Step | Hyp | Ref
| Expression |
1 | | cnlnadjlem.1 |
. . . . . . . 8
⊢ 𝑇 ∈ LinOp |
2 | 1 | lnopfi 30331 |
. . . . . . 7
⊢ 𝑇: ℋ⟶
ℋ |
3 | 2 | ffvelrni 6960 |
. . . . . 6
⊢ (𝑔 ∈ ℋ → (𝑇‘𝑔) ∈ ℋ) |
4 | | hicl 29442 |
. . . . . 6
⊢ (((𝑇‘𝑔) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑔) ·ih 𝑦) ∈
ℂ) |
5 | 3, 4 | sylan 580 |
. . . . 5
⊢ ((𝑔 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑔) ·ih 𝑦) ∈
ℂ) |
6 | 5 | ancoms 459 |
. . . 4
⊢ ((𝑦 ∈ ℋ ∧ 𝑔 ∈ ℋ) → ((𝑇‘𝑔) ·ih 𝑦) ∈
ℂ) |
7 | | cnlnadjlem.3 |
. . . 4
⊢ 𝐺 = (𝑔 ∈ ℋ ↦ ((𝑇‘𝑔) ·ih 𝑦)) |
8 | 6, 7 | fmptd 6988 |
. . 3
⊢ (𝑦 ∈ ℋ → 𝐺:
ℋ⟶ℂ) |
9 | | hvmulcl 29375 |
. . . . . . 7
⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ) → (𝑥
·ℎ 𝑤) ∈ ℋ) |
10 | 1 | lnopaddi 30333 |
. . . . . . . . . . . 12
⊢ (((𝑥
·ℎ 𝑤) ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑤) +ℎ 𝑧)) = ((𝑇‘(𝑥 ·ℎ 𝑤)) +ℎ (𝑇‘𝑧))) |
11 | 10 | 3adant3 1131 |
. . . . . . . . . . 11
⊢ (((𝑥
·ℎ 𝑤) ∈ ℋ ∧ 𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘((𝑥 ·ℎ 𝑤) +ℎ 𝑧)) = ((𝑇‘(𝑥 ·ℎ 𝑤)) +ℎ (𝑇‘𝑧))) |
12 | 11 | oveq1d 7290 |
. . . . . . . . . 10
⊢ (((𝑥
·ℎ 𝑤) ∈ ℋ ∧ 𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘((𝑥 ·ℎ 𝑤) +ℎ 𝑧))
·ih 𝑦) = (((𝑇‘(𝑥 ·ℎ 𝑤)) +ℎ (𝑇‘𝑧)) ·ih 𝑦)) |
13 | 2 | ffvelrni 6960 |
. . . . . . . . . . 11
⊢ ((𝑥
·ℎ 𝑤) ∈ ℋ → (𝑇‘(𝑥 ·ℎ 𝑤)) ∈
ℋ) |
14 | 2 | ffvelrni 6960 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ℋ → (𝑇‘𝑧) ∈ ℋ) |
15 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℋ → 𝑦 ∈
ℋ) |
16 | | ax-his2 29445 |
. . . . . . . . . . 11
⊢ (((𝑇‘(𝑥 ·ℎ 𝑤)) ∈ ℋ ∧ (𝑇‘𝑧) ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑇‘(𝑥 ·ℎ 𝑤)) +ℎ (𝑇‘𝑧)) ·ih 𝑦) = (((𝑇‘(𝑥 ·ℎ 𝑤))
·ih 𝑦) + ((𝑇‘𝑧) ·ih 𝑦))) |
17 | 13, 14, 15, 16 | syl3an 1159 |
. . . . . . . . . 10
⊢ (((𝑥
·ℎ 𝑤) ∈ ℋ ∧ 𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((𝑇‘(𝑥 ·ℎ 𝑤)) +ℎ (𝑇‘𝑧)) ·ih 𝑦) = (((𝑇‘(𝑥 ·ℎ 𝑤))
·ih 𝑦) + ((𝑇‘𝑧) ·ih 𝑦))) |
18 | 12, 17 | eqtrd 2778 |
. . . . . . . . 9
⊢ (((𝑥
