HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  branmfn Structured version   Visualization version   GIF version

Theorem branmfn 28166
Description: The norm of the bra function. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
branmfn (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (norm𝐴))

Proof of Theorem branmfn
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5986 . . . 4 (𝐴 = 0 → (bra‘𝐴) = (bra‘0))
21fveq2d 5990 . . 3 (𝐴 = 0 → (normfn‘(bra‘𝐴)) = (normfn‘(bra‘0)))
3 fveq2 5986 . . 3 (𝐴 = 0 → (norm𝐴) = (norm‘0))
42, 3eqeq12d 2529 . 2 (𝐴 = 0 → ((normfn‘(bra‘𝐴)) = (norm𝐴) ↔ (normfn‘(bra‘0)) = (norm‘0)))
5 brafn 28008 . . . . 5 (𝐴 ∈ ℋ → (bra‘𝐴): ℋ⟶ℂ)
6 nmfnval 27937 . . . . 5 ((bra‘𝐴): ℋ⟶ℂ → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
75, 6syl 17 . . . 4 (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
87adantr 479 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (normfn‘(bra‘𝐴)) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ))
9 nmfnsetre 27938 . . . . . . . 8 ((bra‘𝐴): ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ)
105, 9syl 17 . . . . . . 7 (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ)
11 ressxr 9836 . . . . . . 7 ℝ ⊆ ℝ*
1210, 11syl6ss 3484 . . . . . 6 (𝐴 ∈ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ*)
13 normcl 27184 . . . . . . 7 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
1413rexrd 9842 . . . . . 6 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ*)
1512, 14jca 552 . . . . 5 (𝐴 ∈ ℋ → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (norm𝐴) ∈ ℝ*))
1615adantr 479 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (norm𝐴) ∈ ℝ*))
17 vex 3080 . . . . . . . 8 𝑧 ∈ V
18 eqeq1 2518 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))))
1918anbi2d 735 . . . . . . . . 9 (𝑥 = 𝑧 → (((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
2019rexbidv 2938 . . . . . . . 8 (𝑥 = 𝑧 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))))
2117, 20elab 3223 . . . . . . 7 (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))))
22 id 22 . . . . . . . . . . . . 13 (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))
23 braval 28005 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((bra‘𝐴)‘𝑦) = (𝑦 ·ih 𝐴))
2423fveq2d 5990 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
2524adantr 479 . . . . . . . . . . . . 13 (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) → (abs‘((bra‘𝐴)‘𝑦)) = (abs‘(𝑦 ·ih 𝐴)))
2622, 25sylan9eqr 2570 . . . . . . . . . . . 12 ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 = (abs‘(𝑦 ·ih 𝐴)))
27 bcs2 27241 . . . . . . . . . . . . . . 15 ((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ (norm𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
28273expa 1256 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℋ ∧ 𝐴 ∈ ℋ) ∧ (norm𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
2928ancom1s 842 . . . . . . . . . . . . 13 (((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
3029adantr 479 . . . . . . . . . . . 12 ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → (abs‘(𝑦 ·ih 𝐴)) ≤ (norm𝐴))
3126, 30eqbrtrd 4503 . . . . . . . . . . 11 ((((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) ≤ 1) ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (norm𝐴))
3231exp41 635 . . . . . . . . . 10 (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → ((norm𝑦) ≤ 1 → (𝑧 = (abs‘((bra‘𝐴)‘𝑦)) → 𝑧 ≤ (norm𝐴)))))
3332imp4a 611 . . . . . . . . 9 (𝐴 ∈ ℋ → (𝑦 ∈ ℋ → (((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (norm𝐴))))
3433rexlimdv 2916 . . . . . . . 8 (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦))) → 𝑧 ≤ (norm𝐴)))
3534imp 443 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (abs‘((bra‘𝐴)‘𝑦)))) → 𝑧 ≤ (norm𝐴))
3621, 35sylan2b 490 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}) → 𝑧 ≤ (norm𝐴))
3736ralrimiva 2853 . . . . 5 (𝐴 ∈ ℋ → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (norm𝐴))
3837adantr 479 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (norm𝐴))
3913recnd 9821 . . . . . . . . . . . . . 14 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℂ)
