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Theorem nmopun 30664
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 30570 . . . . 5 (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2 lnopf 30509 . . . . 5 (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
31, 2syl 17 . . . 4 (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
4 nmopval 30506 . . . 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 482 . 2 (( ℋ ≠ 0𝑇 ∈ UniOp) → (normop𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}, ℝ*, < ))
7 nmopsetretHIL 30514 . . . . . . 7 (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ)
8 ressxr 11120 . . . . . . 7 ℝ ⊆ ℝ*
97, 8sstrdi 3944 . . . . . 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 482 . . . 4 (( ℋ ≠ 0𝑇 ∈ UniOp) → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ*)
12 1xr 11135 . . . 4 1 ∈ ℝ*
1311, 12jctir 521 . . 3 (( ℋ ≠ 0𝑇 ∈ UniOp) → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ* ∧ 1 ∈ ℝ*))
14 vex 3445 . . . . . . 7 𝑧 ∈ V
15 eqeq1 2740 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥 = (norm‘(𝑇𝑦)) ↔ 𝑧 = (norm‘(𝑇𝑦))))
1615anbi2d 629 . . . . . . . 8 (𝑥 = 𝑧 → (((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦)))))
1716rexbidv 3171 . . . . . . 7 (𝑥 = 𝑧 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦)))))
1814, 17elab 3619 . . . . . 6 (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦))))
19 unopnorm 30567 . . . . . . . . . . 11 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (norm‘(𝑇𝑦)) = (norm𝑦))
2019eqeq2d 2747 . . . . . . . . . 10 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (𝑧 = (norm‘(𝑇𝑦)) ↔ 𝑧 = (norm𝑦)))
2120anbi2d 629 . . . . . . . . 9 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ 𝑧 = (norm𝑦))))
22 breq1 5095 . . . . . . . . . 10 (𝑧 = (norm𝑦) → (𝑧 ≤ 1 ↔ (norm𝑦) ≤ 1))
2322biimparc 480 . . . . . . . . 9 (((norm𝑦) ≤ 1 ∧ 𝑧 = (norm𝑦)) → 𝑧 ≤ 1)
2421, 23syl6bi 252 . . . . . . . 8 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → (((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦))) → 𝑧 ≤ 1))
2524rexlimdva 3148 . . . . . . 7 (𝑇 ∈ UniOp → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦))) → 𝑧 ≤ 1))
2625imp 407 . . . . . 6 ((𝑇 ∈ UniOp ∧ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦)))) → 𝑧 ≤ 1)
2718, 26sylan2b 594 . . . . 5 ((𝑇 ∈ UniOp ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}) → 𝑧 ≤ 1)
2827ralrimiva 3139 . . . 4 (𝑇 ∈ UniOp → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}𝑧 ≤ 1)
2928adantl 482 . . 3 (( ℋ ≠ 0𝑇 ∈ UniOp) → ∀𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}𝑧 ≤ 1)
30 hne0 30197 . . . . . . . . . . 11 ( ℋ ≠ 0 ↔ ∃𝑦 ∈ ℋ 𝑦 ≠ 0)
31 norm1hex 29901 . . . . . . . . . . 11 (∃𝑦 ∈ ℋ 𝑦 ≠ 0 ↔ ∃𝑦 ∈ ℋ (norm𝑦) = 1)
3230, 31sylbb 218 . . . . . . . . . 10 ( ℋ ≠ 0 → ∃𝑦 ∈ ℋ (norm𝑦) = 1)
3332adantr 481 . . . . . . . . 9 (( ℋ ≠ 0𝑇 ∈ UniOp) → ∃𝑦 ∈ ℋ (norm𝑦) = 1)
34 1le1 11704 . . . . . . . . . . . . . 14 1 ≤ 1
35 breq1 5095 . . . . . . . . . . . . . 14 ((norm𝑦) = 1 → ((norm𝑦) ≤ 1 ↔ 1 ≤ 1))
3634, 35mpbiri 257 . . . . . . . . . . . . 13 ((norm𝑦) = 1 → (norm𝑦) ≤ 1)
3736a1i 11 . . . . . . . . . . . 12 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((norm𝑦) = 1 → (norm𝑦) ≤ 1))
3819adantr 481 . . . . . . . . . . . . . . 15 (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) = 1) → (norm‘(𝑇𝑦)) = (norm𝑦))
39 eqeq2 2748 . . . . . . . . . . . . . . . 16 ((norm𝑦) = 1 → ((norm‘(𝑇𝑦)) = (norm𝑦) ↔ (norm‘(𝑇𝑦)) = 1))
4039adantl 482 . . . . . . . . . . . . . . 15 (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) = 1) → ((norm‘(𝑇𝑦)) = (norm𝑦) ↔ (norm‘(𝑇𝑦)) = 1))
4138, 40mpbid 231 . . . . . . . . . . . . . 14 (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) = 1) → (norm‘(𝑇𝑦)) = 1)
4241eqcomd 2742 . . . . . . . . . . . . 13 (((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) ∧ (norm𝑦) = 1) → 1 = (norm‘(𝑇𝑦)))
4342ex 413 . . . . . . . . . . . 12 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((norm𝑦) = 1 → 1 = (norm‘(𝑇𝑦))))
4437, 43jcad 513 . . . . . . . . . . 11 ((𝑇 ∈ UniOp ∧ 𝑦 ∈ ℋ) → ((norm𝑦) = 1 → ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦)))))
4544adantll 711 . . . . . . . . . 10 ((( ℋ ≠ 0𝑇 ∈ UniOp) ∧ 𝑦 ∈ ℋ) → ((norm𝑦) = 1 → ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦)))))
4645reximdva 3161 . . . . . . . . 9 (( ℋ ≠ 0𝑇 ∈ UniOp) → (∃𝑦 ∈ ℋ (norm𝑦) = 1 → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦)))))
4733, 46mpd 15 . . . . . . . 8 (( ℋ ≠ 0𝑇 ∈ UniOp) → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦))))
48 1ex 11072 . . . . . . . . 9 1 ∈ V
49 eqeq1 2740 . . . . . . . . . . 11 (𝑥 = 1 → (𝑥 = (norm‘(𝑇𝑦)) ↔ 1 = (norm‘(𝑇𝑦))))
5049anbi2d 629 . . . . . . . . . 10 (𝑥 = 1 → (((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦)))))
5150rexbidv 3171 . . . . . . . . 9 (𝑥 = 1 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦)))))
5248, 51elab 3619 . . . . . . . 8 (1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦))))
5347, 52sylibr 233 . . . . . . 7 (( ℋ ≠ 0𝑇 ∈ UniOp) → 1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))})
5453adantr 481 . . . . . 6 ((( ℋ ≠ 0𝑇 ∈ UniOp) ∧ 𝑧 ∈ ℝ) → 1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))})
55 breq2 5096 . . . . . . 7 (𝑤 = 1 → (𝑧 < 𝑤𝑧 < 1))
5655rspcev 3570 . . . . . 6 ((1 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ∧ 𝑧 < 1) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}𝑧 < 𝑤)
5754, 56sylan 580 . . . . 5 (((( ℋ ≠ 0𝑇 ∈ UniOp) ∧ 𝑧 ∈ ℝ) ∧ 𝑧 < 1) → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}𝑧 < 𝑤)
5857ex 413 . . . 4 ((( ℋ ≠ 0𝑇 ∈ UniOp) ∧ 𝑧 ∈ ℝ) → (𝑧 < 1 → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}𝑧 < 𝑤))
5958ralrimiva 3139 . . 3 (( ℋ ≠ 0𝑇 ∈ UniOp) → ∀𝑧 ∈ ℝ (𝑧 < 1 → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}𝑧 < 𝑤))
60 supxr2 13149 . . 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)
6113, 29, 59, 60syl12anc 834 . 2 (( ℋ ≠ 0𝑇 ∈ UniOp) → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}, ℝ*, < ) = 1)
626, 61eqtrd 2776 1 (( ℋ ≠ 0𝑇 ∈ UniOp) → (normop𝑇) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  {cab 2713  wne 2940  wral 3061  wrex 3070  wss 3898   class class class wbr 5092  wf 6475  cfv 6479  supcsup 9297  cr 10971  1c1 10973  *cxr 11109   < clt 11110  cle 11111  chba 29569  normcno 29573  0c0v 29574  0c0h 29585  normopcnop 29595  LinOpclo 29597  UniOpcuo 29599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-pre-sup 11050  ax-hilex 29649  ax-hfvadd 29650  ax-hvcom 29651  ax-hvass 29652  ax-hv0cl 29653  ax-hvaddid 29654  ax-hfvmul 29655  ax-hvmulid 29656  ax-hvmulass 29657  ax-hvdistr1 29658  ax-hvdistr2 29659  ax-hvmul0 29660  ax-hfi 29729  ax-his1 29732  ax-his2 29733  ax-his3 29734  ax-his4 29735
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-er 8569  df-map 8688  df-en 8805  df-dom 8806  df-sdom 8807  df-sup 9299  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-div 11734  df-nn 12075  df-2 12137  df-3 12138  df-4 12139  df-n0 12335  df-z 12421  df-uz 12684  df-rp 12832  df-seq 13823  df-exp 13884  df-cj 14909  df-re 14910  df-im 14911  df-sqrt 15045  df-abs 15046  df-grpo 29143  df-gid 29144  df-ablo 29195  df-vc 29209  df-nv 29242  df-va 29245  df-ba 29246  df-sm 29247  df-0v 29248  df-nmcv 29250  df-hnorm 29618  df-hba 29619  df-hvsub 29621  df-hlim 29622  df-sh 29857  df-ch 29871  df-ch0 29903  df-nmop 30489  df-lnop 30491  df-unop 30493
This theorem is referenced by:  unopbd  30665  unierri  30754
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