·ℎ 𝑤) ∈ ℋ ∧ 𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘((𝑥 ·ℎ 𝑤) +ℎ 𝑧))
·ih 𝑦) = (((𝑇‘(𝑥 ·ℎ 𝑤))
·ih 𝑦) + ((𝑇‘𝑧) ·ih 𝑦))) |
19 | 18 | 3comr 1124 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℋ ∧ (𝑥
·ℎ 𝑤) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑇‘((𝑥 ·ℎ 𝑤) +ℎ 𝑧))
·ih 𝑦) = (((𝑇‘(𝑥 ·ℎ 𝑤))
·ih 𝑦) + ((𝑇‘𝑧) ·ih 𝑦))) |
20 | 19 | 3expa 1117 |
. . . . . . 7
⊢ (((𝑦 ∈ ℋ ∧ (𝑥
·ℎ 𝑤) ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑇‘((𝑥 ·ℎ 𝑤) +ℎ 𝑧))
·ih 𝑦) = (((𝑇‘(𝑥 ·ℎ 𝑤))
·ih 𝑦) + ((𝑇‘𝑧) ·ih 𝑦))) |
21 | 9, 20 | sylanl2 678 |
. . . . . 6
⊢ (((𝑦 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → ((𝑇‘((𝑥 ·ℎ 𝑤) +ℎ 𝑧))
·ih 𝑦) = (((𝑇‘(𝑥 ·ℎ 𝑤))
·ih 𝑦) + ((𝑇‘𝑧) ·ih 𝑦))) |
22 | | hvaddcl 29374 |
. . . . . . . . 9
⊢ (((𝑥
·ℎ 𝑤) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑤) +ℎ 𝑧) ∈
ℋ) |
23 | 9, 22 | sylan 580 |
. . . . . . . 8
⊢ (((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥
·ℎ 𝑤) +ℎ 𝑧) ∈ ℋ) |
24 | | cnlnadjlem.2 |
. . . . . . . . 9
⊢ 𝑇 ∈ ContOp |
25 | 1, 24, 7 | cnlnadjlem1 30429 |
. . . . . . . 8
⊢ (((𝑥
·ℎ 𝑤) +ℎ 𝑧) ∈ ℋ → (𝐺‘((𝑥 ·ℎ 𝑤) +ℎ 𝑧)) = ((𝑇‘((𝑥 ·ℎ 𝑤) +ℎ 𝑧))
·ih 𝑦)) |
26 | 23, 25 | syl 17 |
. . . . . . 7
⊢ (((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (𝐺‘((𝑥 ·ℎ 𝑤) +ℎ 𝑧)) = ((𝑇‘((𝑥 ·ℎ 𝑤) +ℎ 𝑧))
·ih 𝑦)) |
27 | 26 | adantll 711 |
. . . . . 6
⊢ (((𝑦 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝐺‘((𝑥 ·ℎ 𝑤) +ℎ 𝑧)) = ((𝑇‘((𝑥 ·ℎ 𝑤) +ℎ 𝑧))
·ih 𝑦)) |
28 | 2 | ffvelrni 6960 |
. . . . . . . . . . 11
⊢ (𝑤 ∈ ℋ → (𝑇‘𝑤) ∈ ℋ) |
29 | | ax-his3 29446 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℂ ∧ (𝑇‘𝑤) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑥 ·ℎ (𝑇‘𝑤)) ·ih 𝑦) = (𝑥 · ((𝑇‘𝑤) ·ih 𝑦))) |
30 | 28, 29 | syl3an2 1163 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑥
·ℎ (𝑇‘𝑤)) ·ih 𝑦) = (𝑥 · ((𝑇‘𝑤) ·ih 𝑦))) |
31 | 30 | 3comr 1124 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ℋ ∧ 𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ) → ((𝑥
·ℎ (𝑇‘𝑤)) ·ih 𝑦) = (𝑥 · ((𝑇‘𝑤) ·ih 𝑦))) |
