Proof of Theorem nmbdoplbi
| Step | Hyp | Ref
| Expression |
| 1 | | fveq2 6906 |
. . . 4
⊢ (𝐴 = 0ℎ →
(𝑇‘𝐴) = (𝑇‘0ℎ)) |
| 2 | 1 | fveq2d 6910 |
. . 3
⊢ (𝐴 = 0ℎ →
(normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘0ℎ))) |
| 3 | | fveq2 6906 |
. . . 4
⊢ (𝐴 = 0ℎ →
(normℎ‘𝐴) =
(normℎ‘0ℎ)) |
| 4 | 3 | oveq2d 7447 |
. . 3
⊢ (𝐴 = 0ℎ →
((normop‘𝑇) ·
(normℎ‘𝐴)) = ((normop‘𝑇) ·
(normℎ‘0ℎ))) |
| 5 | 2, 4 | breq12d 5156 |
. 2
⊢ (𝐴 = 0ℎ →
((normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) ·
(normℎ‘𝐴)) ↔
(normℎ‘(𝑇‘0ℎ)) ≤
((normop‘𝑇) ·
(normℎ‘0ℎ)))) |
| 6 | | nmbdoplb.1 |
. . . . . . . . . . . 12
⊢ 𝑇 ∈
BndLinOp |
| 7 | | bdopln 31880 |
. . . . . . . . . . . 12
⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp) |
| 8 | 6, 7 | ax-mp 5 |
. . . . . . . . . . 11
⊢ 𝑇 ∈ LinOp |
| 9 | 8 | lnopfi 31988 |
. . . . . . . . . 10
⊢ 𝑇: ℋ⟶
ℋ |
| 10 | 9 | ffvelcdmi 7103 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ) |
| 11 | | normcl 31144 |
. . . . . . . . 9
⊢ ((𝑇‘𝐴) ∈ ℋ →
(normℎ‘(𝑇‘𝐴)) ∈ ℝ) |
| 12 | 10, 11 | syl 17 |
. . . . . . . 8
⊢ (𝐴 ∈ ℋ →
(normℎ‘(𝑇‘𝐴)) ∈ ℝ) |
| 13 | 12 | adantr 480 |
. . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘(𝑇‘𝐴)) ∈ ℝ) |
| 14 | 13 | recnd 11289 |
. . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘(𝑇‘𝐴)) ∈ ℂ) |
| 15 | | normcl 31144 |
. . . . . . . 8
⊢ (𝐴 ∈ ℋ →
(normℎ‘𝐴) ∈ ℝ) |
| 16 | 15 | adantr 480 |
. . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘𝐴) ∈ ℝ) |
| 17 | 16 | recnd 11289 |
. . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘𝐴) ∈ ℂ) |
| 18 | | normne0 31149 |
. . . . . . 7
⊢ (𝐴 ∈ ℋ →
((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠
0ℎ)) |
| 19 | 18 | biimpar 477 |
. . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘𝐴) ≠ 0) |
| 20 | 14, 17, 19 | divrec2d 12047 |
. . . . 5
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = ((1 /
(normℎ‘𝐴)) ·
(normℎ‘(𝑇‘𝐴)))) |
| 21 | 16, 19 | rereccld 12094 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (1 / (normℎ‘𝐴)) ∈ ℝ) |
| 22 | 21 | recnd 11289 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (1 / (normℎ‘𝐴)) ∈ ℂ) |
| 23 | | simpl 482 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ 𝐴 ∈
ℋ) |
| 24 | 8 | lnopmuli 31991 |
. . . . . . . 8
⊢ (((1 /
(normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 /
(normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))) |
| 25 | 22, 23, 24 | syl2anc 584 |
. . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 /
(normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))) |
| 26 | 25 | fveq2d 6910 |
. . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴))) =
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ (𝑇‘𝐴)))) |
| 27 | 10 | adantr 480 |
. . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (𝑇‘𝐴) ∈
ℋ) |
| 28 | | norm-iii 31159 |
. . . . . . 7
⊢ (((1 /
(normℎ‘𝐴)) ∈ ℂ ∧ (𝑇‘𝐴) ∈ ℋ) →
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ (𝑇‘𝐴))) = ((abs‘(1 /
(normℎ‘𝐴))) ·
(normℎ‘(𝑇‘𝐴)))) |
| 29 | 22, 27, 28 | syl2anc 584 |
. . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘((1 /
(normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))) = ((abs‘(1 /
(normℎ‘𝐴))) ·
(normℎ‘(𝑇‘𝐴)))) |
| 30 | | normgt0 31146 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ
↔ 0 < (normℎ‘𝐴))) |
| 31 | 30 | biimpa 476 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ 0 < (normℎ‘𝐴)) |
| 32 | 16, 31 | recgt0d 12202 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ 0 < (1 / (normℎ‘𝐴))) |
| 33 | | 0re 11263 |
. . . . . . . . . 10
⊢ 0 ∈
ℝ |
| 34 | | ltle 11349 |
. . . . . . . . . 10
⊢ ((0
∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 /
(normℎ‘𝐴)) → 0 ≤ (1 /
(normℎ‘𝐴)))) |
| 35 | 33, 34 | mpan 690 |
. . . . . . . . 9
⊢ ((1 /
(normℎ‘𝐴)) ∈ ℝ → (0 < (1 /
(normℎ‘𝐴)) → 0 ≤ (1 /
(normℎ‘𝐴)))) |
| 36 | 21, 32, 35 | sylc 65 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ 0 ≤ (1 / (normℎ‘𝐴))) |
| 37 | 21, 36 | absidd 15461 |
. . . . . . 7
