Proof of Theorem nmcfnlbi
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fveq2 6906 | . . . . . 6
⊢ (𝐴 = 0ℎ →
(𝑇‘𝐴) = (𝑇‘0ℎ)) | 
| 2 |  | nmcfnex.1 | . . . . . . 7
⊢ 𝑇 ∈ LinFn | 
| 3 | 2 | lnfn0i 32061 | . . . . . 6
⊢ (𝑇‘0ℎ) =
0 | 
| 4 | 1, 3 | eqtrdi 2793 | . . . . 5
⊢ (𝐴 = 0ℎ →
(𝑇‘𝐴) = 0) | 
| 5 | 4 | abs00bd 15330 | . . . 4
⊢ (𝐴 = 0ℎ →
(abs‘(𝑇‘𝐴)) = 0) | 
| 6 |  | 0le0 12367 | . . . . 5
⊢ 0 ≤
0 | 
| 7 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝐴 = 0ℎ →
(normℎ‘𝐴) =
(normℎ‘0ℎ)) | 
| 8 |  | norm0 31147 | . . . . . . . 8
⊢
(normℎ‘0ℎ) =
0 | 
| 9 | 7, 8 | eqtrdi 2793 | . . . . . . 7
⊢ (𝐴 = 0ℎ →
(normℎ‘𝐴) = 0) | 
| 10 | 9 | oveq2d 7447 | . . . . . 6
⊢ (𝐴 = 0ℎ →
((normfn‘𝑇) ·
(normℎ‘𝐴)) = ((normfn‘𝑇) · 0)) | 
| 11 |  | nmcfnex.2 | . . . . . . . . 9
⊢ 𝑇 ∈ ContFn | 
| 12 | 2, 11 | nmcfnexi 32070 | . . . . . . . 8
⊢
(normfn‘𝑇) ∈ ℝ | 
| 13 | 12 | recni 11275 | . . . . . . 7
⊢
(normfn‘𝑇) ∈ ℂ | 
| 14 | 13 | mul01i 11451 | . . . . . 6
⊢
((normfn‘𝑇) · 0) = 0 | 
| 15 | 10, 14 | eqtr2di 2794 | . . . . 5
⊢ (𝐴 = 0ℎ → 0
= ((normfn‘𝑇) ·
(normℎ‘𝐴))) | 
| 16 | 6, 15 | breqtrid 5180 | . . . 4
⊢ (𝐴 = 0ℎ → 0
≤ ((normfn‘𝑇) ·
(normℎ‘𝐴))) | 
| 17 | 5, 16 | eqbrtrd 5165 | . . 3
⊢ (𝐴 = 0ℎ →
(abs‘(𝑇‘𝐴)) ≤
((normfn‘𝑇) ·
(normℎ‘𝐴))) | 
| 18 | 17 | adantl 481 | . 2
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 = 0ℎ) →
(abs‘(𝑇‘𝐴)) ≤
((normfn‘𝑇) ·
(normℎ‘𝐴))) | 
| 19 | 2 | lnfnfi 32060 | . . . . . . . . . 10
⊢ 𝑇:
ℋ⟶ℂ | 
| 20 | 19 | ffvelcdmi 7103 | . . . . . . . . 9
⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ) | 
| 21 | 20 | abscld 15475 | . . . . . . . 8
⊢ (𝐴 ∈ ℋ →
(abs‘(𝑇‘𝐴)) ∈
ℝ) | 
| 22 | 21 | adantr 480 | . . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (abs‘(𝑇‘𝐴)) ∈ ℝ) | 
| 23 | 22 | recnd 11289 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (abs‘(𝑇‘𝐴)) ∈ ℂ) | 
| 24 |  | normcl 31144 | . . . . . . . 8
⊢ (𝐴 ∈ ℋ →
(normℎ‘𝐴) ∈ ℝ) | 
| 25 | 24 | adantr 480 | . . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (normℎ‘𝐴) ∈ ℝ) | 
| 26 | 25 | recnd 11289 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (normℎ‘𝐴) ∈ ℂ) | 
| 27 |  | norm-i 31148 | . . . . . . . . 9
⊢ (𝐴 ∈ ℋ →
((normℎ‘𝐴) = 0 ↔ 𝐴 = 0ℎ)) | 
| 28 | 27 | notbid 318 | . . . . . . . 8
⊢ (𝐴 ∈ ℋ → (¬
(normℎ‘𝐴) = 0 ↔ ¬ 𝐴 = 0ℎ)) | 
| 29 | 28 | biimpar 477 | . . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ ¬ (normℎ‘𝐴) = 0) | 
| 30 | 29 | neqned 2947 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (normℎ‘𝐴) ≠ 0) | 
| 31 | 23, 26, 30 | divrec2d 12047 | . . . . 5
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = ((1 /
(normℎ‘𝐴)) · (abs‘(𝑇‘𝐴)))) | 
| 32 | 25, 30 | rereccld 12094 | . . . . . . . . 9
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (1 / (normℎ‘𝐴)) ∈ ℝ) | 
| 33 | 32 | recnd 11289 | . . . . . . . 8
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (1 / (normℎ‘𝐴)) ∈ ℂ) | 
| 34 |  | simpl 482 | . . . . . . . 8
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ 𝐴 ∈
ℋ) | 
| 35 | 2 | lnfnmuli 32063 | . . . . . . . 8
⊢ (((1 /
(normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 /
(normℎ‘𝐴)) · (𝑇‘𝐴))) | 
| 36 | 33, 34, 35 | syl2anc 584 | . . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 /
(normℎ‘𝐴)) · (𝑇‘𝐴))) | 
| 37 | 36 | fveq2d 6910 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (abs‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴))) = (abs‘((1 /
(normℎ‘𝐴)) · (𝑇‘𝐴)))) | 
| 38 | 20 | adantr 480 | . . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (𝑇‘𝐴) ∈
ℂ) | 
| 39 | 33, 38 | absmuld 15493 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (abs‘((1 / (normℎ‘𝐴)) · (𝑇‘𝐴))) = ((abs‘(1 /
(normℎ‘𝐴))) · (abs‘(𝑇‘𝐴)))) | 
| 40 |  | df-ne 2941 | . . . . . . . . . . . 12
⊢ (𝐴 ≠ 0ℎ
↔ ¬ 𝐴 =
0ℎ) | 
| 41 |  | normgt0 31146 | . . . . . . . . . . . 12
⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ
↔ 0 < (normℎ‘𝐴))) | 
| 42 | 40, 41 | bitr3id 285 | . . . . . . . . . . 11
⊢ (𝐴 ∈ ℋ → (¬
𝐴 = 0ℎ
↔ 0 < (normℎ‘𝐴))) | 