32 | 31 | 3expb 1119 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ)) → ((𝑥
·ℎ (𝑇‘𝑤)) ·ih 𝑦) = (𝑥 · ((𝑇‘𝑤) ·ih 𝑦))) |
33 | 1 | lnopmuli 30334 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ) → (𝑇‘(𝑥 ·ℎ 𝑤)) = (𝑥 ·ℎ (𝑇‘𝑤))) |
34 | 33 | oveq1d 7290 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ) → ((𝑇‘(𝑥 ·ℎ 𝑤))
·ih 𝑦) = ((𝑥 ·ℎ (𝑇‘𝑤)) ·ih 𝑦)) |
35 | 34 | adantl 482 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ)) → ((𝑇‘(𝑥 ·ℎ 𝑤))
·ih 𝑦) = ((𝑥 ·ℎ (𝑇‘𝑤)) ·ih 𝑦)) |
36 | 1, 24, 7 | cnlnadjlem1 30429 |
. . . . . . . . . 10
⊢ (𝑤 ∈ ℋ → (𝐺‘𝑤) = ((𝑇‘𝑤) ·ih 𝑦)) |
37 | 36 | oveq2d 7291 |
. . . . . . . . 9
⊢ (𝑤 ∈ ℋ → (𝑥 · (𝐺‘𝑤)) = (𝑥 · ((𝑇‘𝑤) ·ih 𝑦))) |
38 | 37 | ad2antll 726 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ)) → (𝑥 · (𝐺‘𝑤)) = (𝑥 · ((𝑇‘𝑤) ·ih 𝑦))) |
39 | 32, 35, 38 | 3eqtr4rd 2789 |
. . . . . . 7
⊢ ((𝑦 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ)) → (𝑥 · (𝐺‘𝑤)) = ((𝑇‘(𝑥 ·ℎ 𝑤))
·ih 𝑦)) |
40 | 1, 24, 7 | cnlnadjlem1 30429 |
. . . . . . 7
⊢ (𝑧 ∈ ℋ → (𝐺‘𝑧) = ((𝑇‘𝑧) ·ih 𝑦)) |
41 | 39, 40 | oveqan12d 7294 |
. . . . . 6
⊢ (((𝑦 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → ((𝑥 · (𝐺‘𝑤)) + (𝐺‘𝑧)) = (((𝑇‘(𝑥 ·ℎ 𝑤))
·ih 𝑦) + ((𝑇‘𝑧) ·ih 𝑦))) |
42 | 21, 27, 41 | 3eqtr4d 2788 |
. . . . 5
⊢ (((𝑦 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ)) ∧ 𝑧 ∈ ℋ) → (𝐺‘((𝑥 ·ℎ 𝑤) +ℎ 𝑧)) = ((𝑥 · (𝐺‘𝑤)) + (𝐺‘𝑧))) |
43 | 42 | ralrimiva 3103 |
. . . 4
⊢ ((𝑦 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑤 ∈ ℋ)) →
∀𝑧 ∈ ℋ
(𝐺‘((𝑥
·ℎ 𝑤) +ℎ 𝑧)) = ((𝑥 · (𝐺‘𝑤)) + (𝐺‘𝑧))) |
44 | 43 | ralrimivva 3123 |
. . 3
⊢ (𝑦 ∈ ℋ →
∀𝑥 ∈ ℂ
∀𝑤 ∈ ℋ
∀𝑧 ∈ ℋ
(𝐺‘((𝑥
·ℎ 𝑤) +ℎ 𝑧)) = ((𝑥 · (𝐺‘𝑤)) + (𝐺‘𝑧))) |
45 | | ellnfn 30245 |
. . 3
⊢ (𝐺 ∈ LinFn ↔ (𝐺: ℋ⟶ℂ ∧
∀𝑥 ∈ ℂ
∀𝑤 ∈ ℋ
∀𝑧 ∈ ℋ
(𝐺‘((𝑥
·ℎ 𝑤) +ℎ 𝑧)) = ((𝑥 · (𝐺‘𝑤)) + (𝐺‘𝑧)))) |
46 | 8, 44, 45 | sylanbrc 583 |
. 2
⊢ (𝑦 ∈ ℋ → 𝐺 ∈ LinFn) |
47 | 1, 24 | nmcopexi 30389 |
. . . . 5
⊢
(normop‘𝑇) ∈ ℝ |
48 | | normcl 29487 |
. . . . 5
⊢ (𝑦 ∈ ℋ →
(normℎ‘𝑦) ∈ ℝ) |
49 | | remulcl 10956 |
. . . . 5
⊢
(((normop‘𝑇) ∈ ℝ ∧
(normℎ‘𝑦) ∈ ℝ) →
((normop‘𝑇) ·
(normℎ‘𝑦)) ∈ ℝ) |
50 | 47, 48, 49 | sylancr 587 |
. . . 4
⊢ (𝑦 ∈ ℋ →
((normop‘𝑇) ·
(normℎ‘𝑦)) ∈ ℝ) |
51 | 40 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝐺‘𝑧) = ((𝑇‘𝑧) ·ih 𝑦)) |
52 | | hicl 29442 |
. . . . . . . . . . 11
⊢ (((𝑇‘𝑧) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑧) ·ih 𝑦) ∈
ℂ) |
53 | 14, 52 | sylan 580 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑧) ·ih 𝑦) ∈
ℂ) |
54 | 51, 53 | eqeltrd 2839 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝐺‘𝑧) ∈ ℂ) |
55 | 54 | abscld 15148 |
. . . . . . . 8
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) →
(abs‘(𝐺‘𝑧)) ∈
ℝ) |
56 | | normcl 29487 |
. . . . . . . . . 10
⊢ ((𝑇‘𝑧) ∈ ℋ →
(normℎ‘(𝑇‘𝑧)) ∈ ℝ) |
57 | 14, 56 | syl 17 |
. . . . . . . . 9
⊢ (𝑧 ∈ ℋ →
(normℎ‘(𝑇‘𝑧)) ∈ ℝ) |
58 | | remulcl 10956 |
. . . . . . . . 9
⊢
(((normℎ‘(𝑇‘𝑧)) ∈ ℝ ∧
(normℎ‘𝑦) ∈ ℝ) →
((normℎ‘(𝑇‘𝑧)) ·
(normℎ‘𝑦)) ∈ ℝ) |
59 | 57, 48, 58 | syl2an 596 |
. . . . . . . 8
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) →
((normℎ‘(𝑇‘𝑧)) ·
(normℎ‘𝑦)) ∈ ℝ) |
60 | | normcl 29487 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ℋ →
(normℎ‘𝑧) ∈ ℝ) |
61 | | remulcl 10956 |
. . . . . . . . . 10
⊢
(((normop‘𝑇) ∈ ℝ ∧
(normℎ‘𝑧) ∈ ℝ) →
((normop‘𝑇) ·
(normℎ‘𝑧)) ∈ ℝ) |
62 | 47, 60, 61 | sylancr 587 |
. . . . . . . . 9
⊢ (𝑧 ∈ ℋ →
((normop‘𝑇) ·
(normℎ‘𝑧)) ∈ ℝ) |
63 | | remulcl 10956 |
. . . . . . . . 9
⊢
((((normop‘𝑇) ·
(normℎ‘𝑧)) ∈ ℝ ∧
(normℎ‘𝑦) ∈ ℝ) →
(((normop‘𝑇) ·
(normℎ‘𝑧)) ·
(normℎ‘𝑦)) ∈ ℝ) |
64 | 62, 48, 63 | syl2an 596 |
. . . . . . . 8
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) →
(((normop‘𝑇) ·
(normℎ‘𝑧)) ·
(normℎ‘𝑦)) ∈ ℝ) |
65 | 51 | fveq2d 6778 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) →
(abs‘(𝐺‘𝑧)) = (abs‘((𝑇‘𝑧) ·ih 𝑦))) |
66 | | bcs 29543 |
. . . . . . . . . 10
⊢ (((𝑇‘𝑧) ∈ ℋ ∧ 𝑦 ∈ ℋ) → (abs‘((𝑇‘𝑧) ·ih 𝑦)) ≤
((normℎ‘(𝑇‘𝑧)) ·
(normℎ‘𝑦))) |
67 | 14, 66 | sylan 580 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) →
(abs‘((𝑇‘𝑧)
·ih 𝑦)) ≤
((normℎ‘(𝑇‘𝑧)) ·
(normℎ‘𝑦))) |
68 | 65, 67 | eqbrtrd 5096 |
. . . . . . . 8
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) →
(abs‘(𝐺‘𝑧)) ≤
((normℎ‘(𝑇‘𝑧)) ·
(normℎ‘𝑦))) |
69 | 57 | adantr 481 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) →
(normℎ‘(𝑇‘𝑧)) ∈ ℝ) |
70 | 62 | adantr 481 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) →
((normop‘𝑇) ·
(normℎ‘𝑧)) ∈ ℝ) |
71 | | normge0 29488 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ℋ → 0 ≤
(normℎ‘𝑦)) |
72 | 48, 71 | jca 512 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℋ →
((normℎ‘𝑦) ∈ ℝ ∧ 0 ≤
(normℎ‘𝑦))) |
73 | 72 | adantl 482 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) →
((normℎ‘𝑦) ∈ ℝ ∧ 0 ≤
(normℎ‘𝑦))) |
74 | 1, 24 | nmcoplbi 30390 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ℋ →
(normℎ‘(𝑇‘𝑧)) ≤ ((normop‘𝑇) ·
(normℎ‘𝑧))) |
75 | 74 | adantr 481 |
. . . . . . . . 9
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) →
(normℎ‘(𝑇‘𝑧)) ≤ ((normop‘𝑇) ·
(normℎ‘𝑧))) |
76 | | lemul1a 11829 |
. . . . . . . . 9
⊢
((((normℎ‘(𝑇‘𝑧)) ∈ ℝ ∧
((normop‘𝑇) ·
(normℎ‘𝑧)) ∈ ℝ ∧
((normℎ‘𝑦) ∈ ℝ ∧ 0 ≤
(normℎ‘𝑦))) ∧
(normℎ‘(𝑇‘𝑧)) ≤ ((normop‘𝑇) ·
(normℎ‘𝑧))) →
((normℎ‘(𝑇‘𝑧)) ·
(normℎ‘𝑦)) ≤ (((normop‘𝑇) ·
(normℎ‘𝑧)) ·
(normℎ‘𝑦))) |
77 | 69, 70, 73, 75, 76 | syl31anc 1372 |
. . . . . . . 8
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) →
((normℎ‘(𝑇‘𝑧)) ·
(normℎ‘𝑦)) ≤ (((normop‘𝑇) ·
(normℎ‘𝑧)) ·
(normℎ‘𝑦))) |
78 | 55, 59, 64, 68, 77 | letrd 11132 |
. . . . . . 7
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) →
(abs‘(𝐺‘𝑧)) ≤
(((normop‘𝑇) ·
(normℎ‘𝑧)) ·
(normℎ‘𝑦))) |
79 | 60 | recnd 11003 |
. . . . . . . 8
⊢ (𝑧 ∈ ℋ →
(normℎ‘𝑧) ∈ ℂ) |
80 | 48 | recnd 11003 |
. . . . . . . 8
⊢ (𝑦 ∈ ℋ →
(normℎ‘𝑦) ∈ ℂ) |
81 | 47 | recni 10989 |
. . . . . . . . 9
⊢
(normop‘𝑇) ∈ ℂ |
82 | | mul32 11141 |
. . . . . . . . 9
⊢
(((normop‘𝑇) ∈ ℂ ∧
