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Theorem nmfnleub 31954
Description: An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (Revised by Mario Carneiro, 7-Sep-2014.) (New usage is discouraged.)
Assertion
Ref Expression
nmfnleub ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑇

Proof of Theorem nmfnleub
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmfnval 31905 . . . 4 (𝑇: ℋ⟶ℂ → (normfn𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))}, ℝ*, < ))
21adantr 480 . . 3 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → (normfn𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))}, ℝ*, < ))
32breq1d 5158 . 2 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴))
4 nmfnsetre 31906 . . . . 5 (𝑇: ℋ⟶ℂ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))} ⊆ ℝ)
5 ressxr 11303 . . . . 5 ℝ ⊆ ℝ*
64, 5sstrdi 4008 . . . 4 (𝑇: ℋ⟶ℂ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))} ⊆ ℝ*)
7 supxrleub 13365 . . . 4 (({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))} ⊆ ℝ*𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))}𝑧𝐴))
86, 7sylan 580 . . 3 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))}𝑧𝐴))
9 ancom 460 . . . . . . 7 (((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥))) ↔ (𝑦 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1))
10 eqeq1 2739 . . . . . . . 8 (𝑦 = 𝑧 → (𝑦 = (abs‘(𝑇𝑥)) ↔ 𝑧 = (abs‘(𝑇𝑥))))
1110anbi1d 631 . . . . . . 7 (𝑦 = 𝑧 → ((𝑦 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) ↔ (𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
129, 11bitrid 283 . . . . . 6 (𝑦 = 𝑧 → (((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥))) ↔ (𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
1312rexbidv 3177 . . . . 5 (𝑦 = 𝑧 → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥))) ↔ ∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
1413ralab 3700 . . . 4 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
15 ralcom4 3284 . . . . 5 (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧𝑥 ∈ ℋ ((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
16 impexp 450 . . . . . . . 8 (((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ (𝑧 = (abs‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)))
1716albii 1816 . . . . . . 7 (∀𝑧((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧(𝑧 = (abs‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)))
18 fvex 6920 . . . . . . . 8 (abs‘(𝑇𝑥)) ∈ V
19 breq1 5151 . . . . . . . . 9 (𝑧 = (abs‘(𝑇𝑥)) → (𝑧𝐴 ↔ (abs‘(𝑇𝑥)) ≤ 𝐴))
2019imbi2d 340 . . . . . . . 8 (𝑧 = (abs‘(𝑇𝑥)) → (((norm𝑥) ≤ 1 → 𝑧𝐴) ↔ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴)))
2118, 20ceqsalv 3519 . . . . . . 7 (∀𝑧(𝑧 = (abs‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)) ↔ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴))
2217, 21bitri 275 . . . . . 6 (∀𝑧((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴))
2322ralbii 3091 . . . . 5 (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴))
24 r19.23v 3181 . . . . . 6 (∀𝑥 ∈ ℋ ((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ (∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2524albii 1816 . . . . 5 (∀𝑧𝑥 ∈ ℋ ((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2615, 23, 253bitr3i 301 . . . 4 (∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2714, 26bitr4i 278 . . 3 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴))
288, 27bitrdi 287 . 2 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → (sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴)))
293, 28bitrd 279 1 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  wss 3963   class class class wbr 5148  wf 6559  cfv 6563  supcsup 9478  cc 11151  cr 11152  1c1 11154  *cxr 11292   < clt 11293  cle 11294  abscabs 15270  chba 30948  normcno 30952  normfncnmf 30980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-hilex 31028
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-nmfn 31874
This theorem is referenced by:  nmfnleub2  31955
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