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Theorem nmobndseqiALT 28549
Description: Alternate shorter proof of nmobndseqi 28548 based on axioms ax-reg 9048 and ax-ac2 9877 instead of ax-cc 9849. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.) (Proof modification 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
nmobndseqiALT ((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 nmobndseqiALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 impexp 453 . . . . . 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 3339 . . . . . . 7 (∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) ↔ (∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1 → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘))
32imbi2i 338 . . . . . 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 280 . . . . 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 1813 . . . 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 nnex 11636 . . . . . 6 ℕ ∈ V
7 fveq2 6663 . . . . . . . 8 (𝑦 = (𝑓𝑘) → (𝐿𝑦) = (𝐿‘(𝑓𝑘)))
87breq1d 5067 . . . . . . 7 (𝑦 = (𝑓𝑘) → ((𝐿𝑦) ≤ 1 ↔ (𝐿‘(𝑓𝑘)) ≤ 1))
9 fveq2 6663 . . . . . . . . 9 (𝑦 = (𝑓𝑘) → (𝑇𝑦) = (𝑇‘(𝑓𝑘)))
109fveq2d 6667 . . . . . . . 8 (𝑦 = (𝑓𝑘) → (𝑀‘(𝑇𝑦)) = (𝑀‘(𝑇‘(𝑓𝑘))))
1110breq1d 5067 . . . . . . 7 (𝑦 = (𝑓𝑘) → ((𝑀‘(𝑇𝑦)) ≤ 𝑘 ↔ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘))
128, 11imbi12d 347 . . . . . 6 (𝑦 = (𝑓𝑘) → (((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) ↔ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)))
136, 12ac6n 9899 . . . . 5 (∀𝑓(𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) → ∃𝑘 ∈ ℕ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
14 nnre 11637 . . . . . . 7 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
1514anim1i 616 . . . . . 6 ((𝑘 ∈ ℕ ∧ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘)) → (𝑘 ∈ ℝ ∧ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘)))
1615reximi2 3242 . . . . 5 (∃𝑘 ∈ ℕ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘) → ∃𝑘 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
1713, 16syl 17 . . . 4 (∀𝑓(𝑓:ℕ⟶𝑋 → ∃𝑘 ∈ ℕ ((𝐿‘(𝑓𝑘)) ≤ 1 → (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘)) → ∃𝑘 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
185, 17sylbi 219 . . 3 (∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) → ∃𝑘 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘))
19 nmoubi.1 . . . 4 𝑋 = (BaseSet‘𝑈)
20 nmoubi.y . . . 4 𝑌 = (BaseSet‘𝑊)
21 nmoubi.l . . . 4 𝐿 = (normCV𝑈)
22 nmoubi.m . . . 4 𝑀 = (normCV𝑊)
23 nmoubi.3 . . . 4 𝑁 = (𝑈 normOpOLD 𝑊)
24 nmoubi.u . . . 4 𝑈 ∈ NrmCVec
25 nmoubi.w . . . 4 𝑊 ∈ NrmCVec
2619, 20, 21, 22, 23, 24, 25nmobndi 28544 . . 3 (𝑇:𝑋𝑌 → ((𝑁𝑇) ∈ ℝ ↔ ∃𝑘 ∈ ℝ ∀𝑦𝑋 ((𝐿𝑦) ≤ 1 → (𝑀‘(𝑇𝑦)) ≤ 𝑘)))
2718, 26syl5ibr 248 . 2 (𝑇:𝑋𝑌 → (∀𝑓((𝑓:ℕ⟶𝑋 ∧ ∀𝑘 ∈ ℕ (𝐿‘(𝑓𝑘)) ≤ 1) → ∃𝑘 ∈ ℕ (𝑀‘(𝑇‘(𝑓𝑘))) ≤ 𝑘) → (𝑁𝑇) ∈ ℝ))
2827imp 409 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:  wi 4  wa 398  wal 1528   = wceq 1530  wcel 2107  wral 3136  wrex 3137   class class class wbr 5057  wf 6344  cfv 6348  (class class class)co 7148  cr 10528  1c1 10530  cle 10668  cn 11630  NrmCVeccnv 28353  BaseSetcba 28355  normCVcnmcv 28359   normOpOLD cnmoo 28510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-reg 9048  ax-inf2 9096  ax-ac2 9877  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-sup 8898  df-r1 9185  df-rank 9186  df-card 9360  df-ac 9534  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-seq 13362  df-exp 13422  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-grpo 28262  df-gid 28263  df-ginv 28264  df-ablo 28314  df-vc 28328  df-nv 28361  df-va 28364  df-ba 28365  df-sm 28366  df-0v 28367  df-nmcv 28369  df-nmoo 28514
This theorem is referenced by: (None)
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