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Theorem nmfnlb 28644
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 28597 . . . . 5 (𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ⊆ ℝ)
2 ressxr 10030 . . . . 5 ℝ ⊆ ℝ*
31, 2syl6ss 3596 . . . 4 (𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ⊆ ℝ*)
433ad2ant1 1080 . . 3 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ⊆ ℝ*)
5 fveq2 6150 . . . . . . . . 9 (𝑦 = 𝐴 → (norm𝑦) = (norm𝐴))
65breq1d 4625 . . . . . . . 8 (𝑦 = 𝐴 → ((norm𝑦) ≤ 1 ↔ (norm𝐴) ≤ 1))
7 fveq2 6150 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑇𝑦) = (𝑇𝐴))
87fveq2d 6154 . . . . . . . . 9 (𝑦 = 𝐴 → (abs‘(𝑇𝑦)) = (abs‘(𝑇𝐴)))
98eqeq2d 2631 . . . . . . . 8 (𝑦 = 𝐴 → ((abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦)) ↔ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝐴))))
106, 9anbi12d 746 . . . . . . 7 (𝑦 = 𝐴 → (((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦))) ↔ ((norm𝐴) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝐴)))))
11 eqid 2621 . . . . . . . 8 (abs‘(𝑇𝐴)) = (abs‘(𝑇𝐴))
1211biantru 526 . . . . . . 7 ((norm𝐴) ≤ 1 ↔ ((norm𝐴) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝐴))))
1310, 12syl6bbr 278 . . . . . 6 (𝑦 = 𝐴 → (((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦))) ↔ (norm𝐴) ≤ 1))
1413rspcev 3295 . . . . 5 ((𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦))))
15 fvex 6160 . . . . . 6 (abs‘(𝑇𝐴)) ∈ V
16 eqeq1 2625 . . . . . . . 8 (𝑥 = (abs‘(𝑇𝐴)) → (𝑥 = (abs‘(𝑇𝑦)) ↔ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦))))
1716anbi2d 739 . . . . . . 7 (𝑥 = (abs‘(𝑇𝐴)) → (((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦)))))
1817rexbidv 3045 . . . . . 6 (𝑥 = (abs‘(𝑇𝐴)) → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦)))))
1915, 18elab 3334 . . . . 5 ((abs‘(𝑇𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ (abs‘(𝑇𝐴)) = (abs‘(𝑇𝑦))))
2014, 19sylibr 224 . . . 4 ((𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))})
21203adant1 1077 . . 3 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))})
22 supxrub 12100 . . 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 692 . 2 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ≤ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))
24 nmfnval 28596 . . 3 (𝑇: ℋ⟶ℂ → (normfn𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))
25243ad2ant1 1080 . 2 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (normfn𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))
2623, 25breqtrrd 4643 1 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ≤ (normfn𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wrex 2908  wss 3556   class class class wbr 4615  wf 5845  cfv 5849  supcsup 8293  cc 9881  cr 9882  1c1 9884  *cxr 10020   < clt 10021  cle 10022  abscabs 13911  chil 27637  normcno 27641  normfncnmf 27669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960  ax-pre-sup 9961  ax-hilex 27717
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-er 7690  df-map 7807  df-en 7903  df-dom 7904  df-sdom 7905  df-sup 8295  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-div 10632  df-nn 10968  df-2 11026  df-3 11027  df-n0 11240  df-z 11325  df-uz 11635  df-rp 11780  df-seq 12745  df-exp 12804  df-cj 13776  df-re 13777  df-im 13778  df-sqrt 13912  df-abs 13913  df-nmfn 28565
This theorem is referenced by:  nmfnge0  28647  nmbdfnlbi  28769  nmcfnlbi  28772
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