(normℎ‘𝑧) ∈ ℂ ∧
(normℎ‘𝑦) ∈ ℂ) →
(((normop‘𝑇) ·
(normℎ‘𝑧)) ·
(normℎ‘𝑦)) = (((normop‘𝑇) ·
(normℎ‘𝑦)) ·
(normℎ‘𝑧))) |
83 | 81, 82 | mp3an1 1447 |
. . . . . . . 8
⊢
(((normℎ‘𝑧) ∈ ℂ ∧
(normℎ‘𝑦) ∈ ℂ) →
(((normop‘𝑇) ·
(normℎ‘𝑧)) ·
(normℎ‘𝑦)) = (((normop‘𝑇) ·
(normℎ‘𝑦)) ·
(normℎ‘𝑧))) |
84 | 79, 80, 83 | syl2an 596 |
. . . . . . 7
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) →
(((normop‘𝑇) ·
(normℎ‘𝑧)) ·
(normℎ‘𝑦)) = (((normop‘𝑇) ·
(normℎ‘𝑦)) ·
(normℎ‘𝑧))) |
85 | 78, 84 | breqtrd 5100 |
. . . . . 6
⊢ ((𝑧 ∈ ℋ ∧ 𝑦 ∈ ℋ) →
(abs‘(𝐺‘𝑧)) ≤
(((normop‘𝑇) ·
(normℎ‘𝑦)) ·
(normℎ‘𝑧))) |
86 | 85 | ancoms 459 |
. . . . 5
⊢ ((𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) →
(abs‘(𝐺‘𝑧)) ≤
(((normop‘𝑇) ·
(normℎ‘𝑦)) ·
(normℎ‘𝑧))) |
87 | 86 | ralrimiva 3103 |
. . . 4
⊢ (𝑦 ∈ ℋ →
∀𝑧 ∈ ℋ
(abs‘(𝐺‘𝑧)) ≤
(((normop‘𝑇) ·
(normℎ‘𝑦)) ·
(normℎ‘𝑧))) |
88 | | oveq1 7282 |
. . . . . . 7
⊢ (𝑥 =
((normop‘𝑇) ·
(normℎ‘𝑦)) → (𝑥 ·
(normℎ‘𝑧)) = (((normop‘𝑇) ·
(normℎ‘𝑦)) ·
(normℎ‘𝑧))) |
89 | 88 | breq2d 5086 |
. . . . . 6
⊢ (𝑥 =
((normop‘𝑇) ·
(normℎ‘𝑦)) → ((abs‘(𝐺‘𝑧)) ≤ (𝑥 ·
(normℎ‘𝑧)) ↔ (abs‘(𝐺‘𝑧)) ≤ (((normop‘𝑇) ·
(normℎ‘𝑦)) ·
(normℎ‘𝑧)))) |
90 | 89 | ralbidv 3112 |
. . . . 5
⊢ (𝑥 =
((normop‘𝑇) ·
(normℎ‘𝑦)) → (∀𝑧 ∈ ℋ (abs‘(𝐺‘𝑧)) ≤ (𝑥 ·
(normℎ‘𝑧)) ↔ ∀𝑧 ∈ ℋ (abs‘(𝐺‘𝑧)) ≤ (((normop‘𝑇) ·
(normℎ‘𝑦)) ·
(normℎ‘𝑧)))) |
91 | 90 | rspcev 3561 |
. . . 4
⊢
((((normop‘𝑇) ·
(normℎ‘𝑦)) ∈ ℝ ∧ ∀𝑧 ∈ ℋ
(abs‘(𝐺‘𝑧)) ≤
(((normop‘𝑇) ·
(normℎ‘𝑦)) ·
(normℎ‘𝑧))) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ℋ (abs‘(𝐺‘𝑧)) ≤ (𝑥 ·
(normℎ‘𝑧))) |
92 | 50, 87, 91 | syl2anc 584 |
. . 3
⊢ (𝑦 ∈ ℋ →
∃𝑥 ∈ ℝ
∀𝑧 ∈ ℋ
(abs‘(𝐺‘𝑧)) ≤ (𝑥 ·
(normℎ‘𝑧))) |
93 | | lnfncon 30418 |
. . . 4
⊢ (𝐺 ∈ LinFn → (𝐺 ∈ ContFn ↔
∃𝑥 ∈ ℝ
∀𝑧 ∈ ℋ
(abs‘(𝐺‘𝑧)) ≤ (𝑥 ·
(normℎ‘𝑧)))) |
94 | 46, 93 | syl 17 |
. . 3
⊢ (𝑦 ∈ ℋ → (𝐺 ∈ ContFn ↔
∃𝑥 ∈ ℝ
∀𝑧 ∈ ℋ
(abs‘(𝐺‘𝑧)) ≤ (𝑥 ·
(normℎ‘𝑧)))) |
95 | 92, 94 | mpbird 256 |
. 2
⊢ (𝑦 ∈ ℋ → 𝐺 ∈ ContFn) |
96 | 46, 95 | jca 512 |
1
⊢ (𝑦 ∈ ℋ → (𝐺 ∈ LinFn ∧ 𝐺 ∈
ContFn)) |