| 43 | 42 | biimpa 476 | . . . . . . . . . 10
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ 0 < (normℎ‘𝐴)) | 
| 44 | 25, 43 | recgt0d 12202 | . . . . . . . . 9
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ 0 < (1 / (normℎ‘𝐴))) | 
| 45 |  | 0re 11263 | . . . . . . . . . 10
⊢ 0 ∈
ℝ | 
| 46 |  | ltle 11349 | . . . . . . . . . 10
⊢ ((0
∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 /
(normℎ‘𝐴)) → 0 ≤ (1 /
(normℎ‘𝐴)))) | 
| 47 | 45, 46 | mpan 690 | . . . . . . . . 9
⊢ ((1 /
(normℎ‘𝐴)) ∈ ℝ → (0 < (1 /
(normℎ‘𝐴)) → 0 ≤ (1 /
(normℎ‘𝐴)))) | 
| 48 | 32, 44, 47 | sylc 65 | . . . . . . . 8
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ 0 ≤ (1 / (normℎ‘𝐴))) | 
| 49 | 32, 48 | absidd 15461 | . . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (abs‘(1 / (normℎ‘𝐴))) = (1 /
(normℎ‘𝐴))) | 
| 50 | 49 | oveq1d 7446 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ ((abs‘(1 / (normℎ‘𝐴))) · (abs‘(𝑇‘𝐴))) = ((1 /
(normℎ‘𝐴)) · (abs‘(𝑇‘𝐴)))) | 
| 51 | 37, 39, 50 | 3eqtrrd 2782 | . . . . 5
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))) = (abs‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)))) | 
| 52 | 31, 51 | eqtrd 2777 | . . . 4
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = (abs‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)))) | 
| 53 |  | hvmulcl 31032 | . . . . . 6
⊢ (((1 /
(normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 /
(normℎ‘𝐴)) ·ℎ 𝐴) ∈
ℋ) | 
| 54 | 33, 34, 53 | syl2anc 584 | . . . . 5
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈
ℋ) | 
| 55 |  | normcl 31144 | . . . . . . 7
⊢ (((1 /
(normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ →
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ 𝐴)) ∈ ℝ) | 
| 56 | 54, 55 | syl 17 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (normℎ‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) ∈
ℝ) | 
| 57 |  | norm1 31268 | . . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) = 1) | 
| 58 | 40, 57 | sylan2br 595 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (normℎ‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) = 1) | 
| 59 |  | eqle 11363 | . . . . . 6
⊢
(((normℎ‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ ∧
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ 𝐴)) = 1) →
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ 𝐴)) ≤ 1) | 
| 60 | 56, 58, 59 | syl2anc 584 | . . . . 5
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (normℎ‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1) | 
| 61 |  | nmfnlb 31943 | . . . . . 6
⊢ ((𝑇: ℋ⟶ℂ ∧
((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ ∧
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴))) ≤
(normfn‘𝑇)) | 
| 62 | 19, 61 | mp3an1 1450 | . . . . 5
⊢ ((((1 /
(normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ ∧
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴))) ≤
(normfn‘𝑇)) | 
| 63 | 54, 60, 62 | syl2anc 584 | . . . 4
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (abs‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴))) ≤
(normfn‘𝑇)) | 
| 64 | 52, 63 | eqbrtrd 5165 | . . 3
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤
(normfn‘𝑇)) | 
| 65 | 12 | a1i 11 | . . . 4
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (normfn‘𝑇) ∈ ℝ) | 
| 66 |  | ledivmul2 12147 | . . . 4
⊢
(((abs‘(𝑇‘𝐴)) ∈ ℝ ∧
(normfn‘𝑇)
∈ ℝ ∧ ((normℎ‘𝐴) ∈ ℝ ∧ 0 <
(normℎ‘𝐴))) → (((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤
(normfn‘𝑇)
↔ (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) ·
(normℎ‘𝐴)))) | 
| 67 | 22, 65, 25, 43, 66 | syl112anc 1376 | . . 3
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤
(normfn‘𝑇)
↔ (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) ·
(normℎ‘𝐴)))) | 
| 68 | 64, 67 | mpbid 232 | . 2
⊢ ((𝐴 ∈ ℋ ∧ ¬
𝐴 = 0ℎ)
→ (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) ·
(normℎ‘𝐴))) | 
| 69 | 18, 68 | pm2.61dan 813 | 1
⊢ (𝐴 ∈ ℋ →
(abs‘(𝑇‘𝐴)) ≤
((normfn‘𝑇) ·
(normℎ‘𝐴))) |