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Theorem nmoolb 30529
Description: A lower bound for an operator norm. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoolb.1 𝑋 = (BaseSetβ€˜π‘ˆ)
nmoolb.2 π‘Œ = (BaseSetβ€˜π‘Š)
nmoolb.l 𝐿 = (normCVβ€˜π‘ˆ)
nmoolb.m 𝑀 = (normCVβ€˜π‘Š)
nmoolb.3 𝑁 = (π‘ˆ normOpOLD π‘Š)
Assertion
Ref Expression
nmoolb (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) ∧ (𝐴 ∈ 𝑋 ∧ (πΏβ€˜π΄) ≀ 1)) β†’ (π‘€β€˜(π‘‡β€˜π΄)) ≀ (π‘β€˜π‘‡))

Proof of Theorem nmoolb
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmoolb.2 . . . . . 6 π‘Œ = (BaseSetβ€˜π‘Š)
2 nmoolb.m . . . . . 6 𝑀 = (normCVβ€˜π‘Š)
31, 2nmosetre 30522 . . . . 5 ((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ {π‘₯ ∣ βˆƒπ‘¦ ∈ 𝑋 ((πΏβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘¦)))} βŠ† ℝ)
4 ressxr 11259 . . . . 5 ℝ βŠ† ℝ*
53, 4sstrdi 3989 . . . 4 ((π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ {π‘₯ ∣ βˆƒπ‘¦ ∈ 𝑋 ((πΏβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘¦)))} βŠ† ℝ*)
653adant1 1127 . . 3 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ {π‘₯ ∣ βˆƒπ‘¦ ∈ 𝑋 ((πΏβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘¦)))} βŠ† ℝ*)
7 fveq2 6884 . . . . . . . 8 (𝑦 = 𝐴 β†’ (πΏβ€˜π‘¦) = (πΏβ€˜π΄))
87breq1d 5151 . . . . . . 7 (𝑦 = 𝐴 β†’ ((πΏβ€˜π‘¦) ≀ 1 ↔ (πΏβ€˜π΄) ≀ 1))
9 2fveq3 6889 . . . . . . . 8 (𝑦 = 𝐴 β†’ (π‘€β€˜(π‘‡β€˜π‘¦)) = (π‘€β€˜(π‘‡β€˜π΄)))
109eqeq2d 2737 . . . . . . 7 (𝑦 = 𝐴 β†’ ((π‘€β€˜(π‘‡β€˜π΄)) = (π‘€β€˜(π‘‡β€˜π‘¦)) ↔ (π‘€β€˜(π‘‡β€˜π΄)) = (π‘€β€˜(π‘‡β€˜π΄))))
118, 10anbi12d 630 . . . . . 6 (𝑦 = 𝐴 β†’ (((πΏβ€˜π‘¦) ≀ 1 ∧ (π‘€β€˜(π‘‡β€˜π΄)) = (π‘€β€˜(π‘‡β€˜π‘¦))) ↔ ((πΏβ€˜π΄) ≀ 1 ∧ (π‘€β€˜(π‘‡β€˜π΄)) = (π‘€β€˜(π‘‡β€˜π΄)))))
12 eqid 2726 . . . . . . 7 (π‘€β€˜(π‘‡β€˜π΄)) = (π‘€β€˜(π‘‡β€˜π΄))
1312biantru 529 . . . . . 6 ((πΏβ€˜π΄) ≀ 1 ↔ ((πΏβ€˜π΄) ≀ 1 ∧ (π‘€β€˜(π‘‡β€˜π΄)) = (π‘€β€˜(π‘‡β€˜π΄))))
1411, 13bitr4di 289 . . . . 5 (𝑦 = 𝐴 β†’ (((πΏβ€˜π‘¦) ≀ 1 ∧ (π‘€β€˜(π‘‡β€˜π΄)) = (π‘€β€˜(π‘‡β€˜π‘¦))) ↔ (πΏβ€˜π΄) ≀ 1))
1514rspcev 3606 . . . 4 ((𝐴 ∈ 𝑋 ∧ (πΏβ€˜π΄) ≀ 1) β†’ βˆƒπ‘¦ ∈ 𝑋 ((πΏβ€˜π‘¦) ≀ 1 ∧ (π‘€β€˜(π‘‡β€˜π΄)) = (π‘€β€˜(π‘‡β€˜π‘¦))))
16 fvex 6897 . . . . 5 (π‘€β€˜(π‘‡β€˜π΄)) ∈ V
17 eqeq1 2730 . . . . . . 7 (π‘₯ = (π‘€β€˜(π‘‡β€˜π΄)) β†’ (π‘₯ = (π‘€β€˜(π‘‡β€˜π‘¦)) ↔ (π‘€β€˜(π‘‡β€˜π΄)) = (π‘€β€˜(π‘‡β€˜π‘¦))))
1817anbi2d 628 . . . . . 6 (π‘₯ = (π‘€β€˜(π‘‡β€˜π΄)) β†’ (((πΏβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘¦))) ↔ ((πΏβ€˜π‘¦) ≀ 1 ∧ (π‘€β€˜(π‘‡β€˜π΄)) = (π‘€β€˜(π‘‡β€˜π‘¦)))))
