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Theorem nmopun 30956
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 30862 . . . . 5 (𝑇 ∈ UniOp → 𝑇 ∈ LinOp)
2 lnopf 30801 . . . . 5 (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)
31, 2syl 17 . . . 4 (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
4 nmopval 30798 . . . 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 30806 . . . . . . 7 (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ)
8 ressxr 11199 . . . . . . 7 ℝ ⊆ ℝ*
97, 8sstrdi 3956 . . . . . 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 11214 . . . 4 1 ∈ ℝ*
1311, 12jctir 521 . . 3 (( ℋ ≠ 0𝑇 ∈ UniOp) → ({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ* ∧ 1 ∈ ℝ*))
14 vex 3449 . . . . . . 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 3175 . . . . . . 7 (𝑥 = 𝑧 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦)))))
1814, 17elab 3630 . . . . . 6 (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑧 = (norm‘(𝑇𝑦))))
19 unopnorm 30859 . . . . . . . . . . 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 5108 . . . . . . . . . 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 3152 . . . . . . 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 3143 . . . 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 30489 . . . . . . . . . . 11 ( ℋ ≠ 0 ↔ ∃𝑦 ∈ ℋ 𝑦 ≠ 0)
31 norm1hex 30193 . . . . . . . . . . 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 11783 . . . . . . . . . . . . . 14 1 ≤ 1
35 breq1 5108 . . . . . . . . . . . . . 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 712 . . . . . . . . . 10 ((( ℋ ≠ 0𝑇 ∈ UniOp) ∧ 𝑦 ∈ ℋ) → ((norm𝑦) = 1 → ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦)))))
4645reximdva 3165 . . . . . . . . 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 11151 . . . . . . . . 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 3175 . . . . . . . . 9 (𝑥 = 1 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 1 = (norm‘(𝑇𝑦)))))
5248, 51elab 3630 . . . . . . . 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 5109 . . . . . . 7 (𝑤 = 1 → (𝑧 < 𝑤𝑧 < 1))
5655rspcev 3581 . . . . . 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 3143 . . 3 (( ℋ ≠ 0𝑇 ∈ UniOp) → ∀𝑧 ∈ ℝ (𝑧 < 1 → ∃𝑤 ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}𝑧 < 𝑤))
60 supxr2 13233 . . 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 835 . 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 1541  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  wss 3910   class class class wbr 5105  wf 6492  cfv 6496  supcsup 9376  cr 11050  1c1 11052  *cxr 11188   < clt 11189  cle 11190  chba 29861  normcno 29865  0c0v 29866  0c0h 29877  normopcnop 29887  LinOpclo 29889  UniOpcuo 29891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-hilex 29941  ax-hfvadd 29942  ax-hvcom 29943  ax-hvass 29944  ax-hv0cl 29945  ax-hvaddid 29946  ax-hfvmul 29947  ax-hvmulid 29948  ax-hvmulass 29949  ax-hvdistr1 29950  ax-hvdistr2 29951  ax-hvmul0 29952  ax-hfi 30021  ax-his1 30024  ax-his2 30025  ax-his3 30026  ax-his4 30027
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-grpo 29435  df-gid 29436  df-ablo 29487  df-vc 29501  df-nv 29534  df-va 29537  df-ba 29538  df-sm 29539  df-0v 29540  df-nmcv 29542  df-hnorm 29910  df-hba 29911  df-hvsub 29913  df-hlim 29914  df-sh 30149  df-ch 30163  df-ch0 30195  df-nmop 30781  df-lnop 30783  df-unop 30785
This theorem is referenced by:  unopbd  30957  unierri  31046
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