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (abs‘(1 / (normℎ‘𝐴))) = (1 /
(normℎ‘𝐴))) |
| 38 | 37 | oveq1d 7446 |
. . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ ((abs‘(1 / (normℎ‘𝐴))) ·
(normℎ‘(𝑇‘𝐴))) = ((1 /
(normℎ‘𝐴)) ·
(normℎ‘(𝑇‘𝐴)))) |
| 39 | 26, 29, 38 | 3eqtrrd 2782 |
. . . . 5
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ ((1 / (normℎ‘𝐴)) ·
(normℎ‘(𝑇‘𝐴))) = (normℎ‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)))) |
| 40 | 20, 39 | eqtrd 2777 |
. . . 4
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) =
(normℎ‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)))) |
| 41 | | hvmulcl 31032 |
. . . . . 6
⊢ (((1 /
(normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 /
(normℎ‘𝐴)) ·ℎ 𝐴) ∈
ℋ) |
| 42 | 22, 23, 41 | syl2anc 584 |
. . . . 5
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈
ℋ) |
| 43 | | normcl 31144 |
. . . . . . 7
⊢ (((1 /
(normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ →
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ 𝐴)) ∈ ℝ) |
| 44 | 42, 43 | syl 17 |
. . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) ∈
ℝ) |
| 45 | | norm1 31268 |
. . . . . 6
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) = 1) |
| 46 | | eqle 11363 |
. . . . . 6
⊢
(((normℎ‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ ∧
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ 𝐴)) = 1) →
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ 𝐴)) ≤ 1) |
| 47 | 44, 45, 46 | syl2anc 584 |
. . . . 5
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1) |
| 48 | | nmoplb 31926 |
. . . . . 6
⊢ ((𝑇: ℋ⟶ ℋ ∧
((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ ∧
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ 𝐴)) ≤ 1) →
(normℎ‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴))) ≤
(normop‘𝑇)) |
| 49 | 9, 48 | mp3an1 1450 |
. . . . 5
⊢ ((((1 /
(normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ ∧
(normℎ‘((1 / (normℎ‘𝐴))
·ℎ 𝐴)) ≤ 1) →
(normℎ‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴))) ≤
(normop‘𝑇)) |
| 50 | 42, 47, 49 | syl2anc 584 |
. . . 4
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘(𝑇‘((1 /
(normℎ‘𝐴)) ·ℎ 𝐴))) ≤
(normop‘𝑇)) |
| 51 | 40, 50 | eqbrtrd 5165 |
. . 3
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤
(normop‘𝑇)) |
| 52 | | nmopre 31889 |
. . . . . 6
⊢ (𝑇 ∈ BndLinOp →
(normop‘𝑇)
∈ ℝ) |
| 53 | 6, 52 | ax-mp 5 |
. . . . 5
⊢
(normop‘𝑇) ∈ ℝ |
| 54 | 53 | a1i 11 |
. . . 4
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normop‘𝑇) ∈ ℝ) |
| 55 | | ledivmul2 12147 |
. . . 4
⊢
(((normℎ‘(𝑇‘𝐴)) ∈ ℝ ∧
(normop‘𝑇)
∈ ℝ ∧ ((normℎ‘𝐴) ∈ ℝ ∧ 0 <
(normℎ‘𝐴))) →
(((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤
(normop‘𝑇)
↔ (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) ·
(normℎ‘𝐴)))) |
| 56 | 13, 54, 16, 31, 55 | syl112anc 1376 |
. . 3
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤
(normop‘𝑇)
↔ (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) ·
(normℎ‘𝐴)))) |
| 57 | 51, 56 | mpbid 232 |
. 2
⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ)
→ (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) ·
(normℎ‘𝐴))) |
| 58 | | 0le0 12367 |
. . . 4
⊢ 0 ≤
0 |
| 59 | 8 | lnop0i 31989 |
. . . . . 6
⊢ (𝑇‘0ℎ) =
0ℎ |
| 60 | 59 | fveq2i 6909 |
. . . . 5
⊢
(normℎ‘(𝑇‘0ℎ)) =
(normℎ‘0ℎ) |
| 61 | | norm0 31147 |
. . . . 5
⊢
(normℎ‘0ℎ) =
0 |
| 62 | 60, 61 | eqtri 2765 |
. . . 4
⊢
(normℎ‘(𝑇‘0ℎ)) =
0 |
| 63 | 61 | oveq2i 7442 |
. . . . 5
⊢
((normop‘𝑇) ·
(normℎ‘0ℎ)) =
((normop‘𝑇) · 0) |
| 64 | 53 | recni 11275 |
. . . . . 6
⊢
(normop‘𝑇) ∈ ℂ |
| 65 | 64 | mul01i 11451 |
. . . . 5
⊢
((normop‘𝑇) · 0) = 0 |
| 66 | 63, 65 | eqtri 2765 |
. . . 4
⊢
((normop‘𝑇) ·
(normℎ‘0ℎ)) = 0 |
| 67 | 58, 62, 66 | 3brtr4i 5173 |
. . 3
⊢
(normℎ‘(𝑇‘0ℎ)) ≤
((normop‘𝑇) ·
(normℎ‘0ℎ)) |
| 68 | 67 | a1i 11 |
. 2
⊢ (𝐴 ∈ ℋ →
(normℎ‘(𝑇‘0ℎ)) ≤
((normop‘𝑇) ·
(normℎ‘0ℎ))) |
| 69 | 5, 57, 68 | pm2.61ne 3027 |
1
⊢ (𝐴 ∈ ℋ →
(normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) ·
(normℎ‘𝐴))) |