| Step | Hyp | Ref
| Expression |
| 1 | | nmopcoadj.1 |
. . . . . . 7
⊢ 𝑇 ∈
BndLinOp |
| 2 | | adjbdlnb 32103 |
. . . . . . 7
⊢ (𝑇 ∈ BndLinOp ↔
(adjℎ‘𝑇) ∈ BndLinOp) |
| 3 | 1, 2 | mpbi 230 |
. . . . . 6
⊢
(adjℎ‘𝑇) ∈ BndLinOp |
| 4 | | bdopf 31881 |
. . . . . 6
⊢
((adjℎ‘𝑇) ∈ BndLinOp →
(adjℎ‘𝑇): ℋ⟶ ℋ) |
| 5 | 3, 4 | ax-mp 5 |
. . . . 5
⊢
(adjℎ‘𝑇): ℋ⟶ ℋ |
| 6 | | bdopf 31881 |
. . . . . 6
⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶
ℋ) |
| 7 | 1, 6 | ax-mp 5 |
. . . . 5
⊢ 𝑇: ℋ⟶
ℋ |
| 8 | 5, 7 | hocofi 31785 |
. . . 4
⊢
((adjℎ‘𝑇) ∘ 𝑇): ℋ⟶ ℋ |
| 9 | | nmopre 31889 |
. . . . . . 7
⊢ (𝑇 ∈ BndLinOp →
(normop‘𝑇)
∈ ℝ) |
| 10 | 1, 9 | ax-mp 5 |
. . . . . 6
⊢
(normop‘𝑇) ∈ ℝ |
| 11 | 10 | resqcli 14225 |
. . . . 5
⊢
((normop‘𝑇)↑2) ∈ ℝ |
| 12 | | rexr 11307 |
. . . . 5
⊢
(((normop‘𝑇)↑2) ∈ ℝ →
((normop‘𝑇)↑2) ∈
ℝ*) |
| 13 | 11, 12 | ax-mp 5 |
. . . 4
⊢
((normop‘𝑇)↑2) ∈
ℝ* |
| 14 | | nmopub 31927 |
. . . 4
⊢
((((adjℎ‘𝑇) ∘ 𝑇): ℋ⟶ ℋ ∧
((normop‘𝑇)↑2) ∈ ℝ*) →
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ≤ ((normop‘𝑇)↑2) ↔ ∀𝑥 ∈ ℋ
((normℎ‘𝑥) ≤ 1 →
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥)) ≤ ((normop‘𝑇)↑2)))) |
| 15 | 8, 13, 14 | mp2an 692 |
. . 3
⊢
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ≤ ((normop‘𝑇)↑2) ↔ ∀𝑥 ∈ ℋ
((normℎ‘𝑥) ≤ 1 →
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥)) ≤ ((normop‘𝑇)↑2))) |
| 16 | 5, 7 | hocoi 31783 |
. . . . . . . 8
⊢ (𝑥 ∈ ℋ →
(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥) = ((adjℎ‘𝑇)‘(𝑇‘𝑥))) |
| 17 | 16 | fveq2d 6910 |
. . . . . . 7
⊢ (𝑥 ∈ ℋ →
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥)) =
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥)))) |
| 18 | 17 | adantr 480 |
. . . . . 6
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥)) =
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥)))) |
| 19 | 7 | ffvelcdmi 7103 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
| 20 | 5 | ffvelcdmi 7103 |
. . . . . . . . 9
⊢ ((𝑇‘𝑥) ∈ ℋ →
((adjℎ‘𝑇)‘(𝑇‘𝑥)) ∈ ℋ) |
| 21 | | normcl 31144 |
. . . . . . . . 9
⊢
(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ∈ ℋ →
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ∈ ℝ) |
| 22 | 19, 20, 21 | 3syl 18 |
. . . . . . . 8
⊢ (𝑥 ∈ ℋ →
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ∈ ℝ) |
| 23 | 22 | adantr 480 |
. . . . . . 7
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ∈ ℝ) |
| 24 | | nmopre 31889 |
. . . . . . . . . 10
⊢
((adjℎ‘𝑇) ∈ BndLinOp →
(normop‘(adjℎ‘𝑇)) ∈ ℝ) |
| 25 | 3, 24 | ax-mp 5 |
. . . . . . . . 9
⊢
(normop‘(adjℎ‘𝑇)) ∈ ℝ |
| 26 | | normcl 31144 |
. . . . . . . . . 10
⊢ ((𝑇‘𝑥) ∈ ℋ →
(normℎ‘(𝑇‘𝑥)) ∈ ℝ) |
| 27 | 19, 26 | syl 17 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℋ →
(normℎ‘(𝑇‘𝑥)) ∈ ℝ) |
| 28 | | remulcl 11240 |
. . . . . . . . 9
⊢
(((normop‘(adjℎ‘𝑇)) ∈ ℝ ∧
(normℎ‘(𝑇‘𝑥)) ∈ ℝ) →
((normop‘(adjℎ‘𝑇)) ·
(normℎ‘(𝑇‘𝑥))) ∈ ℝ) |
| 29 | 25, 27, 28 | sylancr 587 |
. . . . . . . 8
