Theorem nmobndseqi 31381

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 456	. . . . . 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 3120	. . . . . . 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 1851	. . . 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 14123	. . . . . . . 8 ⊢ ℕ ≈ ω
9		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑓‘𝑘) → (𝐿‘𝑦) = (𝐿‘(𝑓‘𝑘)))
10	9	breq1d 5112	. . . . . . . . . 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 5112	. . . . . . . . . 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 10517	. . . . . . 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 3089	. . . . . . . . 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 1851	. . . . . . 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 1875	. . . . . . 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 3113	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑋 ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘) ↔ ¬ ∃𝑦 ∈ 𝑋 ¬ ((𝐿‘𝑦) ≤ 1 → (𝑀‘(𝑇‘𝑦)) ≤ 𝑘))
23	22	rexbii 3109	. . . . . . 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 3114	. . . . . . 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 12342	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
28	27	anim1i 627	. . . . . 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 3095	. . . . 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 31377	. . 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 412	1 ⊢ ((𝑇:𝑋⟶𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓‘𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓‘𝑘))) ≤ 𝑘)) → (𝑁‘𝑇) ∈ ℝ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   class class class wbr 5102  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  ℝcr 11199  1c1 11201   ≤ cle 11344  ℕcn 12335  NrmCVeccnv 31186  BaseSetcba 31188  norm_CVcnmcv 31192   normOp_OLD cnmoo 31343

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9642  ax-cc 10513  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-sup 9434  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-n0 12607  df-z 12694  df-uz 12966  df-rp 13121  df-seq 14145  df-exp 14205  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-grpo 31095  df-gid 31096  df-ginv 31097  df-ablo 31147  df-vc 31161  df-nv 31194  df-va 31197  df-ba 31198  df-sm 31199  df-0v 31200  df-nmcv 31202  df-nmoo 31347

This theorem is used by: (None)