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Theorem nmfnleub 29616
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 29567 . . . 4 (𝑇: ℋ⟶ℂ → (normfn𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))}, ℝ*, < ))
21adantr 481 . . 3 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → (normfn𝑇) = sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))}, ℝ*, < ))
32breq1d 5073 . 2 ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn𝑇) ≤ 𝐴 ↔ sup({𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))}, ℝ*, < ) ≤ 𝐴))
4 nmfnsetre 29568 . . . . 5 (𝑇: ℋ⟶ℂ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))} ⊆ ℝ)
5 ressxr 10674 . . . . 5 ℝ ⊆ ℝ*
64, 5syl6ss 3983 . . . 4 (𝑇: ℋ⟶ℂ → {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))} ⊆ ℝ*)
7 supxrleub 12709 . . . 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 461 . . . . . . 7 (((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥))) ↔ (𝑦 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1))
10 eqeq1 2830 . . . . . . . 8 (𝑦 = 𝑧 → (𝑦 = (abs‘(𝑇𝑥)) ↔ 𝑧 = (abs‘(𝑇𝑥))))
1110anbi1d 629 . . . . . . 7 (𝑦 = 𝑧 → ((𝑦 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) ↔ (𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
129, 11syl5bb 284 . . . . . 6 (𝑦 = 𝑧 → (((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥))) ↔ (𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
1312rexbidv 3302 . . . . 5 (𝑦 = 𝑧 → (∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥))) ↔ ∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1)))
1413ralab 3688 . . . 4 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
15 ralcom4 3240 . . . . 5 (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧𝑥 ∈ ℋ ((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
16 impexp 451 . . . . . . . 8 (((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ (𝑧 = (abs‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)))
1716albii 1813 . . . . . . 7 (∀𝑧((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧(𝑧 = (abs‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)))
18 fvex 6680 . . . . . . . 8 (abs‘(𝑇𝑥)) ∈ V
19 breq1 5066 . . . . . . . . 9 (𝑧 = (abs‘(𝑇𝑥)) → (𝑧𝐴 ↔ (abs‘(𝑇𝑥)) ≤ 𝐴))
2019imbi2d 342 . . . . . . . 8 (𝑧 = (abs‘(𝑇𝑥)) → (((norm𝑥) ≤ 1 → 𝑧𝐴) ↔ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴)))
2118, 20ceqsalv 3538 . . . . . . 7 (∀𝑧(𝑧 = (abs‘(𝑇𝑥)) → ((norm𝑥) ≤ 1 → 𝑧𝐴)) ↔ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴))
2217, 21bitri 276 . . . . . 6 (∀𝑧((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴))
2322ralbii 3170 . . . . 5 (∀𝑥 ∈ ℋ ∀𝑧((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴))
24 r19.23v 3284 . . . . . 6 (∀𝑥 ∈ ℋ ((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ (∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2524albii 1813 . . . . 5 (∀𝑧𝑥 ∈ ℋ ((𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2615, 23, 253bitr3i 302 . . . 4 (∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴) ↔ ∀𝑧(∃𝑥 ∈ ℋ (𝑧 = (abs‘(𝑇𝑥)) ∧ (norm𝑥) ≤ 1) → 𝑧𝐴))
2714, 26bitr4i 279 . . 3 (∀𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ ℋ ((norm𝑥) ≤ 1 ∧ 𝑦 = (abs‘(𝑇𝑥)))}𝑧𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴))
288, 27syl6bb 288 . 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 280 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 207  wa 396  wal 1528   = wceq 1530  wcel 2107  {cab 2804  wral 3143  wrex 3144  wss 3940   class class class wbr 5063  wf 6348  cfv 6352  supcsup 8893  cc 10524  cr 10525  1c1 10527  *cxr 10663   < clt 10664  cle 10665  abscabs 14583  chba 28610  normcno 28614  normfncnmf 28642
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604  ax-hilex 28690
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-sup 8895  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-seq 13360  df-exp 13420  df-cj 14448  df-re 14449  df-im 14450  df-sqrt 14584  df-abs 14585  df-nmfn 29536
This theorem is referenced by:  nmfnleub2  29617
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