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

Theorem nmopub 30899
Description: An upper bound for an operator norm. (Contributed by NM, 7-Mar-2006.) (New usage is discouraged.)
Assertion
Ref Expression
nmopub ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑇

Proof of Theorem nmopub
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmopval 30847 . . . 4 (𝑇: ℋ⟶ ℋ → (normop𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ))
21adantr 482 . . 3 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → (normop𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ))
32breq1d 5119 . 2 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴))
4 nmopsetretALT 30854 . . . . 5 (𝑇: ℋ⟶ ℋ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))} ⊆ ℝ)
5 ressxr 11207 . . . . 5 ℝ ⊆ ℝ*
64, 5sstrdi 3960 . . . 4 (𝑇: ℋ⟶ ℋ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))} ⊆ ℝ*)
7 supxrleub 13254 . . . 4 (({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))} ⊆ ℝ*𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}𝑧𝐴))
86, 7sylan 581 . . 3 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}𝑧𝐴))
9 ancom 462 . . . . . . 7 (((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥))) ↔ (𝑦 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1))
10 eqeq1 2737 . . . . . . . 8 (𝑦 = 𝑧 → (𝑦 = (norm‘(𝑇𝑥)) ↔ 𝑧 = (norm‘(𝑇𝑥))))
1110anbi1d 631 . . . . . . 7 (𝑦 = 𝑧 → ((𝑦 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) ↔ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
129, 11bitrid 283 . . . . . 6 (𝑦 = 𝑧 → (((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥))) ↔ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
1312rexbidv 3172 . . . . 5 (𝑦 = 𝑧 → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥))) ↔ ∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
1413ralab 3653 . . . 4 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
15 ralcom4 3268 . . . . 5 (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧𝑥 ∈ ℋ ((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
16 impexp 452 . . . . . . . 8 (((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ (𝑧 = (norm‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)))
1716albii 1822 . . . . . . 7 (∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧(𝑧 = (norm‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)))
18 fvex 6859 . . . . . . . 8 (norm‘(𝑇𝑥)) ∈ V
19 breq1 5112 . . . . . . . . 9 (𝑧 = (norm‘(𝑇𝑥)) → (𝑧𝐴 ↔ (norm‘(𝑇𝑥)) ≤ 𝐴))
2019imbi2d 341 . . . . . . . 8 (𝑧 = (norm‘(𝑇𝑥)) → (((norm𝑥) ≤ 1 → 𝑧𝐴) ↔ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴)))
2118, 20ceqsalv 3483 . . . . . . 7 (∀𝑧(𝑧 = (norm‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)) ↔ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
2217, 21bitri 275 . . . . . 6 (∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
2322ralbii 3093 . . . . 5 (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
24 r19.23v 3176 . . . . . 6 (∀𝑥 ∈ ℋ ((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ (∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2524albii 1822 . . . . 5 (∀𝑧𝑥 ∈ ℋ ((𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2615, 23, 253bitr3i 301 . . . 4 (∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (norm‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2714, 26bitr4i 278 . . 3 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴))
288, 27bitrdi 287 . 2 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (norm‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴)))
293, 28bitrd 279 1 ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wrex 3070  wss 3914   class class class wbr 5109  wf 6496  cfv 6500  supcsup 9384  cr 11058  1c1 11060  *cxr 11196   < clt 11197  cle 11198  chba 29910  normcno 29914  normopcnop 29936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137  ax-hilex 29990  ax-hv0cl 29994  ax-hvmul0 30001  ax-hfi 30070  ax-his1 30073  ax-his3 30075  ax-his4 30076
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-sup 9386  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-n0 12422  df-z 12508  df-uz 12772  df-rp 12924  df-seq 13916  df-exp 13977  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-hnorm 29959  df-nmop 30830
This theorem is referenced by:  nmopub2tALT  30900  nmophmi  31022  nmopadjlem  31080  nmoptrii  31085  nmopcoi  31086  nmopcoadji  31092
  Copyright terms: Public domain W3C validator