Theorem nmobndseqi 29141

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 = (normCV‘𝑈)
nmoubi.m	⊢ 𝑀 = (normCV‘𝑊)
nmoubi.3	⊢ 𝑁 = (𝑈 normOpOLD 𝑊)
nmoubi.u	⊢ 𝑈 ∈ NrmCVec
nmoubi.w	⊢ 𝑊 ∈ NrmCVec

Assertion

Ref	Expression
nmobndseqi	⊢ ((𝑇:𝑋⟶𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) → (𝑁‘𝑇) ∈ 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.

Step	Hyp	Ref	Expression
1		impexp 451	. . . . . 6 ⊢ (((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘) ↔ (𝑓:ℕ⟶𝑋 → (∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1 → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)))
2		r19.35 3271	. . . . . . 7 ⊢ (∃𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘) ↔ (∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1 → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘))
3	2	imbi2i 336	. . . . . 6 ⊢ ((𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) ↔ (𝑓:ℕ⟶𝑋 → (∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1 → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)))
4	1, 3	bitr4i 277	. . . . 5 ⊢ (((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘) ↔ (𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)))
5	4	albii 1822	. . . 4 ⊢ (∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘) ↔ ∀𝑓(𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)))
6		nmoubi.1	. . . . . . . . 9 ⊢ 𝑋 = (BaseSet‘𝑈)
7	6	fvexi 6788	. . . . . . . 8 ⊢ 𝑋 ∈ V
8		nnenom 13700	. . . . . . . 8 ⊢ ℕ ≈ ω
9		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑓‘𝑘) → (𝐿‘𝑦) = (𝐿‘(𝑓‘𝑘)))
10	9	breq1d 5084	. . . . . . . . . 10 ⊢ (𝑦 = (𝑓‘𝑘) → ((𝐿‘𝑦) ≤ 1 ↔ (𝐿‘(𝑓‘𝑘)) ≤ 1))
11		2fveq3 6779	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑓‘𝑘) → (𝑀‘(𝑇‘𝑦)) = (𝑀‘(𝑇‘(𝑓‘𝑘))))
12	11	breq1d 5084	. . . . . . . . . 10 ⊢ (𝑦 = (𝑓‘𝑘) → ((𝑀‘(𝑇‘𝑦)) ≤ 𝑘 ↔ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘))
13	10, 12	imbi12d 345	. . . . . . . . 9 ⊢ (𝑦 = (𝑓‘𝑘) → (((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘) ↔ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)))
14	13	notbid 318	. . . . . . . 8 ⊢ (𝑦 = (𝑓‘𝑘) → (¬ ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘) ↔ ¬ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)))
15	7, 8, 14	axcc4 10195	. . . . . . 7 ⊢ (∀𝑘 ∈ ℕ ∃𝑦 ∈ 𝑋 ¬ ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘) → ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)))
16	15	con3i 154	. . . . . 6 ⊢ (¬ ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) → ¬ ∀𝑘 ∈ ℕ ∃𝑦 ∈ 𝑋 ¬ ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘))
17		dfrex2 3170	. . . . . . . . 9 ⊢ (∃𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘) ↔ ¬ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘))
18	17	imbi2i 336	. . . . . . . 8 ⊢ ((𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) ↔ (𝑓:ℕ⟶𝑋 → ¬ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)))
19	18	albii 1822	. . . . . . 7 ⊢ (∀𝑓(𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) ↔ ∀𝑓(𝑓:ℕ⟶𝑋 → ¬ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)))
20		alinexa 1845	. . . . . . 7 ⊢ (∀𝑓(𝑓:ℕ⟶𝑋 → ¬ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) ↔ ¬ ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)))
21	19, 20	bitri 274	. . . . . 6 ⊢ (∀𝑓(𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) ↔ ¬ ∃𝑓(𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ ¬ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)))
22		dfral2 3168	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘) ↔ ¬ ∃𝑦 ∈ 𝑋 ¬ ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘))
23	22	rexbii 3181	. . . . . . 7 ⊢ (∃𝑘 ∈ ℕ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘) ↔ ∃𝑘 ∈ ℕ ¬ ∃𝑦 ∈ 𝑋 ¬ ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘))
24		rexnal 3169	. . . . . . 7 ⊢ (∃𝑘 ∈ ℕ ¬ ∃𝑦 ∈ 𝑋 ¬ ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘) ↔ ¬ ∀𝑘 ∈ ℕ ∃𝑦 ∈ 𝑋 ¬ ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘))
25	23, 24	bitri 274	. . . . . 6 ⊢ (∃𝑘 ∈ ℕ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘) ↔ ¬ ∀𝑘 ∈ ℕ ∃𝑦 ∈ 𝑋 ¬ ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘))
26	16, 21, 25	3imtr4i 292	. . . . 5 ⊢ (∀𝑓(𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) → ∃𝑘 ∈ ℕ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘))
27		nnre 11980	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
28	27	anim1i 615	. . . . . 6 ⊢ ((𝑘 ∈ ℕ ∧ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘)) → (𝑘 ∈ ℝ ∧ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘)))
29	28	reximi2 3175	. . . . 5 ⊢ (∃𝑘 ∈ ℕ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘) → ∃𝑘 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘))
30	26, 29	syl 17	. . . 4 ⊢ (∀𝑓(𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) → ∃𝑘 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘))
31	5, 30	sylbi 216	. . 3 ⊢ (∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘) → ∃𝑘 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘))
32		nmoubi.y	. . . 4 ⊢ 𝑌 = (BaseSet‘𝑊)
33		nmoubi.l	. . . 4 ⊢ 𝐿 = (normCV‘𝑈)
34		nmoubi.m	. . . 4 ⊢ 𝑀 = (normCV‘𝑊)
35		nmoubi.3	. . . 4 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)
36		nmoubi.u	. . . 4 ⊢ 𝑈 ∈ NrmCVec
37		nmoubi.w	. . . 4 ⊢ 𝑊 ∈ NrmCVec
38	6, 32, 33, 34, 35, 36, 37	nmobndi 29137	. . 3 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑘 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘)))
39	31, 38	syl5ibr 245	. 2 ⊢ (𝑇:𝑋⟶𝑌 → (∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘) → (𝑁‘𝑇) ∈ ℝ))
40	39	imp 407	1 ⊢ ((𝑇:𝑋⟶𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) → (𝑁‘𝑇) ∈ ℝ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   class class class wbr 5074  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  ℝcr 10870  1c1 10872   ≤ cle 11010  ℕcn 11973  NrmCVeccnv 28946  BaseSetcba 28948  norm_CVcnmcv 28952   normOp_OLD cnmoo 29103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cc 10191  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-seq 13722  df-exp 13783  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-grpo 28855  df-gid 28856  df-ginv 28857  df-ablo 28907  df-vc 28921  df-nv 28954  df-va 28957  df-ba 28958  df-sm 28959  df-0v 28960  df-nmcv 28962  df-nmoo 29107

This theorem is referenced by: (None)