4039adantr 479 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℂ)
41 normne0 27189 . . . . . . . . . . . . . 14 (𝐴 ∈ ℋ → ((norm𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
4241biimpar 500 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ≠ 0)
4340, 42reccld 10541 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℂ)
44 simpl 471 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℋ)
45 hvmulcl 27072 . . . . . . . . . . . 12 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
4643, 44, 45syl2anc 690 . . . . . . . . . . 11 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
47 norm1 27308 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
48 1le1 10402 . . . . . . . . . . . 12 1 ≤ 1
4947, 48syl6eqbr 4520 . . . . . . . . . . 11 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
50 ax-his3 27143 . . . . . . . . . . . . 13 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
5143, 44, 44, 50syl3anc 1317 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
5213adantr 479 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℝ)
5352, 42rereccld 10599 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℝ)
54 hiidrcl 27154 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℋ → (𝐴 ·ih 𝐴) ∈ ℝ)
5554adantr 479 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝐴 ·ih 𝐴) ∈ ℝ)
5653, 55remulcld 9823 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)) ∈ ℝ)
5751, 56eqeltrd 2592 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴) ∈ ℝ)
58 normgt0 27186 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
5958biimpa 499 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (norm𝐴))
6052, 59recgt0d 10706 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (1 / (norm𝐴)))
61 0re 9793 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ
62 ltle 9874 . . . . . . . . . . . . . . . . 17 ((0 ∈ ℝ ∧ (1 / (norm𝐴)) ∈ ℝ) → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
6361, 62mpan 701 . . . . . . . . . . . . . . . 16 ((1 / (norm𝐴)) ∈ ℝ → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
6453, 60, 63sylc 62 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (1 / (norm𝐴)))
65 hiidge0 27157 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℋ → 0 ≤ (𝐴 ·ih 𝐴))
6665adantr 479 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (𝐴 ·ih 𝐴))
6753, 55, 64, 66mulge0d 10351 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
6867, 51breqtrrd 4509 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))
6957, 68absidd 13864 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)) = (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))
7040, 42recid2d 10544 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · (norm𝐴)) = 1)
7170oveq2d 6441 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · ((1 / (norm𝐴)) · (norm𝐴))) = ((norm𝐴) · 1))
7240, 43, 40mul12d 9994 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · ((1 / (norm𝐴)) · (norm𝐴))) = ((1 / (norm𝐴)) · ((norm𝐴) · (norm𝐴))))
7339sqvald 12732 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℋ → ((norm𝐴)↑2) = ((norm𝐴) · (norm𝐴)))
74 normsq 27193 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ ℋ → ((norm𝐴)↑2) = (𝐴 ·ih 𝐴))
7573, 74eqtr3d 2550 . . . . . . . . . . . . . . . 16 (𝐴 ∈ ℋ → ((norm𝐴) · (norm𝐴)) = (𝐴 ·ih 𝐴))
7675adantr 479 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · (norm𝐴)) = (𝐴 ·ih 𝐴))
7776oveq2d 6441 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · ((norm𝐴) · (norm𝐴))) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
7872, 77eqtrd 2548 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · ((1 / (norm𝐴)) · (norm𝐴))) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
7939mulid1d 9810 . . . . . . . . . . . . . 14 (𝐴 ∈ ℋ → ((norm𝐴) · 1) = (norm𝐴))
8079adantr 479 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm𝐴) · 1) = (norm𝐴))
8171, 78, 803eqtr3rd 2557 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) = ((1 / (norm𝐴)) · (𝐴 ·ih 𝐴)))
8251, 69, 813eqtr4rd 2559 . . . . . . . . . . 11 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)))
83 fveq2 5986 . . . . . . . . . . . . . 14 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (norm𝑦) = (norm‘((1 / (norm𝐴)) · 𝐴)))
8483breq1d 4491 . . . . . . . . . . . . 13 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → ((norm𝑦) ≤ 1 ↔ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1))
85 oveq1 6432 . . . . . . . . . . . . . . 15 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (𝑦 ·ih 𝐴) = (((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))
8685fveq2d 5990 . . . . . . . . . . . . . 14 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (abs‘(𝑦 ·ih 𝐴)) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)))
8786eqeq2d 2524 . . . . . . . . . . . . 13 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → ((norm𝐴) = (abs‘(𝑦 ·ih 𝐴)) ↔ (norm𝐴) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴))))
8884, 87anbi12d 742 . . . . . . . . . . . 12 (𝑦 = ((1 / (norm𝐴)) · 𝐴) → (((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴))) ↔ ((norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1 ∧ (norm𝐴) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)))))
8988rspcev 3186 . . . . . . . . . . 11 ((((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ ((norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1 ∧ (norm𝐴) = (abs‘(((1 / (norm𝐴)) · 𝐴) ·ih 𝐴)))) → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴))))
9046, 49, 82, 89syl12anc 1315 . . . . . . . . . 10 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴))))
9124eqeq2d 2524 . . . . . . . . . . . . 13 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((norm𝐴) = (abs‘((bra‘𝐴)‘𝑦)) ↔ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴))))
9291anbi2d 735 . . . . . . . . . . . 12 ((𝐴 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
9392rexbidva 2935 . . . . . . . . . . 11 (𝐴 ∈ ℋ → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
9493adantr 479 . . . . . . . . . 10 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘(𝑦 ·ih 𝐴)))))
9590, 94mpbird 245 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
96 fvex 5996 . . . . . . . . . 10 (norm𝐴) ∈ V
97 eqeq1 2518 . . . . . . . . . . . 12 (𝑥 = (norm𝐴) → (𝑥 = (abs‘((bra‘𝐴)‘𝑦)) ↔ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
9897anbi2d 735 . . . . . . . . . . 11 (𝑥 = (norm𝐴) → (((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
9998rexbidv 2938 . . . . . . . . . 10 (𝑥 = (norm𝐴) → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦)))))
10096, 99elab 3223 . . . . . . . . 9 ((norm𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (norm𝐴) = (abs‘((bra‘𝐴)‘𝑦))))
10195, 100sylibr 222 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))})
102 breq2 4485 . . . . . . . . 9 (𝑤 = (norm𝐴) → (𝑧 < 𝑤𝑧 < (norm𝐴)))
103102rspcev 3186 . . . . . . . 8 (((norm𝐴) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ∧ 𝑧 < (norm𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
104101, 103sylan 486 . . . . . . 7 (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ 𝑧 < (norm𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
105104adantlr 746 . . . . . 6 ((((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < (norm𝐴)) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤)
106105ex 448 . . . . 5 (((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) ∧ 𝑧 ∈ ℝ) → (𝑧 < (norm𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
107106ralrimiva 2853 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ∀𝑧 ∈ ℝ (𝑧 < (norm𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))
108 supxr2 11878 . . . 4 ((({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))} ⊆ ℝ* ∧ (norm𝐴) ∈ ℝ*) ∧ (∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 ≤ (norm𝐴) ∧ ∀𝑧 ∈ ℝ (𝑧 < (norm𝐴) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}𝑧 < 𝑤))) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ) = (norm𝐴))
10916, 38, 107, 108syl12anc 1315 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘((bra‘𝐴)‘𝑦)))}, ℝ*, < ) = (norm𝐴))
1108, 109eqtrd 2548 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (normfn‘(bra‘𝐴)) = (norm𝐴))
