MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nmoubi Structured version   Visualization version   GIF version

Theorem nmoubi 30595
Description: An upper bound for an operator norm. (Contributed by NM, 11-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoubi.1 𝑋 = (BaseSetβ€˜π‘ˆ)
nmoubi.y π‘Œ = (BaseSetβ€˜π‘Š)
nmoubi.l 𝐿 = (normCVβ€˜π‘ˆ)
nmoubi.m 𝑀 = (normCVβ€˜π‘Š)
nmoubi.3 𝑁 = (π‘ˆ normOpOLD π‘Š)
nmoubi.u π‘ˆ ∈ NrmCVec
nmoubi.w π‘Š ∈ NrmCVec
Assertion
Ref Expression
nmoubi ((𝑇:π‘‹βŸΆπ‘Œ ∧ 𝐴 ∈ ℝ*) β†’ ((π‘β€˜π‘‡) ≀ 𝐴 ↔ βˆ€π‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 β†’ (π‘€β€˜(π‘‡β€˜π‘₯)) ≀ 𝐴)))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐿   π‘₯,π‘ˆ   π‘₯,π‘Š   π‘₯,π‘Œ   π‘₯,𝑀   π‘₯,𝑇   π‘₯,𝑋
Allowed substitution hint:   𝑁(π‘₯)

Proof of Theorem nmoubi
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmoubi.u . . . . . 6 π‘ˆ ∈ NrmCVec
2 nmoubi.w . . . . . 6 π‘Š ∈ NrmCVec
3 nmoubi.1 . . . . . . 7 𝑋 = (BaseSetβ€˜π‘ˆ)
4 nmoubi.y . . . . . . 7 π‘Œ = (BaseSetβ€˜π‘Š)
5 nmoubi.l . . . . . . 7 𝐿 = (normCVβ€˜π‘ˆ)
6 nmoubi.m . . . . . . 7 𝑀 = (normCVβ€˜π‘Š)
7 nmoubi.3 . . . . . . 7 𝑁 = (π‘ˆ normOpOLD π‘Š)
83, 4, 5, 6, 7nmooval 30586 . . . . . 6 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ (π‘β€˜π‘‡) = sup({𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)))}, ℝ*, < ))
91, 2, 8mp3an12 1448 . . . . 5 (𝑇:π‘‹βŸΆπ‘Œ β†’ (π‘β€˜π‘‡) = sup({𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)))}, ℝ*, < ))
109breq1d 5158 . . . 4 (𝑇:π‘‹βŸΆπ‘Œ β†’ ((π‘β€˜π‘‡) ≀ 𝐴 ↔ sup({𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)))}, ℝ*, < ) ≀ 𝐴))
1110adantr 480 . . 3 ((𝑇:π‘‹βŸΆπ‘Œ ∧ 𝐴 ∈ ℝ*) β†’ ((π‘β€˜π‘‡) ≀ 𝐴 ↔ sup({𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)))}, ℝ*, < ) ≀ 𝐴))
124, 6nmosetre 30587 . . . . . 6 ((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)))} βŠ† ℝ)
132, 12mpan 689 . . . . 5 (𝑇:π‘‹βŸΆπ‘Œ β†’ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)))} βŠ† ℝ)
14 ressxr 11289 . . . . 5 ℝ βŠ† ℝ*
1513, 14sstrdi 3992 . . . 4 (𝑇:π‘‹βŸΆπ‘Œ β†’ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)))} βŠ† ℝ*)
16 supxrleub 13338 . . . 4 (({𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)))} βŠ† ℝ* ∧ 𝐴 ∈ ℝ*) β†’ (sup({𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)))}, ℝ*, < ) ≀ 𝐴 ↔ βˆ€π‘§ ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)))}𝑧 ≀ 𝐴))
1715, 16sylan 579 . . 3 ((𝑇:π‘‹βŸΆπ‘Œ ∧ 𝐴 ∈ ℝ*) β†’ (sup({𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)))}, ℝ*, < ) ≀ 𝐴 ↔ βˆ€π‘§ ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)))}𝑧 ≀ 𝐴))
1811, 17bitrd 279 . 2 ((𝑇:π‘‹βŸΆπ‘Œ ∧ 𝐴 ∈ ℝ*) β†’ ((π‘β€˜π‘‡) ≀ 𝐴 ↔ βˆ€π‘§ ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)))}𝑧 ≀ 𝐴))
19 eqeq1 2732 . . . . . 6 (𝑦 = 𝑧 β†’ (𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)) ↔ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯))))
2019anbi2d 629 . . . . 5 (𝑦 = 𝑧 β†’ (((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯))) ↔ ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯)))))
2120rexbidv 3175 . . . 4 (𝑦 = 𝑧 β†’ (βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯))) ↔ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯)))))
2221ralab 3686 . . 3 (βˆ€π‘§ ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)))}𝑧 ≀ 𝐴 ↔ βˆ€π‘§(βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯))) β†’ 𝑧 ≀ 𝐴))
23 ralcom4 3280 . . . 4 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘§(((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯))) β†’ 𝑧 ≀ 𝐴) ↔ βˆ€π‘§βˆ€π‘₯ ∈ 𝑋 (((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯))) β†’ 𝑧 ≀ 𝐴))
