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Theorem nmfnlb 29472
Description: A lower bound for a functional norm. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
nmfnlb ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ≤ (normfn𝑇))

Proof of Theorem nmfnlb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmfnsetre 29425 . . . . 5 (𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ⊆ ℝ)
2 ressxr 10476 . . . . 5 ℝ ⊆ ℝ*
31, 2syl6ss 3866 . . . 4 (𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ⊆ ℝ*)
433ad2ant1 1113 . . 3 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ⊆ ℝ*)
5 fveq2 6493 . . . . . . . . 9 (𝑦 = 𝐴 → (norm𝑦) = (norm𝐴))
65breq1d 4933 . . . . . . . 8 (𝑦 = 𝐴 → ((norm𝑦) ≤ 1 ↔ (norm𝐴) ≤ 1))
7 2fveq3 6498 . . . . . . . . 9 (𝑦 = 𝐴 → (abs‘(𝑇𝑦)) = (abs‘(𝑇𝐴)))
87eqeq2d 2782 . . . . . . . 8 (𝑦 = 𝐴 → ((abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦)) ↔ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝐴))))
96, 8anbi12d 621 . . . . . . 7 (𝑦 = 𝐴 → (((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦))) ↔ ((norm𝐴) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝐴)))))
10 eqid 2772 . . . . . . . 8 (abs‘(𝑇𝐴)) = (abs‘(𝑇𝐴))
1110biantru 522 . . . . . . 7 ((norm𝐴) ≤ 1 ↔ ((norm𝐴) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝐴))))
129, 11syl6bbr 281 . . . . . 6 (𝑦 = 𝐴 → (((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦))) ↔ (norm𝐴) ≤ 1))
1312rspcev 3529 . . . . 5 ((𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦))))
14 fvex 6506 . . . . . 6 (abs‘(𝑇𝐴)) ∈ V
15 eqeq1 2776 . . . . . . . 8 (𝑥 = (abs‘(𝑇𝐴)) → (𝑥 = (abs‘(𝑇𝑦)) ↔ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦))))
1615anbi2d 619 . . . . . . 7 (𝑥 = (abs‘(𝑇𝐴)) → (((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦)))))
1716rexbidv 3236 . . . . . 6 (𝑥 = (abs‘(𝑇𝐴)) → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦)))))
1814, 17elab 3576 . . . . 5 ((abs‘(𝑇𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦))))
1913, 18sylibr 226 . . . 4 ((𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))})
20193adant1 1110 . . 3 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))})
21 supxrub 12526 . . 3 (({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ⊆ ℝ* ∧ (abs‘(𝑇𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}) → (abs‘(𝑇𝐴)) ≤ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))
224, 20, 21syl2anc 576 . 2 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ≤ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))
23 nmfnval 29424 . . 3 (𝑇: ℋ⟶ℂ → (normfn𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))
24233ad2ant1 1113 . 2 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (normfn𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))
2522, 24breqtrrd 4951 1 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ≤ (normfn𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2048  {cab 2753  wrex 3083  wss 3825   class class class wbr 4923  wf 6178  cfv 6182  supcsup 8691  cc 10325  cr 10326  1c1 10328  *cxr 10465   < clt 10466  cle 10467  abscabs 14444  chba 28465  normcno 28469  normfncnmf 28497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404  ax-pre-sup 10405  ax-hilex 28545
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-er 8081  df-map 8200  df-en 8299  df-dom 8300  df-sdom 8301  df-sup 8693  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-div 11091  df-nn 11432  df-2 11496  df-3 11497  df-n0 11701  df-z 11787  df-uz 12052  df-rp 12198  df-seq 13178  df-exp 13238  df-cj 14309  df-re 14310  df-im 14311  df-sqrt 14445  df-abs 14446  df-nmfn 29393
This theorem is referenced by:  nmfnge0  29475  nmbdfnlbi  29597  nmcfnlbi  29600
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