|
adantlr |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ≤ ( abs ‘ ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ) ) |
14 |
|
hmopf |
⊢ ( 𝑇 ∈ HrmOp → 𝑇 : ℋ ⟶ ℋ ) |
15 |
|
ffvelrn |
⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑥 ∈ ℋ ) → ( 𝑇 ‘ 𝑥 ) ∈ ℋ ) |
16 |
|
normcl |
⊢ ( ( 𝑇 ‘ 𝑥 ) ∈ ℋ → ( normℎ ‘ ( 𝑇 ‘ 𝑥 ) ) ∈ ℝ ) |
17 |
15 16
|
syl |
⊢ ( ( 𝑇 : ℋ ⟶ ℋ ∧ 𝑥 ∈ ℋ ) → ( normℎ ‘ ( 𝑇 ‘ 𝑥 ) ) ∈ ℝ ) |
18 |
14 17
|
sylan |
⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( normℎ ‘ ( 𝑇 ‘ 𝑥 ) ) ∈ ℝ ) |
19 |
18
|
adantlr |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( normℎ ‘ ( 𝑇 ‘ 𝑥 ) ) ∈ ℝ ) |
20 |
|
normcl |
⊢ ( 𝑥 ∈ ℋ → ( normℎ ‘ 𝑥 ) ∈ ℝ ) |
21 |
20
|
adantl |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( normℎ ‘ 𝑥 ) ∈ ℝ ) |
22 |
19 21
|
remulcld |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( normℎ ‘ ( 𝑇 ‘ 𝑥 ) ) · ( normℎ ‘ 𝑥 ) ) ∈ ℝ ) |
23 |
14 15
|
sylan |
⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( 𝑇 ‘ 𝑥 ) ∈ ℋ ) |
24 |
|
bcs |
⊢ ( ( ( 𝑇 ‘ 𝑥 ) ∈ ℋ ∧ 𝑥 ∈ ℋ ) → ( abs ‘ ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ) ≤ ( ( normℎ ‘ ( 𝑇 ‘ 𝑥 ) ) · ( normℎ ‘ 𝑥 ) ) ) |
25 |
23 24
|
sylancom |
⊢ ( ( 𝑇 ∈ HrmOp ∧ 𝑥 ∈ ℋ ) → ( abs ‘ ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ) ≤ ( ( normℎ ‘ ( 𝑇 ‘ 𝑥 ) ) · ( normℎ ‘ 𝑥 ) ) ) |
26 |
25
|
adantlr |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( abs ‘ ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ) ≤ ( ( normℎ ‘ ( 𝑇 ‘ 𝑥 ) ) · ( normℎ ‘ 𝑥 ) ) ) |
27 |
|
remulcl |
⊢ ( ( ( normop ‘ 𝑇 ) ∈ ℝ ∧ ( normℎ ‘ 𝑥 ) ∈ ℝ ) → ( ( normop ‘ 𝑇 ) · ( normℎ ‘ 𝑥 ) ) ∈ ℝ ) |
28 |
20 27
|
sylan2 |
⊢ ( ( ( normop ‘ 𝑇 ) ∈ ℝ ∧ 𝑥 ∈ ℋ ) → ( ( normop ‘ 𝑇 ) · ( normℎ ‘ 𝑥 ) ) ∈ ℝ ) |
29 |
28
|
adantll |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( normop ‘ 𝑇 ) · ( normℎ ‘ 𝑥 ) ) ∈ ℝ ) |
30 |
|
normge0 |
⊢ ( 𝑥 ∈ ℋ → 0 ≤ ( normℎ ‘ 𝑥 ) ) |
31 |
20 30
|
jca |
⊢ ( 𝑥 ∈ ℋ → ( ( normℎ ‘ 𝑥 ) ∈ ℝ ∧ 0 ≤ ( normℎ ‘ 𝑥 ) ) ) |
32 |
31
|
adantl |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( normℎ ‘ 𝑥 ) ∈ ℝ ∧ 0 ≤ ( normℎ ‘ 𝑥 ) ) ) |
33 |
|
hmoplin |
⊢ ( 𝑇 ∈ HrmOp → 𝑇 ∈ LinOp ) |
34 |
|
elbdop2 |
⊢ ( 𝑇 ∈ BndLinOp ↔ ( 𝑇 ∈ LinOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ) |
35 |
34
|
biimpri |
⊢ ( ( 𝑇 ∈ LinOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) → 𝑇 ∈ BndLinOp ) |
36 |
33 35
|
sylan |
⊢ ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) → 𝑇 ∈ BndLinOp ) |
37 |
|
nmbdoplb |
⊢ ( ( 𝑇 ∈ BndLinOp ∧ 𝑥 ∈ ℋ ) → ( normℎ ‘ ( 𝑇 ‘ 𝑥 ) ) ≤ ( ( normop ‘ 𝑇 ) · ( normℎ ‘ 𝑥 ) ) ) |
38 |
36 37
|
sylan |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( normℎ ‘ ( 𝑇 ‘ 𝑥 ) ) ≤ ( ( normop ‘ 𝑇 ) · ( normℎ ‘ 𝑥 ) ) ) |
39 |
|
lemul1a |
⊢ ( ( ( ( normℎ ‘ ( 𝑇 ‘ 𝑥 ) ) ∈ ℝ ∧ ( ( normop ‘ 𝑇 ) · ( normℎ ‘ 𝑥 ) ) ∈ ℝ ∧ ( ( normℎ ‘ 𝑥 ) ∈ ℝ ∧ 0 ≤ ( normℎ ‘ 𝑥 ) ) ) ∧ ( normℎ ‘ ( 𝑇 ‘ 𝑥 ) ) ≤ ( ( normop ‘ 𝑇 ) · ( normℎ ‘ 𝑥 ) ) ) → ( ( normℎ ‘ ( 𝑇 ‘ 𝑥 ) ) · ( normℎ ‘ 𝑥 ) ) ≤ ( ( ( normop ‘ 𝑇 ) · ( normℎ ‘ 𝑥 ) ) · ( normℎ ‘ 𝑥 ) ) ) |
40 |
19 29 32 38 39
|
syl31anc |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( normℎ ‘ ( 𝑇 ‘ 𝑥 ) ) · ( normℎ ‘ 𝑥 ) ) ≤ ( ( ( normop ‘ 𝑇 ) · ( normℎ ‘ 𝑥 ) ) · ( normℎ ‘ 𝑥 ) ) ) |
41 |
|
recn |
⊢ ( ( normop ‘ 𝑇 ) ∈ ℝ → ( normop ‘ 𝑇 ) ∈ ℂ ) |
42 |
41
|
ad2antlr |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( normop ‘ 𝑇 ) ∈ ℂ ) |
43 |
21
|
recnd |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( normℎ ‘ 𝑥 ) ∈ ℂ ) |
44 |
42 43 43
|
mulassd |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( ( normop ‘ 𝑇 ) · ( normℎ ‘ 𝑥 ) ) · ( normℎ ‘ 𝑥 ) ) = ( ( normop ‘ 𝑇 ) · ( ( normℎ ‘ 𝑥 ) · ( normℎ ‘ 𝑥 ) ) ) ) |
45 |
|
simpr |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → 𝑥 ∈ ℋ ) |
46 |
|
ax-his3 |
⊢ ( ( ( normop ‘ 𝑇 ) ∈ ℂ ∧ 𝑥 ∈ ℋ ∧ 𝑥 ∈ ℋ ) → ( ( ( normop ‘ 𝑇 ) ·ℎ 𝑥 ) ·ih 𝑥 ) = ( ( normop ‘ 𝑇 ) · ( 𝑥 ·ih 𝑥 ) ) ) |
47 |
42 45 45 46
|
syl3anc |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( ( normop ‘ 𝑇 ) ·ℎ 𝑥 ) ·ih 𝑥 ) = ( ( normop ‘ 𝑇 ) · ( 𝑥 ·ih 𝑥 ) ) ) |
48 |
20
|
recnd |
⊢ ( 𝑥 ∈ ℋ → ( normℎ ‘ 𝑥 ) ∈ ℂ ) |
49 |
48
|
sqvald |
⊢ ( 𝑥 ∈ ℋ → ( ( normℎ ‘ 𝑥 ) ↑ 2 ) = ( ( normℎ ‘ 𝑥 ) · ( normℎ ‘ 𝑥 ) ) ) |
50 |
|
normsq |
⊢ ( 𝑥 ∈ ℋ → ( ( normℎ ‘ 𝑥 ) ↑ 2 ) = ( 𝑥 ·ih 𝑥 ) ) |
51 |
49 50
|
eqtr3d |
⊢ ( 𝑥 ∈ ℋ → ( ( normℎ ‘ 𝑥 ) · ( normℎ ‘ 𝑥 ) ) = ( 𝑥 ·ih 𝑥 ) ) |
52 |
51
|
oveq2d |
⊢ ( 𝑥 ∈ ℋ → ( ( normop ‘ 𝑇 ) · ( ( normℎ ‘ 𝑥 ) · ( normℎ ‘ 𝑥 ) ) ) = ( ( normop ‘ 𝑇 ) · ( 𝑥 ·ih 𝑥 ) ) ) |
53 |
52
|
adantl |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( normop ‘ 𝑇 ) · ( ( normℎ ‘ 𝑥 ) · ( normℎ ‘ 𝑥 ) ) ) = ( ( normop ‘ 𝑇 ) · ( 𝑥 ·ih 𝑥 ) ) ) |
54 |
47 53
|
eqtr4d |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( ( normop ‘ 𝑇 ) ·ℎ 𝑥 ) ·ih 𝑥 ) = ( ( normop ‘ 𝑇 ) · ( ( normℎ ‘ 𝑥 ) · ( normℎ ‘ 𝑥 ) ) ) ) |
55 |
44 54
|
eqtr4d |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( ( normop ‘ 𝑇 ) · ( normℎ ‘ 𝑥 ) ) · ( normℎ ‘ 𝑥 ) ) = ( ( ( normop ‘ 𝑇 ) ·ℎ 𝑥 ) ·ih 𝑥 ) ) |
56 |
|
hoif |
⊢ Iop : ℋ –1-1-onto→ ℋ |
57 |
