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Theorem nmobndseqi 30712
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 449 . . . . . 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 3098 . . . . . . 7 (∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) ↔ (∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1 → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘))
32imbi2i 335 . . . . . 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 277 . . . . 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 1814 . . . 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 6915 . . . . . . . 8 𝑋 ∈ V
8 nnenom 14000 . . . . . . . 8 ℕ ≈ ω
9 fveq2 6901 . . . . . . . . . . 11 (𝑦 = (𝑓𝑘) → (𝐿𝑦) = (𝐿‘(𝑓𝑘)))
109breq1d 5163 . . . . . . . . . 10 (𝑦 = (𝑓𝑘) → ((𝐿𝑦) ≤ 1 ↔ (𝐿‘(𝑓𝑘)) ≤ 1))
11 2fveq3 6906 . . . . . . . . . . 11 (𝑦 = (𝑓𝑘) → (𝑀‘(𝑇𝑦)) = (𝑀‘(𝑇‘(𝑓𝑘))))
1211breq1d 5163 . . . . . . . . . 10 (𝑦 = (𝑓𝑘) → ((𝑀‘(𝑇𝑦)) ≤ 𝑘 ↔ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘))
1310, 12imbi12d 343 . . . . . . . . 9 (𝑦 = (𝑓𝑘) → (((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
1413notbid 317 . . . . . . . 8 (𝑦 = (𝑓𝑘) → (¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
157, 8, 14axcc4 10482 . . . . . . 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 3063 . . . . . . . . 9 (∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) ↔ ¬ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘))
1817imbi2i 335 . . . . . . . 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 1814 . . . . . . 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 1838 . . . . . . 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 274 . . . . . 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 3089 . . . . . . . 8 (∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ¬ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
2322rexbii 3084 . . . . . . 7 (∃𝑘 ∈ ℕ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ∃𝑘 ∈ ℕ ¬ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
24 rexnal 3090 . . . . . . 7 (∃𝑘 ∈ ℕ ¬ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ¬ ∀𝑘 ∈ ℕ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
2523, 24bitri 274 . . . . . 6 (∃𝑘 ∈ ℕ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ¬ ∀𝑘 ∈ ℕ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
2616, 21, 253imtr4i 291 . . . . 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 613 . . . . . 6 ((𝑘 ∈ ℕ ∧ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘)) → (𝑘 ∈ ℝ ∧ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘)))
2928reximi2 3069 . . . . 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 216 . . 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 30708 . . 3 (𝑇:𝑋𝑌 → ((𝑁𝑇) ∈ ℝ ↔ ∃𝑘 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘)))
3931, 38imbitrrid 245 . 2 (𝑇:𝑋𝑌 → (∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) → (𝑁𝑇) ∈ ℝ))
4039imp 405 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 394  wal 1532   = wceq 1534  wex 1774  wcel 2099  wral 3051  wrex 3060   class class class wbr 5153  wf 6550  cfv 6554  (class class class)co 7424  cr 11157  1c1 11159  cle 11299  cn 12264  NrmCVeccnv 30517  BaseSetcba 30519  normCVcnmcv 30523   normOpOLD cnmoo 30674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-cc 10478  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-sup 9485  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12611  df-uz 12875  df-rp 13029  df-seq 14022  df-exp 14082  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-grpo 30426  df-gid 30427  df-ginv 30428  df-ablo 30478  df-vc 30492  df-nv 30525  df-va 30528  df-ba 30529  df-sm 30530  df-0v 30531  df-nmcv 30533  df-nmoo 30678
This theorem is referenced by: (None)
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