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Theorem nmobndseqi 27495
 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 462 . . . . . 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 3076 . . . . . . 7 (∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) ↔ (∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1 → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘))
32imbi2i 326 . . . . . 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 267 . . . . 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 1744 . . . 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)
7 fvex 6160 . . . . . . . . 9 (BaseSet‘𝑈) ∈ V
86, 7eqeltri 2694 . . . . . . . 8 𝑋 ∈ V
9 nnenom 12722 . . . . . . . 8 ℕ ≈ ω
10 fveq2 6150 . . . . . . . . . . 11 (𝑦 = (𝑓𝑘) → (𝐿𝑦) = (𝐿‘(𝑓𝑘)))
1110breq1d 4625 . . . . . . . . . 10 (𝑦 = (𝑓𝑘) → ((𝐿𝑦) ≤ 1 ↔ (𝐿‘(𝑓𝑘)) ≤ 1))
12 fveq2 6150 . . . . . . . . . . . 12 (𝑦 = (𝑓𝑘) → (𝑇𝑦) = (𝑇‘(𝑓𝑘)))
1312fveq2d 6154 . . . . . . . . . . 11 (𝑦 = (𝑓𝑘) → (𝑀‘(𝑇𝑦)) = (𝑀‘(𝑇‘(𝑓𝑘))))
1413breq1d 4625 . . . . . . . . . 10 (𝑦 = (𝑓𝑘) → ((𝑀‘(𝑇𝑦)) ≤ 𝑘 ↔ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘))
1511, 14imbi12d 334 . . . . . . . . 9 (𝑦 = (𝑓𝑘) → (((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
1615notbid 308 . . . . . . . 8 (𝑦 = (𝑓𝑘) → (¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
178, 9, 16axcc4 9208 . . . . . . 7 (∀𝑘 ∈ ℕ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
1817con3i 150 . . . . . 6 (¬ ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) → ¬ ∀𝑘 ∈ ℕ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
19 dfrex2 2990 . . . . . . . . 9 (∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) ↔ ¬ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘))
2019imbi2i 326 . . . . . . . 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)))
2120albii 1744 . . . . . . 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)))
22 alinexa 1767 . . . . . . 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)))
2321, 22bitri 264 . . . . . 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)))
24 dfral2 2988 . . . . . . . 8 (∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ¬ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
2524rexbii 3034 . . . . . . 7 (∃𝑘 ∈ ℕ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ∃𝑘 ∈ ℕ ¬ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
26 rexnal 2989 . . . . . . 7 (∃𝑘 ∈ ℕ ¬ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ¬ ∀𝑘 ∈ ℕ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
2725, 26bitri 264 . . . . . 6 (∃𝑘 ∈ ℕ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ¬ ∀𝑘 ∈ ℕ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
2818, 23, 273imtr4i 281 . . . . 5 (∀𝑓(𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) → ∃𝑘 ∈ ℕ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
29 nnre 10974 . . . . . . 7 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
3029anim1i 591 . . . . . 6 ((𝑘 ∈ ℕ ∧ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘)) → (𝑘 ∈ ℝ ∧ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘)))
3130reximi2 3004 . . . . 5 (∃𝑘 ∈ ℕ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) → ∃𝑘 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
3228, 31syl 17 . . . 4 (∀𝑓(𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) → ∃𝑘 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
335, 32sylbi 207 . . 3 (∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) → ∃𝑘 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
34 nmoubi.y . . . 4 𝑌 = (BaseSet‘𝑊)
35 nmoubi.l . . . 4 𝐿 = (normCV𝑈)
36 nmoubi.m . . . 4 𝑀 = (normCV𝑊)
37 nmoubi.3 . . . 4 𝑁 = (𝑈 normOpOLD 𝑊)
38 nmoubi.u . . . 4 𝑈 ∈ NrmCVec
39 nmoubi.w . . . 4 𝑊 ∈ NrmCVec
406, 34, 35, 36, 37, 38, 39nmobndi 27491 . . 3 (𝑇:𝑋𝑌 → ((𝑁𝑇) ∈ ℝ ↔ ∃𝑘 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘)))
4133, 40syl5ibr 236 . 2 (𝑇:𝑋𝑌 → (∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) → (𝑁𝑇) ∈ ℝ))
4241imp 445 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 384  ∀wal 1478   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  Vcvv 3186   class class class wbr 4615  ⟶wf 5845  ‘cfv 5849  (class class class)co 6607  ℝcr 9882  1c1 9884   ≤ cle 10022  ℕcn 10967  NrmCVeccnv 27300  BaseSetcba 27302  normCVcnmcv 27306   normOpOLD cnmoo 27457 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-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-inf2 8485  ax-cc 9204  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 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-1st 7116  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-grpo 27208  df-gid 27209  df-ginv 27210  df-ablo 27260  df-vc 27275  df-nv 27308  df-va 27311  df-ba 27312  df-sm 27313  df-0v 27314  df-nmcv 27316  df-nmoo 27461 This theorem is referenced by: (None)
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