Proof of Theorem nmbdfnlbi
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fveq2 6905 | . . . . . 6
⊢ (𝐴 = 0ℎ →
(𝑇‘𝐴) = (𝑇‘0ℎ)) | 
| 2 |  | nmbdfnlb.1 | . . . . . . . 8
⊢ (𝑇 ∈ LinFn ∧
(normfn‘𝑇)
∈ ℝ) | 
| 3 | 2 | simpli 483 | . . . . . . 7
⊢ 𝑇 ∈ LinFn | 
| 4 | 3 | lnfn0i 32062 | . . . . . 6
⊢ (𝑇‘0ℎ) =
0 | 
| 5 | 1, 4 | eqtrdi 2792 | . . . . 5
⊢ (𝐴 = 0ℎ →
(𝑇‘𝐴) = 0) | 
| 6 | 5 | abs00bd 15331 | . . . 4
⊢ (𝐴 = 0ℎ →
(abs‘(𝑇‘𝐴)) = 0) | 
| 7 |  | 0le0 12368 | . . . . 5
⊢ 0 ≤
0 | 
| 8 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝐴 = 0ℎ →
(normℎ‘𝐴) =
(normℎ‘0ℎ)) | 
| 9 |  | norm0 31148 | . . . . . . . 8
⊢
(normℎ‘0ℎ) =
0 | 
| 10 | 8, 9 | eqtrdi 2792 | . . . . . . 7
⊢ (𝐴 = 0ℎ →
(normℎ‘𝐴) = 0) | 
| 11 | 10 | oveq2d 7448 | . . . . . 6
⊢ (𝐴 = 0ℎ →
((normfn‘𝑇) ·
(normℎ‘𝐴)) = ((normfn‘𝑇) · 0)) | 
| 12 | 2 | simpri 485 | . . . . . . . 8
⊢
(normfn‘𝑇) ∈ ℝ | 
| 13 | 12 | recni 11276 | . . . . . . 7
⊢
(normfn‘𝑇) ∈ ℂ | 
| 14 | 13 | mul01i 11452 | . . . . . 6
⊢
((normfn‘𝑇) · 0) = 0 | 
| 15 | 11, 14 | eqtr2di 2793 | . . . . 5
⊢ (𝐴 = 0ℎ → 0
= ((normfn‘𝑇) ·
(normℎ‘𝐴))) | 
| 16 | 7, 15 | breqtrid 5179 | . . . 4
⊢ (𝐴 = 0ℎ → 0
≤ ((normfn‘𝑇) ·
(normℎ‘𝐴))) | 
| 17 | 6, 16 | eqbrtrd 5164 | . . 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 | 3 | lnfnfi 32061 | . . . . . . . . . 10
⊢ 𝑇:
ℋ⟶ℂ | 
| 20 | 19 | ffvelcdmi 7102 | . . . . . . . . 9
⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ) | 
| 21 | 20 | abscld 15476 | . . . . . . . 8
⊢ (𝐴 ∈ ℋ →
(abs‘(𝑇‘𝐴)) ∈
ℝ) | 
| 22 | 21 | adantr 480 | . . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (abs‘(𝑇‘𝐴)) ∈ ℝ) | 
| 23 | 22 | recnd 11290 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (abs‘(𝑇‘𝐴)) ∈ ℂ) | 
| 24 |  | normcl 31145 | . . . . . . . 8
⊢ (𝐴 ∈ ℋ →
(normℎ‘𝐴) ∈ ℝ) | 
| 25 | 24 | adantr 480 | . . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘𝐴) ∈ ℝ) | 
| 26 | 25 | recnd 11290 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘𝐴) ∈ ℂ) | 
| 27 |  | normne0 31150 | . . . . . . 7
⊢ (𝐴 ∈ ℋ →
((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠
0ℎ)) | 
| 28 | 27 | biimpar 477 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘𝐴) ≠ 0) | 
| 29 | 23, 26, 28 | divrec2d 12048 | . . . . 5
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = ((1 /
(normℎ‘𝐴)) · (abs‘(𝑇‘𝐴)))) | 
| 30 | 25, 28 | rereccld 12095 | . . . . . . . . 9
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (1 / (normℎ‘𝐴)) ∈ ℝ) | 
| 31 | 30 | recnd 11290 | . . . . . . . 8
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (1 / (normℎ‘𝐴)) ∈ ℂ) | 
| 32 |  | simpl 482 | . . . . . . . 8
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ 𝐴 ∈
ℋ) | 
| 33 | 3 | lnfnmuli 32064 | . . . . . . . 8
⊢ (((1 /
(normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 /
(normℎ‘𝐴)) · (𝑇‘𝐴))) | 
| 34 | 31, 32, 33 | syl2anc 584 | . . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 /
(normℎ‘𝐴)) · (𝑇‘𝐴))) | 
| 35 | 34 | fveq2d 6909 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (abs‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴))) = (abs‘((1 /
(normℎ‘𝐴)) · (𝑇‘𝐴)))) | 
| 36 | 20 | adantr 480 | . . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (𝑇‘𝐴) ∈
ℂ) | 
| 37 | 31, 36 | absmuld 15494 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (abs‘((1 / (normℎ‘𝐴)) · (𝑇‘𝐴))) = ((abs‘(1 /
(normℎ‘𝐴))) · (abs‘(𝑇‘𝐴)))) | 
| 38 |  | normgt0 31147 | . . . . . . . . . . 11
⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ
↔ 0 < (normℎ‘𝐴))) | 
| 39 | 38 | biimpa 476 | . . . . . . . . . 10
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ 0 < (normℎ‘𝐴)) | 
| 40 | 25, 39 | recgt0d 12203 | . . . . . . . . 9
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ 0 < (1 / (normℎ‘𝐴))) | 
| 41 |  | 0re 11264 | . . . . . . . . . 10
⊢ 0 ∈
ℝ | 
| 42 |  | ltle 11350 | . . . . . . . . . 10
⊢ ((0
∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 /
(normℎ‘𝐴)) → 0 ≤ (1 /
(normℎ‘𝐴)))) | 
| 43 | 41, 42 | mpan 690 | . . . . . . . . 9
⊢ ((1 /
(normℎ‘𝐴)) ∈ ℝ → (0 < (1 /
(normℎ‘𝐴)) → 0 ≤ (1 /
(normℎ‘𝐴)))) | 
| 44 | 30, 40, 43 | sylc 65 | . . . . . . . 8
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ 0 ≤ (1 / (normℎ‘𝐴))) | 
| 45 | 30, 44 | absidd 15462 | . . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (abs‘(1 / (normℎ‘𝐴))) = (1 /
(normℎ‘𝐴))) | 
| 46 | 45 | oveq1d 7447 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ ((abs‘(1 / (normℎ‘𝐴))) · (abs‘(𝑇‘𝐴))) = ((1 /
(normℎ‘𝐴)) · (abs‘(𝑇‘𝐴)))) | 
| 47 | 35, 37, 46 | 3eqtrrd 2781 | . . . . 5
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))) = (abs‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)))) | 
| 48 | 29, 47 | eqtrd 2776 | . . . 4
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = (abs‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)))) | 
| 49 |  | hvmulcl 31033 | . . . . . 6
⊢ (((1 /
(normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 /
(normℎ‘𝐴)) ·ℎ 𝐴) ∈
ℋ) | 
| 50 | 31, 32, 49 | syl2anc 584 | . . . . 5
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈
ℋ) | 
| 51 |  | normcl 31145 | . . . . . . 7
⊢ (((1 /
(normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ →
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ 𝐴)) ∈ ℝ) | 
| 52 | 50, 51 | syl 17 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) ∈
ℝ) | 
| 53 |  | norm1 31269 | . . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) = 1) | 
| 54 |  | eqle 11364 | . . . . . 6
⊢
(((normℎ‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ ∧
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ 𝐴)) = 1) →
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ 𝐴)) ≤ 1) | 
| 55 | 52, 53, 54 | syl2anc 584 | . . . . 5
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1) | 
| 56 |  | nmfnlb 31944 | . . . . . 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)) | 
| 57 | 19, 56 | mp3an1 1449 | . . . . 5
⊢ ((((1 /
(normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ ∧
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴))) ≤
(normfn‘𝑇)) | 
| 58 | 50, 55, 57 | syl2anc 584 | . . . 4
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (abs‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴))) ≤
(normfn‘𝑇)) | 
| 59 | 48, 58 | eqbrtrd 5164 | . . 3
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤
(normfn‘𝑇)) | 
| 60 | 12 | a1i 11 | . . . 4
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normfn‘𝑇) ∈ ℝ) | 
| 61 |  | ledivmul2 12148 | . . . 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)))) | 
| 62 | 22, 60, 25, 39, 61 | syl112anc 1375 | . . 3
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤
(normfn‘𝑇)
↔ (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) ·
(normℎ‘𝐴)))) | 
| 63 | 59, 62 | mpbid 232 | . 2
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) ·
(normℎ‘𝐴))) | 
| 64 | 18, 63 | pm2.61dane 3028 | 1
⊢ (𝐴 ∈ ℋ →
(abs‘(𝑇‘𝐴)) ≤
((normfn‘𝑇) ·
(normℎ‘𝐴))) |