MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nmobndseqi Structured version   Visualization version   GIF version

Theorem nmobndseqi 29370
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 451 . . . . . 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 3107 . . . . . . 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 1820 . . . 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 6833 . . . . . . . 8 𝑋 ∈ V
8 nnenom 13793 . . . . . . . 8 ℕ ≈ ω
9 fveq2 6819 . . . . . . . . . . 11 (𝑦 = (𝑓𝑘) → (𝐿𝑦) = (𝐿‘(𝑓𝑘)))
109breq1d 5099 . . . . . . . . . 10 (𝑦 = (𝑓𝑘) → ((𝐿𝑦) ≤ 1 ↔ (𝐿‘(𝑓𝑘)) ≤ 1))
11 2fveq3 6824 . . . . . . . . . . 11 (𝑦 = (𝑓𝑘) → (𝑀‘(𝑇𝑦)) = (𝑀‘(𝑇‘(𝑓𝑘))))
1211breq1d 5099 . . . . . . . . . 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 317 . . . . . . . 8 (𝑦 = (𝑓𝑘) → (¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ¬ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
157, 8, 14axcc4 10288 . . . . . . 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 3073 . . . . . . . . 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 1820 . . . . . . 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 1844 . . . . . . 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 3098 . . . . . . . 8 (∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ¬ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
2322rexbii 3093 . . . . . . 7 (∃𝑘 ∈ ℕ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ∃𝑘 ∈ ℕ ¬ ∃𝑦𝑋 ¬ ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
24 rexnal 3099 . . . . . . 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 12073 . . . . . . 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 3078 . . . . 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 29366 . . 3 (𝑇:𝑋𝑌 → ((𝑁𝑇) ∈ ℝ ↔ ∃𝑘 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘)))
3931, 38syl5ibr 245 . 2 (𝑇:𝑋𝑌 → (∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) → (𝑁𝑇) ∈ ℝ))
4039imp 407 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 396  wal 1538   = wceq 1540  wex 1780  wcel 2105  wral 3061  wrex 3070   class class class wbr 5089  wf 6469  cfv 6473  (class class class)co 7329  cr 10963  1c1 10965  cle 11103  cn 12066  NrmCVeccnv 29175  BaseSetcba 29177  normCVcnmcv 29181   normOpOLD cnmoo 29332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-inf2 9490  ax-cc 10284  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041  ax-pre-sup 11042
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-1st 7891  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-er 8561  df-map 8680  df-en 8797  df-dom 8798  df-sdom 8799  df-sup 9291  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-div 11726  df-nn 12067  df-2 12129  df-3 12130  df-n0 12327  df-z 12413  df-uz 12676  df-rp 12824  df-seq 13815  df-exp 13876  df-cj 14901  df-re 14902  df-im 14903  df-sqrt 15037  df-abs 15038  df-grpo 29084  df-gid 29085  df-ginv 29086  df-ablo 29136  df-vc 29150  df-nv 29183  df-va 29186  df-ba 29187  df-sm 29188  df-0v 29189  df-nmcv 29191  df-nmoo 29336
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator