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Theorem nmopun 29001
Description: Norm of a unitary Hilbert space operator. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
nmopun (( ℋ ≠ 0𝑇 ∈ UniOp) → (normop𝑇) = 1)

Proof of Theorem nmopun
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unoplin 28907 . . . . 5 (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2 lnopf 28846 . . . . 5 (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
31, 2syl 17 . . . 4 (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
4 nmopval 28843 . . . 4 (𝑇: ℋ⟶ ℋ → (normop𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}, ℝ*, < ))
53, 4syl 17 . . 3 (𝑇 ∈ UniOp → (normop𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}, ℝ*, < ))
65adantl 481 . 2 (( ℋ ≠ 0𝑇 ∈ UniOp) → (normop𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}, ℝ*, < ))
7 nmopsetretHIL 28851 . . . . . . 7 (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ)
8 ressxr 10121 . . . . . . 7 ℝ ⊆ ℝ*
97, 8syl6ss 3648 . . . . . 6 (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ*)
103, 9syl 17 . . . . 5 (𝑇 ∈ UniOp → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ*)
1110adantl 481 . . . 4 (( ℋ ≠ 0𝑇 ∈ UniOp) → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ*)
12 1re 10077 . . . . 5 1 ∈ ℝ
1312rexri 10135 . . . 4 1 ∈ ℝ*
1411, 13jctir 560 . . 3 (( ℋ ≠ 0𝑇 ∈ UniOp) → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ* ∧ 1 ∈ ℝ*))
15 vex 3234 . . . . . . 7 𝑧 ∈ V
16 eqeq1 2655 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 = (norm‘(𝑇𝑦)) ↔ 𝑧 = (norm‘(𝑇𝑦))))
1716anbi2d 740 . . . . . . . 8 (𝑥 = 𝑧 → (((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦)))))
1817rexbidv 3081 . . . . . . 7 (𝑥 = 𝑧 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦)))))
1915, 18elab 3382 . . . . . 6 (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦))))
20 unopnorm 28904 . . . . . . . . . . 11 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (norm‘(𝑇𝑦)) = (norm𝑦))
2120eqeq2d 2661 . . . . . . . . . 10 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (𝑧 = (norm‘(𝑇𝑦)) ↔ 𝑧 = (norm𝑦)))
2221anbi2d 740 . . . . . . . . 9 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ 𝑧 = (norm𝑦))))
23 breq1 4688 . . . . . . . . . 10 (𝑧 = (norm𝑦) → (𝑧 ≤ 1 ↔ (norm𝑦) ≤ 1))
2423biimparc 503 . . . . . . . . 9 (((norm𝑦) ≤ 1 ∧ 𝑧 = (norm𝑦)) → 𝑧 ≤ 1)
2522, 24syl6bi 243 . . . . . . . 8 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦))) → 𝑧 ≤ 1))
2625rexlimdva 3060 . . . . . . 7 (𝑇 ∈ UniOp → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦))) → 𝑧 ≤ 1))
2726imp 444 . . . . . 6 ((𝑇 ∈ UniOp ∧ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦)))) → 𝑧 ≤ 1)
2819, 27sylan2b 491 . . . . 5 ((𝑇 ∈ UniOp ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}) → 𝑧 ≤ 1)
2928ralrimiva 2995 . . . 4 (𝑇 ∈ UniOp → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}𝑧 ≤ 1)
3029adantl 481 . . 3 (( ℋ ≠ 0𝑇 ∈ UniOp) → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}𝑧 ≤ 1)
31 hne0 28534 . . . . . . . . . . 11 ( ℋ ≠ 0 ↔ ∃𝑦 ∈ ℋ 𝑦 ≠ 0)
32 norm1hex 28236 . . . . . . . . . . 11 (∃𝑦 ∈ ℋ 𝑦 ≠ 0 ↔ ∃𝑦 ∈ ℋ (norm𝑦) = 1)
3331, 32sylbb 209 . . . . . . . . . 10 ( ℋ ≠ 0 → ∃𝑦 ∈ ℋ (norm𝑦) = 1)
3433adantr 480 . . . . . . . . 9 (( ℋ ≠ 0𝑇 ∈ UniOp) → ∃𝑦 ∈ ℋ (norm𝑦) = 1)
35 1le1 10693 . . . . . . . . . . . . . 14 1 ≤ 1
36 breq1 4688 . . . . . . . . . . . . . 14 ((norm𝑦) = 1 → ((norm𝑦) ≤ 1 ↔ 1 ≤ 1))
3735, 36mpbiri 248 . . . . . . . . . . . . 13 ((norm𝑦) = 1 → (norm𝑦) ≤ 1)
3837a1i 11 . . . . . . . . . . . 12 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((norm𝑦) = 1 → (norm𝑦) ≤ 1))
3920adantr 480 . . . . . . . . . . . . . . 15 (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) = 1) → (norm‘(𝑇𝑦)) = (norm𝑦))
40 eqeq2 2662 . . . . . . . . . . . . . . . 16 ((norm𝑦) = 1 → ((norm‘(𝑇𝑦)) = (norm𝑦) ↔ (norm‘(𝑇𝑦)) = 1))