1918rexbidv 3172 . . . . 5 (π‘₯ = (π‘€β€˜(π‘‡β€˜π΄)) β†’ (βˆƒπ‘¦ ∈ 𝑋 ((πΏβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘¦))) ↔ βˆƒπ‘¦ ∈ 𝑋 ((πΏβ€˜π‘¦) ≀ 1 ∧ (π‘€β€˜(π‘‡β€˜π΄)) = (π‘€β€˜(π‘‡β€˜π‘¦)))))
2016, 19elab 3663 . . . 4 ((π‘€β€˜(π‘‡β€˜π΄)) ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ 𝑋 ((πΏβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘¦)))} ↔ βˆƒπ‘¦ ∈ 𝑋 ((πΏβ€˜π‘¦) ≀ 1 ∧ (π‘€β€˜(π‘‡β€˜π΄)) = (π‘€β€˜(π‘‡β€˜π‘¦))))
2115, 20sylibr 233 . . 3 ((𝐴 ∈ 𝑋 ∧ (πΏβ€˜π΄) ≀ 1) β†’ (π‘€β€˜(π‘‡β€˜π΄)) ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ 𝑋 ((πΏβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘¦)))})
22 supxrub 13306 . . 3 (({π‘₯ ∣ βˆƒπ‘¦ ∈ 𝑋 ((πΏβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘¦)))} βŠ† ℝ* ∧ (π‘€β€˜(π‘‡β€˜π΄)) ∈ {π‘₯ ∣ βˆƒπ‘¦ ∈ 𝑋 ((πΏβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘¦)))}) β†’ (π‘€β€˜(π‘‡β€˜π΄)) ≀ sup({π‘₯ ∣ βˆƒπ‘¦ ∈ 𝑋 ((πΏβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘¦)))}, ℝ*, < ))
236, 21, 22syl2an 595 . 2 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) ∧ (𝐴 ∈ 𝑋 ∧ (πΏβ€˜π΄) ≀ 1)) β†’ (π‘€β€˜(π‘‡β€˜π΄)) ≀ sup({π‘₯ ∣ βˆƒπ‘¦ ∈ 𝑋 ((πΏβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘¦)))}, ℝ*, < ))
24 nmoolb.1 . . . 4 𝑋 = (BaseSetβ€˜π‘ˆ)
25 nmoolb.l . . . 4 𝐿 = (normCVβ€˜π‘ˆ)
26 nmoolb.3 . . . 4 𝑁 = (π‘ˆ normOpOLD π‘Š)
2724, 1, 25, 2, 26nmooval 30521 . . 3 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ (π‘β€˜π‘‡) = sup({π‘₯ ∣ βˆƒπ‘¦ ∈ 𝑋 ((πΏβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘¦)))}, ℝ*, < ))
2827adantr 480 . 2 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) ∧ (𝐴 ∈ 𝑋 ∧ (πΏβ€˜π΄) ≀ 1)) β†’ (π‘β€˜π‘‡) = sup({π‘₯ ∣ βˆƒπ‘¦ ∈ 𝑋 ((πΏβ€˜π‘¦) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‡β€˜π‘¦)))}, ℝ*, < ))
2923, 28breqtrrd 5169 1 (((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) ∧ (𝐴 ∈ 𝑋 ∧ (πΏβ€˜π΄) ≀ 1)) β†’ (π‘€β€˜(π‘‡β€˜π΄)) ≀ (π‘β€˜π‘‡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆƒwrex 3064   βŠ† wss 3943   class class class wbr 5141  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404  supcsup 9434  β„cr 11108  1c1 11110  β„*cxr 11248   < clt 11249   ≀ cle 11250  NrmCVeccnv 30342  BaseSetcba 30344  normCVcnmcv 30348   normOpOLD cnmoo 30499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-sup 9436  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-vc 30317  df-nv 30350  df-va 30353  df-ba 30354  df-sm 30355  df-0v 30356  df-nmcv 30358  df-nmoo 30503
This theorem is referenced by:  nmblolbii  30557
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