⊢ (𝑥 ∈ ℋ →
((normop‘(adjℎ‘𝑇)) ·
(normℎ‘(𝑇‘𝑥))) ∈ ℝ) |
| 30 | 29 | adantr 480 |
. . . . . . 7
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normop‘(adjℎ‘𝑇)) ·
(normℎ‘(𝑇‘𝑥))) ∈ ℝ) |
| 31 | 25, 10 | remulcli 11277 |
. . . . . . . 8
⊢
((normop‘(adjℎ‘𝑇)) ·
(normop‘𝑇)) ∈ ℝ |
| 32 | 31 | a1i 11 |
. . . . . . 7
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normop‘(adjℎ‘𝑇)) · (normop‘𝑇)) ∈
ℝ) |
| 33 | 3 | nmbdoplbi 32043 |
. . . . . . . . 9
⊢ ((𝑇‘𝑥) ∈ ℋ →
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ≤
((normop‘(adjℎ‘𝑇)) ·
(normℎ‘(𝑇‘𝑥)))) |
| 34 | 19, 33 | syl 17 |
. . . . . . . 8
⊢ (𝑥 ∈ ℋ →
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ≤
((normop‘(adjℎ‘𝑇)) ·
(normℎ‘(𝑇‘𝑥)))) |
| 35 | 34 | adantr 480 |
. . . . . . 7
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ≤
((normop‘(adjℎ‘𝑇)) ·
(normℎ‘(𝑇‘𝑥)))) |
| 36 | 27 | adantr 480 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normℎ‘(𝑇‘𝑥)) ∈ ℝ) |
| 37 | 10 | a1i 11 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normop‘𝑇)
∈ ℝ) |
| 38 | | normcl 31144 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℋ →
(normℎ‘𝑥) ∈ ℝ) |
| 39 | | remulcl 11240 |
. . . . . . . . . . 11
⊢
(((normop‘𝑇) ∈ ℝ ∧
(normℎ‘𝑥) ∈ ℝ) →
((normop‘𝑇) ·
(normℎ‘𝑥)) ∈ ℝ) |
| 40 | 10, 38, 39 | sylancr 587 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℋ →
((normop‘𝑇) ·
(normℎ‘𝑥)) ∈ ℝ) |
| 41 | 40 | adantr 480 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normop‘𝑇) ·
(normℎ‘𝑥)) ∈ ℝ) |
| 42 | 1 | nmbdoplbi 32043 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℋ →
(normℎ‘(𝑇‘𝑥)) ≤ ((normop‘𝑇) ·
(normℎ‘𝑥))) |
| 43 | 42 | adantr 480 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normℎ‘(𝑇‘𝑥)) ≤ ((normop‘𝑇) ·
(normℎ‘𝑥))) |
| 44 | | 1re 11261 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℝ |
| 45 | | nmopge0 31930 |
. . . . . . . . . . . . . . 15
⊢ (𝑇: ℋ⟶ ℋ →
0 ≤ (normop‘𝑇)) |
| 46 | 1, 6, 45 | mp2b 10 |
. . . . . . . . . . . . . 14
⊢ 0 ≤
(normop‘𝑇) |
| 47 | 10, 46 | pm3.2i 470 |
. . . . . . . . . . . . 13
⊢
((normop‘𝑇) ∈ ℝ ∧ 0 ≤
(normop‘𝑇)) |
| 48 | | lemul2a 12122 |
. . . . . . . . . . . . 13
⊢
((((normℎ‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧
((normop‘𝑇) ∈ ℝ ∧ 0 ≤
(normop‘𝑇))) ∧
(normℎ‘𝑥) ≤ 1) →
((normop‘𝑇) ·
(normℎ‘𝑥)) ≤ ((normop‘𝑇) · 1)) |
| 49 | 47, 48 | mp3anl3 1459 |
. . . . . . . . . . . 12
⊢
((((normℎ‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ)
∧ (normℎ‘𝑥) ≤ 1) →
((normop‘𝑇) ·
(normℎ‘𝑥)) ≤ ((normop‘𝑇) · 1)) |
| 50 | 44, 49 | mpanl2 701 |
. . . . . . . . . . 11
⊢
(((normℎ‘𝑥) ∈ ℝ ∧
(normℎ‘𝑥) ≤ 1) →
((normop‘𝑇) ·
(normℎ‘𝑥)) ≤ ((normop‘𝑇) · 1)) |
| 51 | 38, 50 | sylan 580 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normop‘𝑇) ·
(normℎ‘𝑥)) ≤ ((normop‘𝑇) · 1)) |
| 52 | 10 | recni 11275 |
. . . . . . . . . . 11
⊢
(normop‘𝑇) ∈ ℂ |
| 53 | 52 | mulridi 11265 |
. . . . . . . . . 10
⊢
((normop‘𝑇) · 1) =
(normop‘𝑇) |
| 54 | 51, 53 | breqtrdi 5184 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normop‘𝑇) ·
(normℎ‘𝑥)) ≤ (normop‘𝑇)) |
| 55 | 36, 41, 37, 43, 54 | letrd 11418 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇)) |
| 56 | | nmopge0 31930 |
. . . . . . . . . . 11
⊢
((adjℎ‘𝑇): ℋ⟶ ℋ → 0 ≤
(normop‘(adjℎ‘𝑇))) |
| 57 | 3, 4, 56 | mp2b 10 |
. . . . . . . . . 10
⊢ 0 ≤
(normop‘(adjℎ‘𝑇)) |
| 58 | 25, 57 | pm3.2i 470 |
. . . . . . . . 9
⊢
((normop‘(adjℎ‘𝑇)) ∈ ℝ ∧ 0 ≤
(normop‘(adjℎ‘𝑇))) |
| 59 | | lemul2a 12122 |
. . . . . . . . 9
⊢
((((normℎ‘(𝑇‘𝑥)) ∈ ℝ ∧
(normop‘𝑇)
∈ ℝ ∧
((normop‘(adjℎ‘𝑇)) ∈ ℝ ∧ 0 ≤
(normop‘(adjℎ‘𝑇)))) ∧
(normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇)) →
((normop‘(adjℎ‘𝑇)) ·
(normℎ‘(𝑇‘𝑥))) ≤
((normop‘(adjℎ‘𝑇)) · (normop‘𝑇))) |
| 60 | 58, 59 | mp3anl3 1459 |
. . . . . . . 8
⊢
((((normℎ‘(𝑇‘𝑥)) ∈ ℝ ∧
(normop‘𝑇)
∈ ℝ) ∧ (normℎ‘(𝑇‘𝑥)) ≤ (normop‘𝑇)) →
((normop‘(adjℎ‘𝑇)) ·
(normℎ‘(𝑇‘𝑥))) ≤
((normop‘(adjℎ‘𝑇)) · (normop‘𝑇))) |
| 61 | 36, 37, 55, 60 | syl21anc 838 |
. . . . . . 7
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normop‘(adjℎ‘𝑇)) ·
(normℎ‘(𝑇‘𝑥))) ≤
((normop‘(adjℎ‘𝑇)) · (normop‘𝑇))) |
| 62 | 23, 30, 32, 35, 61 | letrd 11418 |
. . . . . 6
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ≤
((normop‘(adjℎ‘𝑇)) · (normop‘𝑇))) |
| 63 | 18, 62 | eqbrtrd 5165 |
. . . . 5
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥)) ≤
((normop‘(adjℎ‘𝑇)) · (normop‘𝑇))) |
| 64 | 1 | nmopadji 32109 |
. . . . . . 7
⊢
(normop‘(adjℎ‘𝑇)) = (normop‘𝑇) |
| 65 | 64 | oveq1i 7441 |
. . . . . 6
⊢
((normop‘(adjℎ‘𝑇)) ·
(normop‘𝑇)) = ((normop‘𝑇) ·
(normop‘𝑇)) |
| 66 | 52 | sqvali 14219 |
. . . . . 6
⊢
((normop‘𝑇)↑2) = ((normop‘𝑇) ·
(normop‘𝑇)) |
| 67 | 65, 66 | eqtr4i 2768 |
. . . . 5
⊢
((normop‘(adjℎ‘𝑇)) ·
(normop‘𝑇)) = ((normop‘𝑇)↑2) |
| 68 | 63, 67 | breqtrdi 5184 |
. . . 4
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥)) ≤ ((normop‘𝑇)↑2)) |
| 69 | 68 | ex 412 |
. . 3
⊢ (𝑥 ∈ ℋ →
((normℎ‘𝑥) ≤ 1 →
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥)) ≤ ((normop‘𝑇)↑2))) |
| 70 | 15, 69 | mprgbir 3068 |
. 2
⊢
(normop‘((adjℎ‘𝑇) ∘ 𝑇)) ≤ ((normop‘𝑇)↑2) |
| 71 | | nmopge0 31930 |
. . . . . . . 8
⊢
(((adjℎ‘𝑇) ∘ 𝑇): ℋ⟶ ℋ → 0 ≤
(normop‘((adjℎ‘𝑇) ∘ 𝑇))) |
| 72 | 8, 71 | ax-mp 5 |
. . . . . . 7
⊢ 0 ≤
(normop‘((adjℎ‘𝑇) ∘ 𝑇)) |
| 73 | 3, 1 | bdopcoi 32117 |
. . . . . . . . 9
⊢
((adjℎ‘𝑇) ∘ 𝑇) ∈ BndLinOp |
| 74 | | nmopre 31889 |
. . . . . . . . 9
⊢
(((adjℎ‘𝑇) ∘ 𝑇) ∈ BndLinOp →
(normop‘((adjℎ‘𝑇) ∘ 𝑇)) ∈ ℝ) |
| 75 | 73, 74 | ax-mp 5 |
. . . . . . . 8
⊢
(normop‘((adjℎ‘𝑇) ∘ 𝑇)) ∈ ℝ |
| 76 | 75 | sqrtcli 15410 |
. . . . . . 7
⊢ (0 ≤
(normop‘((adjℎ‘𝑇) ∘ 𝑇)) →
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) ∈ ℝ) |
| 77 | | rexr 11307 |
. . . . . . 7
⊢
((√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) ∈ ℝ →
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) ∈ ℝ*) |
| 78 | 72, 76, 77 | mp2b 10 |
. . . . . 6
⊢
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) ∈ ℝ* |
| 79 | | nmopub 31927 |
. . . . . 6
⊢ ((𝑇: ℋ⟶ ℋ ∧
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) ∈ ℝ*) →
((normop‘𝑇) ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) ↔ ∀𝑥 ∈ ℋ
((normℎ‘𝑥) ≤ 1 →
(normℎ‘(𝑇‘𝑥)) ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇)))))) |
| 80 | 7, 78, 79 | mp2an 692 |
. . . . 5
⊢
((normop‘𝑇) ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) ↔ ∀𝑥 ∈ ℋ
((normℎ‘𝑥) ≤ 1 →
(normℎ‘(𝑇‘𝑥)) ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))))) |
| 81 | 19, 20 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℋ →
((adjℎ‘𝑇)‘(𝑇‘𝑥)) ∈ ℋ) |
| 82 | | hicl 31099 |
. . . . . . . . . . . 12
⊢
((((adjℎ‘𝑇)‘(𝑇‘𝑥)) ∈ ℋ ∧ 𝑥 ∈ ℋ) →
(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥) ∈
ℂ) |
| 83 | 81, 82 | mpancom 688 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℋ →
(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥) ∈
ℂ) |
| 84 | 83 | abscld 15475 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℋ →
(abs‘(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥)) ∈
ℝ) |
| 85 | 84 | adantr 480 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(abs‘(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥)) ∈
ℝ) |
| 86 | 22, 38 | remulcld 11291 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℋ →
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ·
(normℎ‘𝑥)) ∈ ℝ) |
| 87 | 86 | adantr 480 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ·
(normℎ‘𝑥)) ∈ ℝ) |
| 88 | 75 | a1i 11 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normop‘((adjℎ‘𝑇) ∘ 𝑇)) ∈ ℝ) |
| 89 | | bcs 31200 |
. . . . . . . . . . 11
⊢
((((adjℎ‘𝑇)‘(𝑇‘𝑥)) ∈ ℋ ∧ 𝑥 ∈ ℋ) →
(abs‘(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥)) ≤
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ·
(normℎ‘𝑥))) |
| 90 | 81, 89 | mpancom 688 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℋ →
(abs‘(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥)) ≤
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ·
(normℎ‘𝑥))) |
| 91 | 90 | adantr 480 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(abs‘(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥)) ≤
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ·
(normℎ‘𝑥))) |
| 92 | 5, 7 | hococli 31784 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℋ →
(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥) ∈ ℋ) |
| 93 | | normcl 31144 |
. . . . . . . . . . . 12
⊢
((((adjℎ‘𝑇) ∘ 𝑇)‘𝑥) ∈ ℋ →
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥)) ∈ ℝ) |
| 94 | 92, 93 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℋ →
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥)) ∈ ℝ) |
| 95 | 94 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥)) ∈ ℝ) |
| 96 | 38 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normℎ‘𝑥) ∈ ℝ) |
| 97 | | normge0 31145 |
. . . . . . . . . . . . . . 15
⊢
(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ∈ ℋ → 0 ≤