4140adantl 481 . . . . . . . . . . . . . . 15 (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) = 1) → ((norm‘(𝑇𝑦)) = (norm𝑦) ↔ (norm‘(𝑇𝑦)) = 1))
4239, 41mpbid 222 . . . . . . . . . . . . . 14 (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) = 1) → (norm‘(𝑇𝑦)) = 1)
4342eqcomd 2657 . . . . . . . . . . . . 13 (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) = 1) → 1 = (norm‘(𝑇𝑦)))
4443ex 449 . . . . . . . . . . . 12 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((norm𝑦) = 1 → 1 = (norm‘(𝑇𝑦))))
4538, 44jcad 554 . . . . . . . . . . 11 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((norm𝑦) = 1 → ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦)))))
4645adantll 750 . . . . . . . . . 10 ((( ℋ ≠ 0𝑇 ∈ UniOp) ∧ 𝑦 ∈ ℋ) → ((norm𝑦) = 1 → ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦)))))
4746reximdva 3046 . . . . . . . . 9 (( ℋ ≠ 0𝑇 ∈ UniOp) → (∃𝑦 ∈ ℋ (norm𝑦) = 1 → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦)))))
4834, 47mpd 15 . . . . . . . 8 (( ℋ ≠ 0𝑇 ∈ UniOp) → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦))))
49 1ex 10073 . . . . . . . . 9 1 ∈ V
50 eqeq1 2655 . . . . . . . . . . 11 (𝑥 = 1 → (𝑥 = (norm‘(𝑇𝑦)) ↔ 1 = (norm‘(𝑇𝑦))))
5150anbi2d 740 . . . . . . . . . 10 (𝑥 = 1 → (((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦)))))
5251rexbidv 3081 . . . . . . . . 9 (𝑥 = 1 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦)))))
5349, 52elab 3382 . . . . . . . 8 (1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦))))
5448, 53sylibr 224 . . . . . . 7 (( ℋ ≠ 0𝑇 ∈ UniOp) → 1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))})
5554adantr 480 . . . . . 6 ((( ℋ ≠ 0𝑇 ∈ UniOp) ∧ 𝑧 ∈ ℝ) → 1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))})
56 breq2 4689 . . . . . . 7 (𝑤 = 1 → (𝑧 < 𝑤𝑧 < 1))
5756rspcev 3340 . . . . . 6 ((1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ∧ 𝑧 < 1) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}𝑧 < 𝑤)
5855, 57sylan 487 . . . . 5 (((( ℋ ≠ 0𝑇 ∈ UniOp) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < 1) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}𝑧 < 𝑤)
5958ex 449 . . . 4 ((( ℋ ≠ 0𝑇 ∈ UniOp) ∧ 𝑧 ∈ ℝ) → (𝑧 < 1 → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}𝑧 < 𝑤))
6059ralrimiva 2995 . . 3 (( ℋ ≠ 0𝑇 ∈ UniOp) → ∀𝑧 ∈ ℝ (𝑧 < 1 → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}𝑧 < 𝑤))
61 supxr2 12182 . . 3 ((({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ* ∧ 1 ∈ ℝ*) ∧ (∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}𝑧 ≤ 1 ∧ ∀𝑧 ∈ ℝ (𝑧 < 1 → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}𝑧 < 𝑤))) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}, ℝ*, < ) = 1)
6214, 30, 60, 61syl12anc 1364 . 2 (( ℋ ≠ 0𝑇 ∈ UniOp) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}, ℝ*, < ) = 1)
636, 62eqtrd 2685 1 (( ℋ ≠ 0𝑇 ∈ UniOp) → (normop𝑇) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  wss 3607   class class class wbr 4685  wf 5922  cfv 5926  supcsup 8387  cr 9973  1c1 9975  *cxr 10111   < clt 10112  cle 10113  chil 27904  normcno 27908  0c0v 27909  0c0h 27920  normopcnop 27930  LinOpclo 27932  UniOpcuo 27934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-hilex 27984  ax-hfvadd 27985  ax-hvcom 27986  ax-hvass 27987  ax-hv0cl 27988  ax-hvaddid 27989  ax-hfvmul 27990  ax-hvmulid 27991  ax-hvmulass 27992  ax-hvdistr1 27993  ax-hvdistr2 27994  ax-hvmul0 27995  ax-hfi 28064  ax-his1 28067  ax-his2 28068  ax-his3 28069  ax-his4 28070
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-grpo 27475  df-gid 27476  df-ablo 27527  df-vc 27542  df-nv 27575  df-va 27578  df-ba 27579  df-sm 27580  df-0v 27581  df-nmcv 27583  df-hnorm 27953  df-hba 27954  df-hvsub 27956  df-hlim 27957  df-sh 28192  df-ch 28206  df-ch0 28238  df-nmop 28826  df-lnop 28828  df-unop 28830
This theorem is referenced by:  unopbd  29002  unierri  29091
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