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Theorem nmfnlb 28167
 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 28120 . . . . 5 (𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ⊆ ℝ)
2 ressxr 9962 . . . . 5 ℝ ⊆ ℝ*
31, 2syl6ss 3580 . . . 4 (𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ⊆ ℝ*)
433ad2ant1 1075 . . 3 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ⊆ ℝ*)
5 fveq2 6103 . . . . . . . . 9 (𝑦 = 𝐴 → (norm𝑦) = (norm𝐴))
65breq1d 4593 . . . . . . . 8 (𝑦 = 𝐴 → ((norm𝑦) ≤ 1 ↔ (norm𝐴) ≤ 1))
7 fveq2 6103 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑇𝑦) = (𝑇𝐴))
87fveq2d 6107 . . . . . . . . 9 (𝑦 = 𝐴 → (abs‘(𝑇𝑦)) = (abs‘(𝑇𝐴)))
98eqeq2d 2620 . . . . . . . 8 (𝑦 = 𝐴 → ((abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦)) ↔ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝐴))))
106, 9anbi12d 743 . . . . . . 7 (𝑦 = 𝐴 → (((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦))) ↔ ((norm𝐴) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝐴)))))
11 eqid 2610 . . . . . . . 8 (abs‘(𝑇𝐴)) = (abs‘(𝑇𝐴))
1211biantru 525 . . . . . . 7 ((norm𝐴) ≤ 1 ↔ ((norm𝐴) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝐴))))
1310, 12syl6bbr 277 . . . . . 6 (𝑦 = 𝐴 → (((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦))) ↔ (norm𝐴) ≤ 1))
1413rspcev 3282 . . . . 5 ((𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦))))
15 fvex 6113 . . . . . 6 (abs‘(𝑇𝐴)) ∈ V
16 eqeq1 2614 . . . . . . . 8 (𝑥 = (abs‘(𝑇𝐴)) → (𝑥 = (abs‘(𝑇𝑦)) ↔ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦))))
1716anbi2d 736 . . . . . . 7 (𝑥 = (abs‘(𝑇𝐴)) → (((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦)))))
1817rexbidv 3034 . . . . . 6 (𝑥 = (abs‘(𝑇𝐴)) → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦)))))
1915, 18elab 3319 . . . . 5 ((abs‘(𝑇𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦))))
2014, 19sylibr 223 . . . 4 ((𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))})
21203adant1 1072 . . 3 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))})
22 supxrub 12026 . . 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*, < ))
234, 21, 22syl2anc 691 . 2 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ≤ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))
24 nmfnval 28119 . . 3 (𝑇: ℋ⟶ℂ → (normfn𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))
25243ad2ant1 1075 . 2 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (normfn𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))
2623, 25breqtrrd 4611 1 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ≤ (normfn𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897   ⊆ wss 3540   class class class wbr 4583  ⟶wf 5800  ‘cfv 5804  supcsup 8229  ℂcc 9813  ℝcr 9814  1c1 9816  ℝ*cxr 9952   < clt 9953   ≤ cle 9954  abscabs 13822   ℋchil 27160  normℎcno 27164  normfncnmf 27192 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-hilex 27240 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-nmfn 28088 This theorem is referenced by:  nmfnge0  28170  nmbdfnlbi  28292  nmcfnlbi  28295
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