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥)))) |
| 98 | 19, 20, 97 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℋ → 0 ≤
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥)))) |
| 99 | 22, 98 | jca 511 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℋ →
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ∈ ℝ ∧ 0 ≤
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))))) |
| 100 | 99 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ∈ ℝ ∧ 0 ≤
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))))) |
| 101 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normℎ‘𝑥) ≤ 1) |
| 102 | | lemul2a 12122 |
. . . . . . . . . . . . 13
⊢
((((normℎ‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ∈ ℝ ∧ 0 ≤
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))))) ∧
(normℎ‘𝑥) ≤ 1) →
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ·
(normℎ‘𝑥)) ≤
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) · 1)) |
| 103 | 44, 102 | mp3anl2 1458 |
. . . . . . . . . . . 12
⊢
((((normℎ‘𝑥) ∈ ℝ ∧
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ∈ ℝ ∧ 0 ≤
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))))) ∧
(normℎ‘𝑥) ≤ 1) →
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ·
(normℎ‘𝑥)) ≤
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) · 1)) |
| 104 | 96, 100, 101, 103 | syl21anc 838 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ·
(normℎ‘𝑥)) ≤
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) · 1)) |
| 105 | 22 | recnd 11289 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℋ →
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ∈ ℂ) |
| 106 | 105 | mulridd 11278 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℋ →
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) · 1) =
(normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥)))) |
| 107 | 106, 17 | eqtr4d 2780 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℋ →
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) · 1) =
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥))) |
| 108 | 107 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) · 1) =
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥))) |
| 109 | 104, 108 | breqtrd 5169 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ·
(normℎ‘𝑥)) ≤
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥))) |
| 110 | | remulcl 11240 |
. . . . . . . . . . . . 13
⊢
(((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ∈ ℝ ∧
(normℎ‘𝑥) ∈ ℝ) →
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ·
(normℎ‘𝑥)) ∈ ℝ) |
| 111 | 75, 38, 110 | sylancr 587 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℋ →
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ·
(normℎ‘𝑥)) ∈ ℝ) |
| 112 | 111 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ·
(normℎ‘𝑥)) ∈ ℝ) |
| 113 | 73 | nmbdoplbi 32043 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℋ →
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥)) ≤
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ·
(normℎ‘𝑥))) |
| 114 | 113 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥)) ≤
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ·
(normℎ‘𝑥))) |
| 115 | 75, 72 | pm3.2i 470 |
. . . . . . . . . . . . . . 15
⊢
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ∈ ℝ ∧ 0 ≤
(normop‘((adjℎ‘𝑇) ∘ 𝑇))) |
| 116 | | lemul2a 12122 |
. . . . . . . . . . . . . . 15
⊢
((((normℎ‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ∈ ℝ ∧ 0 ≤
(normop‘((adjℎ‘𝑇) ∘ 𝑇)))) ∧
(normℎ‘𝑥) ≤ 1) →
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ·
(normℎ‘𝑥)) ≤
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) · 1)) |
| 117 | 115, 116 | mp3anl3 1459 |
. . . . . . . . . . . . . 14
⊢
((((normℎ‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ)
∧ (normℎ‘𝑥) ≤ 1) →
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ·
(normℎ‘𝑥)) ≤
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) · 1)) |
| 118 | 44, 117 | mpanl2 701 |
. . . . . . . . . . . . 13
⊢
(((normℎ‘𝑥) ∈ ℝ ∧
(normℎ‘𝑥) ≤ 1) →
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ·
(normℎ‘𝑥)) ≤
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) · 1)) |
| 119 | 38, 118 | sylan 580 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ·
(normℎ‘𝑥)) ≤
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) · 1)) |
| 120 | 75 | recni 11275 |
. . . . . . . . . . . . 13
⊢
(normop‘((adjℎ‘𝑇) ∘ 𝑇)) ∈ ℂ |
| 121 | 120 | mulridi 11265 |
. . . . . . . . . . . 12
⊢
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) · 1) =
(normop‘((adjℎ‘𝑇) ∘ 𝑇)) |
| 122 | 119, 121 | breqtrdi 5184 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ·
(normℎ‘𝑥)) ≤
(normop‘((adjℎ‘𝑇) ∘ 𝑇))) |
| 123 | 95, 112, 88, 114, 122 | letrd 11418 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normℎ‘(((adjℎ‘𝑇) ∘ 𝑇)‘𝑥)) ≤
(normop‘((adjℎ‘𝑇) ∘ 𝑇))) |
| 124 | 87, 95, 88, 109, 123 | letrd 11418 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normℎ‘((adjℎ‘𝑇)‘(𝑇‘𝑥))) ·
(normℎ‘𝑥)) ≤
(normop‘((adjℎ‘𝑇) ∘ 𝑇))) |
| 125 | 85, 87, 88, 91, 124 | letrd 11418 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(abs‘(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥)) ≤
(normop‘((adjℎ‘𝑇) ∘ 𝑇))) |
| 126 | | resqcl 14164 |
. . . . . . . . . . . 12
⊢
((normℎ‘(𝑇‘𝑥)) ∈ ℝ →
((normℎ‘(𝑇‘𝑥))↑2) ∈ ℝ) |
| 127 | | sqge0 14176 |
. . . . . . . . . . . 12
⊢
((normℎ‘(𝑇‘𝑥)) ∈ ℝ → 0 ≤
((normℎ‘(𝑇‘𝑥))↑2)) |
| 128 | 126, 127 | absidd 15461 |
. . . . . . . . . . 11
⊢
((normℎ‘(𝑇‘𝑥)) ∈ ℝ →
(abs‘((normℎ‘(𝑇‘𝑥))↑2)) =
((normℎ‘(𝑇‘𝑥))↑2)) |
| 129 | 19, 26, 128 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℋ →
(abs‘((normℎ‘(𝑇‘𝑥))↑2)) =
((normℎ‘(𝑇‘𝑥))↑2)) |
| 130 | | normsq 31153 |
. . . . . . . . . . . . 13
⊢ ((𝑇‘𝑥) ∈ ℋ →
((normℎ‘(𝑇‘𝑥))↑2) = ((𝑇‘𝑥) ·ih (𝑇‘𝑥))) |
| 131 | 19, 130 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℋ →
((normℎ‘(𝑇‘𝑥))↑2) = ((𝑇‘𝑥) ·ih (𝑇‘𝑥))) |
| 132 | | bdopadj 32101 |
. . . . . . . . . . . . . . . 16
⊢
((adjℎ‘𝑇) ∈ BndLinOp →
(adjℎ‘𝑇) ∈ dom
adjℎ) |
| 133 | 3, 132 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
(adjℎ‘𝑇) ∈ dom
adjℎ |
| 134 | | adj2 31953 |
. . . . . . . . . . . . . . 15
⊢
(((adjℎ‘𝑇) ∈ dom adjℎ ∧
(𝑇‘𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) →
(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥) = ((𝑇‘𝑥) ·ih
((adjℎ‘(adjℎ‘𝑇))‘𝑥))) |
| 135 | 133, 134 | mp3an1 1450 |
. . . . . . . . . . . . . 14
⊢ (((𝑇‘𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ) →
(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥) = ((𝑇‘𝑥) ·ih
((adjℎ‘(adjℎ‘𝑇))‘𝑥))) |
| 136 | 19, 135 | mpancom 688 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℋ →
(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥) = ((𝑇‘𝑥) ·ih
((adjℎ‘(adjℎ‘𝑇))‘𝑥))) |
| 137 | | bdopadj 32101 |
. . . . . . . . . . . . . . . 16
⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ dom
adjℎ) |
| 138 | | adjadj 31955 |
. . . . . . . . . . . . . . . 16
⊢ (𝑇 ∈ dom
adjℎ →
(adjℎ‘(adjℎ‘𝑇)) = 𝑇) |
| 139 | 1, 137, 138 | mp2b 10 |
. . . . . . . . . . . . . . 15
⊢
(adjℎ‘(adjℎ‘𝑇)) = 𝑇 |
| 140 | 139 | fveq1i 6907 |
. . . . . . . . . . . . . 14
⊢
((adjℎ‘(adjℎ‘𝑇))‘𝑥) = (𝑇‘𝑥) |
| 141 | 140 | oveq2i 7442 |
. . . . . . . . . . . . 13
⊢ ((𝑇‘𝑥) ·ih
((adjℎ‘(adjℎ‘𝑇))‘𝑥)) = ((𝑇‘𝑥) ·ih (𝑇‘𝑥)) |
| 142 | 136, 141 | eqtr2di 2794 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℋ → ((𝑇‘𝑥) ·ih (𝑇‘𝑥)) = (((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥)) |
| 143 | 131, 142 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℋ →
((normℎ‘(𝑇‘𝑥))↑2) =
(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥)) |
| 144 | 143 | fveq2d 6910 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℋ →
(abs‘((normℎ‘(𝑇‘𝑥))↑2)) =
(abs‘(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥))) |
| 145 | 129, 144 | eqtr3d 2779 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℋ →
((normℎ‘(𝑇‘𝑥))↑2) =
(abs‘(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥))) |
| 146 | 145 | adantr 480 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normℎ‘(𝑇‘𝑥))↑2) =
(abs‘(((adjℎ‘𝑇)‘(𝑇‘𝑥)) ·ih 𝑥))) |
| 147 | 75 | sqsqrti 15414 |
. . . . . . . . . 10
⊢ (0 ≤
(normop‘((adjℎ‘𝑇) ∘ 𝑇)) →
((√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇)))↑2) =
(normop‘((adjℎ‘𝑇) ∘ 𝑇))) |
| 148 | 8, 71, 147 | mp2b 10 |
. . . . . . . . 9
⊢
((√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇)))↑2) =
(normop‘((adjℎ‘𝑇) ∘ 𝑇)) |
| 149 | 148 | a1i 11 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇)))↑2) =
(normop‘((adjℎ‘𝑇) ∘ 𝑇))) |
| 150 | 125, 146,
149 | 3brtr4d 5175 |
. . . . . . 7
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normℎ‘(𝑇‘𝑥))↑2) ≤
((√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇)))↑2)) |
| 151 | | normge0 31145 |
. . . . . . . . . 10
⊢ ((𝑇‘𝑥) ∈ ℋ → 0 ≤
(normℎ‘(𝑇‘𝑥))) |