|
f1of |
⊢ ( Iop : ℋ –1-1-onto→ ℋ → Iop : ℋ ⟶ ℋ ) |
58 |
56 57
|
mp1i |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → Iop : ℋ ⟶ ℋ ) |
59 |
|
homval |
⊢ ( ( ( normop ‘ 𝑇 ) ∈ ℂ ∧ Iop : ℋ ⟶ ℋ ∧ 𝑥 ∈ ℋ ) → ( ( ( normop ‘ 𝑇 ) ·op Iop ) ‘ 𝑥 ) = ( ( normop ‘ 𝑇 ) ·ℎ ( Iop ‘ 𝑥 ) ) ) |
60 |
42 58 45 59
|
syl3anc |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( ( normop ‘ 𝑇 ) ·op Iop ) ‘ 𝑥 ) = ( ( normop ‘ 𝑇 ) ·ℎ ( Iop ‘ 𝑥 ) ) ) |
61 |
|
hoival |
⊢ ( 𝑥 ∈ ℋ → ( Iop ‘ 𝑥 ) = 𝑥 ) |
62 |
61
|
oveq2d |
⊢ ( 𝑥 ∈ ℋ → ( ( normop ‘ 𝑇 ) ·ℎ ( Iop ‘ 𝑥 ) ) = ( ( normop ‘ 𝑇 ) ·ℎ 𝑥 ) ) |
63 |
62
|
adantl |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( normop ‘ 𝑇 ) ·ℎ ( Iop ‘ 𝑥 ) ) = ( ( normop ‘ 𝑇 ) ·ℎ 𝑥 ) ) |
64 |
60 63
|
eqtrd |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( ( normop ‘ 𝑇 ) ·op Iop ) ‘ 𝑥 ) = ( ( normop ‘ 𝑇 ) ·ℎ 𝑥 ) ) |
65 |
64
|
oveq1d |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( ( ( normop ‘ 𝑇 ) ·op Iop ) ‘ 𝑥 ) ·ih 𝑥 ) = ( ( ( normop ‘ 𝑇 ) ·ℎ 𝑥 ) ·ih 𝑥 ) ) |
66 |
55 65
|
eqtr4d |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( ( normop ‘ 𝑇 ) · ( normℎ ‘ 𝑥 ) ) · ( normℎ ‘ 𝑥 ) ) = ( ( ( ( normop ‘ 𝑇 ) ·op Iop ) ‘ 𝑥 ) ·ih 𝑥 ) ) |
67 |
40 66
|
breqtrd |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( normℎ ‘ ( 𝑇 ‘ 𝑥 ) ) · ( normℎ ‘ 𝑥 ) ) ≤ ( ( ( ( normop ‘ 𝑇 ) ·op Iop ) ‘ 𝑥 ) ·ih 𝑥 ) ) |
68 |
5 22 11 26 67
|
letrd |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( abs ‘ ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ) ≤ ( ( ( ( normop ‘ 𝑇 ) ·op Iop ) ‘ 𝑥 ) ·ih 𝑥 ) ) |
69 |
2 5 11 13 68
|
letrd |
⊢ ( ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) ∧ 𝑥 ∈ ℋ ) → ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ≤ ( ( ( ( normop ‘ 𝑇 ) ·op Iop ) ‘ 𝑥 ) ·ih 𝑥 ) ) |
70 |
69
|
ralrimiva |
⊢ ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) → ∀ 𝑥 ∈ ℋ ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ≤ ( ( ( ( normop ‘ 𝑇 ) ·op Iop ) ‘ 𝑥 ) ·ih 𝑥 ) ) |
71 |
|
leop2 |
⊢ ( ( 𝑇 ∈ HrmOp ∧ ( ( normop ‘ 𝑇 ) ·op Iop ) ∈ HrmOp ) → ( 𝑇 ≤op ( ( normop ‘ 𝑇 ) ·op Iop ) ↔ ∀ 𝑥 ∈ ℋ ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ≤ ( ( ( ( normop ‘ 𝑇 ) ·op Iop ) ‘ 𝑥 ) ·ih 𝑥 ) ) ) |
72 |
8 71
|
sylan2 |
⊢ ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) → ( 𝑇 ≤op ( ( normop ‘ 𝑇 ) ·op Iop ) ↔ ∀ 𝑥 ∈ ℋ ( ( 𝑇 ‘ 𝑥 ) ·ih 𝑥 ) ≤ ( ( ( ( normop ‘ 𝑇 ) ·op Iop ) ‘ 𝑥 ) ·ih 𝑥 ) ) ) |
73 |
70 72
|
mpbird |
⊢ ( ( 𝑇 ∈ HrmOp ∧ ( normop ‘ 𝑇 ) ∈ ℝ ) → 𝑇 ≤op ( ( normop ‘ 𝑇 ) ·op Iop ) ) |