24 ancomst 464 . . . . . . . 8 ((((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯))) β†’ 𝑧 ≀ 𝐴) ↔ ((𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯)) ∧ (πΏβ€˜π‘₯) ≀ 1) β†’ 𝑧 ≀ 𝐴))
25 impexp 450 . . . . . . . 8 (((𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯)) ∧ (πΏβ€˜π‘₯) ≀ 1) β†’ 𝑧 ≀ 𝐴) ↔ (𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯)) β†’ ((πΏβ€˜π‘₯) ≀ 1 β†’ 𝑧 ≀ 𝐴)))
2624, 25bitri 275 . . . . . . 7 ((((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯))) β†’ 𝑧 ≀ 𝐴) ↔ (𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯)) β†’ ((πΏβ€˜π‘₯) ≀ 1 β†’ 𝑧 ≀ 𝐴)))
2726albii 1814 . . . . . 6 (βˆ€π‘§(((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯))) β†’ 𝑧 ≀ 𝐴) ↔ βˆ€π‘§(𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯)) β†’ ((πΏβ€˜π‘₯) ≀ 1 β†’ 𝑧 ≀ 𝐴)))
28 fvex 6910 . . . . . . 7 (π‘€β€˜(π‘‡β€˜π‘₯)) ∈ V
29 breq1 5151 . . . . . . . 8 (𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯)) β†’ (𝑧 ≀ 𝐴 ↔ (π‘€β€˜(π‘‡β€˜π‘₯)) ≀ 𝐴))
3029imbi2d 340 . . . . . . 7 (𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯)) β†’ (((πΏβ€˜π‘₯) ≀ 1 β†’ 𝑧 ≀ 𝐴) ↔ ((πΏβ€˜π‘₯) ≀ 1 β†’ (π‘€β€˜(π‘‡β€˜π‘₯)) ≀ 𝐴)))
3128, 30ceqsalv 3509 . . . . . 6 (βˆ€π‘§(𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯)) β†’ ((πΏβ€˜π‘₯) ≀ 1 β†’ 𝑧 ≀ 𝐴)) ↔ ((πΏβ€˜π‘₯) ≀ 1 β†’ (π‘€β€˜(π‘‡β€˜π‘₯)) ≀ 𝐴))
3227, 31bitri 275 . . . . 5 (βˆ€π‘§(((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯))) β†’ 𝑧 ≀ 𝐴) ↔ ((πΏβ€˜π‘₯) ≀ 1 β†’ (π‘€β€˜(π‘‡β€˜π‘₯)) ≀ 𝐴))
3332ralbii 3090 . . . 4 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘§(((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯))) β†’ 𝑧 ≀ 𝐴) ↔ βˆ€π‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 β†’ (π‘€β€˜(π‘‡β€˜π‘₯)) ≀ 𝐴))
34 r19.23v 3179 . . . . 5 (βˆ€π‘₯ ∈ 𝑋 (((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯))) β†’ 𝑧 ≀ 𝐴) ↔ (βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯))) β†’ 𝑧 ≀ 𝐴))
3534albii 1814 . . . 4 (βˆ€π‘§βˆ€π‘₯ ∈ 𝑋 (((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯))) β†’ 𝑧 ≀ 𝐴) ↔ βˆ€π‘§(βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯))) β†’ 𝑧 ≀ 𝐴))
3623, 33, 353bitr3i 301 . . 3 (βˆ€π‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 β†’ (π‘€β€˜(π‘‡β€˜π‘₯)) ≀ 𝐴) ↔ βˆ€π‘§(βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑧 = (π‘€β€˜(π‘‡β€˜π‘₯))) β†’ 𝑧 ≀ 𝐴))
3722, 36bitr4i 278 . 2 (βˆ€π‘§ ∈ {𝑦 ∣ βˆƒπ‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 ∧ 𝑦 = (π‘€β€˜(π‘‡β€˜π‘₯)))}𝑧 ≀ 𝐴 ↔ βˆ€π‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 β†’ (π‘€β€˜(π‘‡β€˜π‘₯)) ≀ 𝐴))
3818, 37bitrdi 287 1 ((𝑇:π‘‹βŸΆπ‘Œ ∧ 𝐴 ∈ ℝ*) β†’ ((π‘β€˜π‘‡) ≀ 𝐴 ↔ βˆ€π‘₯ ∈ 𝑋 ((πΏβ€˜π‘₯) ≀ 1 β†’ (π‘€β€˜(π‘‡β€˜π‘₯)) ≀ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395  βˆ€wal 1532   = wceq 1534   ∈ wcel 2099  {cab 2705  βˆ€wral 3058  βˆƒwrex 3067   βŠ† wss 3947   class class class wbr 5148  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420  supcsup 9464  β„cr 11138  1c1 11140  β„*cxr 11278   < clt 11279   ≀ cle 11280  NrmCVeccnv 30407  BaseSetcba 30409  normCVcnmcv 30413   normOpOLD cnmoo 30564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216  ax-pre-sup 11217
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9466  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-vc 30382  df-nv 30415  df-va 30418  df-ba 30419  df-sm 30420  df-0v 30421  df-nmcv 30423  df-nmoo 30568
This theorem is referenced by:  nmoub3i  30596  nmobndi  30598  ubthlem2  30694
  Copyright terms: Public domain W3C validator