Theorem nmobndseqi 31101

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 455	. . . . . 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 3130	. . . . . . 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 339	. . . . . 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 281	. . . . 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 1847	. . . 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 6899	. . . . . . . 8 ⊢ 𝑋 ∈ V
8		nnenom 14019	. . . . . . . 8 ⊢ ℕ ≈ ω
9		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑓‘𝑘) → (𝐿‘𝑦) = (𝐿‘(𝑓‘𝑘)))
10	9	breq1d 5124	. . . . . . . . . 10 ⊢ (𝑦 = (𝑓‘𝑘) → ((𝐿‘𝑦) ≤ 1 ↔ (𝐿‘(𝑓‘𝑘)) ≤ 1))
11		2fveq3 6890	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑓‘𝑘) → (𝑀‘(𝑇‘𝑦)) = (𝑀‘(𝑇‘(𝑓‘𝑘))))
12	11	breq1d 5124	. . . . . . . . . 10 ⊢ (𝑦 = (𝑓‘𝑘) → ((𝑀‘(𝑇‘𝑦)) ≤ 𝑘 ↔ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘))
13	10, 12	imbi12d 347	. . . . . . . . 9 ⊢ (𝑦 = (𝑓‘𝑘) → (((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘) ↔ ((𝐿‘(𝑓‘𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)))
14	13	notbid 321	. . . . . . . 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 10426	. . . . . . 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 155	. . . . . 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 3099	. . . . . . . . 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 339	. . . . . . . 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 1847	. . . . . . 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 1871	. . . . . . 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 278	. . . . . 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 3123	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘) ↔ ¬ ∃𝑦 ∈ 𝑋 ¬ ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘))
23	22	rexbii 3119	. . . . . . 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 3124	. . . . . . 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 278	. . . . . 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 295	. . . . 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 12243	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
28	27	anim1i 626	. . . . . 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 3105	. . . . 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 18	. . . 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 220	. . 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 31097	. . 3 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑁‘𝑇) ∈ ℝ ↔ ∃𝑘 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘)))
39	31, 38	imbitrrid 249	. 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 411	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 400  ∀wal 1566   = wceq 1568  ∃wex 1807   ∈ wcel 2150  ∀wral 3086  ∃wrex 3096   class class class wbr 5114  ⟶wf 6536  ‘cfv 6540  (class class class)co 7414  ℝcr 11102  1c1 11104   ≤ cle 11247  ℕcn 12236  NrmCVeccnv 30906  BaseSetcba 30908  norm_CVcnmcv 30912   normOp_OLD cnmoo 31063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736  ax-inf2 9613  ax-cc 10422  ax-cnex 11159  ax-resscn 11160  ax-1cn 11161  ax-icn 11162  ax-addcl 11163  ax-addrcl 11164  ax-mulcl 11165  ax-mulrcl 11166  ax-mulcom 11167  ax-addass 11168  ax-mulass 11169  ax-distr 11170  ax-i2m1 11171  ax-1ne0 11172  ax-1rid 11173  ax-rnegex 11174  ax-rrecex 11175  ax-cnre 11176  ax-pre-lttri 11177  ax-pre-lttrn 11178  ax-pre-ltadd 11179  ax-pre-mulgt0 11180  ax-pre-sup 11181

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-nel 3072  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5560  df-eprel 5565  df-po 5573  df-so 5574  df-fr 5618  df-we 5620  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-er 8697  df-map 8829  df-en 8947  df-dom 8948  df-sdom 8949  df-sup 9405  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11446  df-neg 11447  df-div 11875  df-nn 12237  df-2 12306  df-3 12307  df-n0 12508  df-z 12595  df-uz 12866  df-rp 13020  df-seq 14041  df-exp 14101  df-cj 15153  df-re 15154  df-im 15155  df-sqrt 15289  df-abs 15290  df-grpo 30815  df-gid 30816  df-ginv 30817  df-ablo 30867  df-vc 30881  df-nv 30914  df-va 30917  df-ba 30918  df-sm 30919  df-0v 30920  df-nmcv 30922  df-nmoo 31067

This theorem is referenced by: (None)