| 152 | 19, 151 | syl 17 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℋ → 0 ≤
(normℎ‘(𝑇‘𝑥))) |
| 153 | 8, 71, 76 | mp2b 10 |
. . . . . . . . . 10
⊢
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) ∈ ℝ |
| 154 | 75 | sqrtge0i 15415 |
. . . . . . . . . . 11
⊢ (0 ≤
(normop‘((adjℎ‘𝑇) ∘ 𝑇)) → 0 ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇)))) |
| 155 | 8, 71, 154 | mp2b 10 |
. . . . . . . . . 10
⊢ 0 ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) |
| 156 | | le2sq 14174 |
. . . . . . . . . 10
⊢
((((normℎ‘(𝑇‘𝑥)) ∈ ℝ ∧ 0 ≤
(normℎ‘(𝑇‘𝑥))) ∧
((√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) ∈ ℝ ∧ 0 ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))))) →
((normℎ‘(𝑇‘𝑥)) ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) ↔
((normℎ‘(𝑇‘𝑥))↑2) ≤
((√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇)))↑2))) |
| 157 | 153, 155,
156 | mpanr12 705 |
. . . . . . . . 9
⊢
(((normℎ‘(𝑇‘𝑥)) ∈ ℝ ∧ 0 ≤
(normℎ‘(𝑇‘𝑥))) →
((normℎ‘(𝑇‘𝑥)) ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) ↔
((normℎ‘(𝑇‘𝑥))↑2) ≤
((√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇)))↑2))) |
| 158 | 27, 152, 157 | syl2anc 584 |
. . . . . . . 8
⊢ (𝑥 ∈ ℋ →
((normℎ‘(𝑇‘𝑥)) ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) ↔
((normℎ‘(𝑇‘𝑥))↑2) ≤
((√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇)))↑2))) |
| 159 | 158 | adantr 480 |
. . . . . . 7
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
((normℎ‘(𝑇‘𝑥)) ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) ↔
((normℎ‘(𝑇‘𝑥))↑2) ≤
((√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇)))↑2))) |
| 160 | 150, 159 | mpbird 257 |
. . . . . 6
⊢ ((𝑥 ∈ ℋ ∧
(normℎ‘𝑥) ≤ 1) →
(normℎ‘(𝑇‘𝑥)) ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇)))) |
| 161 | 160 | ex 412 |
. . . . 5
⊢ (𝑥 ∈ ℋ →
((normℎ‘𝑥) ≤ 1 →
(normℎ‘(𝑇‘𝑥)) ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))))) |
| 162 | 80, 161 | mprgbir 3068 |
. . . 4
⊢
(normop‘𝑇) ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) |
| 163 | 10, 153 | le2sqi 14229 |
. . . . 5
⊢ ((0 ≤
(normop‘𝑇)
∧ 0 ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇)))) → ((normop‘𝑇) ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) ↔ ((normop‘𝑇)↑2) ≤
((√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇)))↑2))) |
| 164 | 46, 155, 163 | mp2an 692 |
. . . 4
⊢
((normop‘𝑇) ≤
(√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇))) ↔ ((normop‘𝑇)↑2) ≤
((√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇)))↑2)) |
| 165 | 162, 164 | mpbi 230 |
. . 3
⊢
((normop‘𝑇)↑2) ≤
((√‘(normop‘((adjℎ‘𝑇) ∘ 𝑇)))↑2) |
| 166 | 165, 148 | breqtri 5168 |
. 2
⊢
((normop‘𝑇)↑2) ≤
(normop‘((adjℎ‘𝑇) ∘ 𝑇)) |
| 167 | 75, 11 | letri3i 11377 |
. 2
⊢
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) = ((normop‘𝑇)↑2) ↔
((normop‘((adjℎ‘𝑇) ∘ 𝑇)) ≤ ((normop‘𝑇)↑2) ∧
((normop‘𝑇)↑2) ≤
(normop‘((adjℎ‘𝑇) ∘ 𝑇)))) |
| 168 | 70, 166, 167 | mpbir2an 711 |
1
⊢
(normop‘((adjℎ‘𝑇) ∘ 𝑇)) = ((normop‘𝑇)↑2) |