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Theorem nmobndseqi 30808
Description: A bounded sequence determines a bounded operator. (Contributed by NM, 18-Jan-2008.) (Revised by Mario Carneiro, 7-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoubi.1 𝑋 = (BaseSet‘𝑈)
nmoubi.y 𝑌 = (BaseSet‘𝑊)
nmoubi.l 𝐿 = (normCV𝑈)
nmoubi.m 𝑀 = (normCV𝑊)
nmoubi.3 𝑁 = (𝑈 normOpOLD 𝑊)
nmoubi.u 𝑈 ∈ NrmCVec
nmoubi.w 𝑊 ∈ NrmCVec
Assertion
Ref Expression
nmobndseqi ((𝑇:𝑋𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) → (𝑁𝑇) ∈ ℝ)
Distinct variable groups:   𝑓,𝑘,𝐿   𝑘,𝑌   𝑓,𝑀,𝑘   𝑇,𝑓,𝑘   𝑓,𝑋,𝑘   𝑘,𝑁
Allowed substitution hints:   𝑈(𝑓,𝑘)   𝑁(𝑓)   𝑊(𝑓,𝑘)   𝑌(𝑓)

Proof of Theorem nmobndseqi
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 impexp 450 . . . . . 6 (((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) ↔ (𝑓:ℕ⟶𝑋 → (∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1 → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
2 r19.35 3106 . . . . . . 7 (∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) ↔ (∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1 → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘))
32imbi2i 336 . . . . . 6 ((𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) ↔ (𝑓:ℕ⟶𝑋 → (∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1 → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
41, 3bitr4i 278 . . . . 5 (((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) ↔ (𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
54albii 1816 . . . 4 (∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) ↔ ∀𝑓(𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
6 nmoubi.1 . . . . . . . . 9 𝑋 = (BaseSet‘𝑈)
76fvexi 6921 . . . . . . . 8 𝑋 ∈ V
8 nnenom 14018 . . . . . . . 8 ℕ ≈ ω
9 fveq2 6907 . . . . . . . . . . 11 (𝑦 = (𝑓𝑘) → (𝐿𝑦) = (𝐿‘(𝑓𝑘)))
109breq1d 5158 . . . . . . . . . 10 (𝑦 = (𝑓𝑘) → ((𝐿𝑦) ≤ 1 ↔ (𝐿‘(𝑓𝑘)) ≤ 1))
11 2fveq3 6912 . . . . . . . . . . 11 (𝑦 = (𝑓𝑘) → (𝑀‘(𝑇𝑦)) = (𝑀‘(𝑇‘(𝑓𝑘))))
1211breq1d 5158 . . . . . . . . . 10 (𝑦 = (𝑓𝑘) → ((𝑀‘(𝑇𝑦)) ≤ 𝑘 ↔ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘))
1310, 12imbi12d 344 . . . . . . . . 9 (𝑦 = (𝑓𝑘) → (((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
1413notbid 318 . . . . . . . 8 (𝑦 = (𝑓𝑘) → (¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
157, 8, 14axcc4 10477 . . . . . . 7 (∀𝑘 ∈ ℕ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
1615con3i 154 . . . . . 6 (¬ ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) → ¬ ∀𝑘 ∈ ℕ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
17 dfrex2 3071 . . . . . . . . 9 (∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) ↔ ¬ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘))
1817imbi2i 336 . . . . . . . 8 ((𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) ↔ (𝑓:ℕ⟶𝑋 → ¬ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
1918albii 1816 . . . . . . 7 (∀𝑓(𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) ↔ ∀𝑓(𝑓:ℕ⟶𝑋 → ¬ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
20 alinexa 1840 . . . . . . 7 (∀𝑓(𝑓:ℕ⟶𝑋 → ¬ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) ↔ ¬ ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
2119, 20bitri 275 . . . . . 6 (∀𝑓(𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) ↔ ¬ ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
22 dfral2 3097 . . . . . . . 8 (∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ¬ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
2322rexbii 3092 . . . . . . 7 (∃𝑘 ∈ ℕ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ∃𝑘 ∈ ℕ ¬ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
24 rexnal 3098 . . . . . . 7 (∃𝑘 ∈ ℕ ¬ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ¬ ∀𝑘 ∈ ℕ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
2523, 24bitri 275 . . . . . 6 (∃𝑘 ∈ ℕ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ¬ ∀𝑘 ∈ ℕ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
2616, 21, 253imtr4i 292 . . . . 5 (∀𝑓(𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) → ∃𝑘 ∈ ℕ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
27 nnre 12271 . . . . . . 7 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
2827anim1i 615 . . . . . 6 ((𝑘 ∈ ℕ ∧ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘)) → (𝑘 ∈ ℝ ∧ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘)))
2928reximi2 3077 . . . . 5 (∃𝑘 ∈ ℕ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) → ∃𝑘 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
3026, 29syl 17 . . . 4 (∀𝑓(𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) → ∃𝑘 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
315, 30sylbi 217 . . 3 (∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) → ∃𝑘 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
32 nmoubi.y . . . 4 𝑌 = (BaseSet‘𝑊)
33 nmoubi.l . . . 4 𝐿 = (normCV𝑈)
34 nmoubi.m . . . 4 𝑀 = (normCV𝑊)
35 nmoubi.3 . . . 4 𝑁 = (𝑈 normOpOLD 𝑊)
36 nmoubi.u . . . 4 𝑈 ∈ NrmCVec
37 nmoubi.w . . . 4 𝑊 ∈ NrmCVec
386, 32, 33, 34, 35, 36, 37nmobndi 30804 . . 3 (𝑇:𝑋𝑌 → ((𝑁𝑇) ∈ ℝ ↔ ∃𝑘 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘)))
3931, 38imbitrrid 246 . 2 (𝑇:𝑋𝑌 → (∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) → (𝑁𝑇) ∈ ℝ))
4039imp 406 1 ((𝑇:𝑋𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) → (𝑁𝑇) ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068   class class class wbr 5148  wf 6559  cfv 6563  (class class class)co 7431  cr 11152  1c1 11154  cle 11294  cn 12264  NrmCVeccnv 30613  BaseSetcba 30615  normCVcnmcv 30619   normOpOLD cnmoo 30770
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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cc 10473  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
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-1st 8013  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-grpo 30522  df-gid 30523  df-ginv 30524  df-ablo 30574  df-vc 30588  df-nv 30621  df-va 30624  df-ba 30625  df-sm 30626  df-0v 30627  df-nmcv 30629  df-nmoo 30774
This theorem is referenced by: (None)
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