111 nmfn0 28048 . . . 4 (normfn‘( ℋ × {0})) = 0
112 bra0 28011 . . . . 5 (bra‘0) = ( ℋ × {0})
113112fveq2i 5989 . . . 4 (normfn‘(bra‘0)) = (normfn‘( ℋ × {0}))
114 norm0 27187 . . . 4 (norm‘0) = 0
115111, 113, 1143eqtr4i 2546 . . 3 (normfn‘(bra‘0)) = (norm‘0)
116115a1i 11 . 2 (𝐴 ∈ ℋ → (normfn‘(bra‘0)) = (norm‘0))
1174, 110, 116pm2.61ne 2771 1 (𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (norm𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1938  {cab 2500  wne 2684  wral 2800  wrex 2801  wss 3444  {csn 4028   class class class wbr 4481   × cxp 4930  wf 5685  cfv 5689  (class class class)co 6425  supcsup 8103  cc 9687  cr 9688  0cc0 9689  1c1 9690   · cmul 9694  *cxr 9826   < clt 9827  cle 9828   / cdiv 10431  2c2 10823  cexp 12587  abscabs 13677  chil 26978   · csm 26980   ·ih csp 26981  normcno 26982  0c0v 26983  normfncnmf 27010  bracbr 27015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721  ax-inf2 8295  ax-cnex 9745  ax-resscn 9746  ax-1cn 9747  ax-icn 9748  ax-addcl 9749  ax-addrcl 9750  ax-mulcl 9751  ax-mulrcl 9752  ax-mulcom 9753  ax-addass 9754  ax-mulass 9755  ax-distr 9756  ax-i2m1 9757  ax-1ne0 9758  ax-1rid 9759  ax-rnegex 9760  ax-rrecex 9761  ax-cnre 9762  ax-pre-lttri 9763  ax-pre-lttrn 9764  ax-pre-ltadd 9765  ax-pre-mulgt0 9766  ax-pre-sup 9767  ax-addf 9768  ax-mulf 9769  ax-hilex 27058  ax-hfvadd 27059  ax-hvcom 27060  ax-hvass 27061  ax-hv0cl 27062  ax-hvaddid 27063  ax-hfvmul 27064  ax-hvmulid 27065  ax-hvmulass 27066  ax-hvdistr1 27067  ax-hvdistr2 27068  ax-hvmul0 27069  ax-hfi 27138  ax-his1 27141  ax-his2 27142  ax-his3 27143  ax-his4 27144
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-iin 4356  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-se 4892  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-isom 5698  df-riota 6387  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-of 6669  df-om 6832  df-1st 6932  df-2nd 6933  df-supp 7056  df-wrecs 7167  df-recs 7229  df-rdg 7267  df-1o 7321  df-2o 7322  df-oadd 7325  df-er 7503  df-map 7620  df-ixp 7669  df-en 7716  df-dom 7717  df-sdom 7718  df-fin 7719  df-fsupp 8033  df-fi 8074  df-sup 8105  df-inf 8106  df-oi 8172  df-card 8522  df-cda 8747  df-pnf 9829  df-mnf 9830  df-xr 9831  df-ltxr 9832  df-le 9833  df-sub 10017  df-neg 10018  df-div 10432  df-nn 10774  df-2 10832  df-3 10833  df-4 10834  df-5 10835  df-6 10836  df-7 10837  df-8 10838  df-9 10839  df-10OLD 10840  df-n0 11046  df-z 11117  df-dec 11232  df-uz 11424  df-q 11527  df-rp 11571  df-xneg 11684  df-xadd 11685  df-xmul 11686  df-ioo 11916  df-icc 11919  df-fz 12063  df-fzo 12200  df-seq 12529  df-exp 12588  df-hash 12845  df-cj 13542  df-re 13543  df-im 13544  df-sqrt 13678  df-abs 13679  df-clim 13929  df-sum 14130  df-struct 15602  df-ndx 15603  df-slot 15604  df-base 15605  df-sets 15606  df-ress 15607  df-plusg 15686  df-mulr 15687  df-starv 15688  df-sca 15689  df-vsca 15690  df-ip 15691  df-tset 15692  df-ple 15693  df-ds 15696  df-unif 15697  df-hom 15698  df-cco 15699  df-rest 15811  df-topn 15812  df-0g 15830  df-gsum 15831  df-topgen 15832  df-pt 15833  df-prds 15836  df-xrs 15890  df-qtop 15896  df-imas 15897  df-xps 15900  df-mre 15982  df-mrc 15983  df-acs 15985  df-mgm 16978  df-sgrp 17020  df-mnd 17031  df-submnd 17072  df-mulg 17277  df-cntz 17486  df-cmn 17947  df-psmet 19484  df-xmet 19485  df-met 19486  df-bl 19487  df-mopn 19488  df-cnfld 19493  df-top 20445  df-bases 20446  df-topon 20447  df-topsp 20448  df-cld 20557  df-ntr 20558  df-cls 20559  df-cn 20765  df-cnp 20766  df-t1 20852  df-haus 20853  df-tx 21099  df-hmeo 21292  df-xms 21858  df-ms 21859  df-tms 21860  df-grpo 26499  df-gid 26500  df-ginv 26501  df-gdiv 26502  df-ablo 26554  df-vc 26569  df-nv 26617  df-va 26620  df-ba 26621  df-sm 26622  df-0v 26623  df-vs 26624  df-nmcv 26625  df-ims 26626  df-dip 26743  df-ph 26860  df-hnorm 27027  df-hba 27028  df-hvsub 27030  df-nmfn 27906  df-lnfn 27909  df-bra 27911
This theorem is referenced by:  brabn  28167
  Copyright terms: Public domain W3C validator