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Theorem hlimi 28957
Description: Express the predicate: The limit of vector sequence 𝐹 in a Hilbert space is 𝐴, i.e. 𝐹 converges to 𝐴. This means that for any real 𝑥, no matter how small, there always exists an integer 𝑦 such that the norm of any later vector in the sequence minus the limit is less than 𝑥. Definition of converge in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
Hypothesis
Ref Expression
hlim.1 𝐴 ∈ V
Assertion
Ref Expression
hlimi (𝐹𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐹   𝑥,𝐴,𝑦,𝑧

Proof of Theorem hlimi
Dummy variables 𝑤 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-hlim 28741 . . . 4 𝑣 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝑓𝑧) − 𝑤)) < 𝑥)}
21relopabi 5687 . . 3 Rel ⇝𝑣
32brrelex1i 5601 . 2 (𝐹𝑣 𝐴𝐹 ∈ V)
4 nnex 11636 . . . 4 ℕ ∈ V
5 fex 6981 . . . 4 ((𝐹:ℕ⟶ ℋ ∧ ℕ ∈ V) → 𝐹 ∈ V)
64, 5mpan2 689 . . 3 (𝐹:ℕ⟶ ℋ → 𝐹 ∈ V)
76ad2antrr 724 . 2 (((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥) → 𝐹 ∈ V)
8 hlim.1 . . 3 𝐴 ∈ V
9 feq1 6488 . . . . . 6 (𝑓 = 𝐹 → (𝑓:ℕ⟶ ℋ ↔ 𝐹:ℕ⟶ ℋ))
10 eleq1 2898 . . . . . 6 (𝑤 = 𝐴 → (𝑤 ∈ ℋ ↔ 𝐴 ∈ ℋ))
119, 10bi2anan9 637 . . . . 5 ((𝑓 = 𝐹𝑤 = 𝐴) → ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ↔ (𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ)))
12 fveq1 6662 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑧) = (𝐹𝑧))
13 oveq12 7157 . . . . . . . . . 10 (((𝑓𝑧) = (𝐹𝑧) ∧ 𝑤 = 𝐴) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐴))
1412, 13sylan 582 . . . . . . . . 9 ((𝑓 = 𝐹𝑤 = 𝐴) → ((𝑓𝑧) − 𝑤) = ((𝐹𝑧) − 𝐴))
1514fveq2d 6667 . . . . . . . 8 ((𝑓 = 𝐹𝑤 = 𝐴) → (norm‘((𝑓𝑧) − 𝑤)) = (norm‘((𝐹𝑧) − 𝐴)))
1615breq1d 5067 . . . . . . 7 ((𝑓 = 𝐹𝑤 = 𝐴) → ((norm‘((𝑓𝑧) − 𝑤)) < 𝑥 ↔ (norm‘((𝐹𝑧) − 𝐴)) < 𝑥))
1716rexralbidv 3299 . . . . . 6 ((𝑓 = 𝐹𝑤 = 𝐴) → (∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝑓𝑧) − 𝑤)) < 𝑥 ↔ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥))
1817ralbidv 3195 . . . . 5 ((𝑓 = 𝐹𝑤 = 𝐴) → (∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝑓𝑧) − 𝑤)) < 𝑥 ↔ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥))
1911, 18anbi12d 632 . . . 4 ((𝑓 = 𝐹𝑤 = 𝐴) → (((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝑓𝑧) − 𝑤)) < 𝑥) ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥)))
2019, 1brabga 5412 . . 3 ((𝐹 ∈ V ∧ 𝐴 ∈ V) → (𝐹𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥)))
218, 20mpan2 689 . 2 (𝐹 ∈ V → (𝐹𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥)))
223, 7, 21pm5.21nii 382 1 (𝐹𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ𝑦)(norm‘((𝐹𝑧) − 𝐴)) < 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1531  wcel 2108  wral 3136  wrex 3137  Vcvv 3493   class class class wbr 5057  wf 6344  cfv 6348  (class class class)co 7148   < clt 10667  cn 11630  cuz 12235  +crp 12381  chba 28688  normcno 28692   cmv 28694  𝑣 chli 28696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-1cn 10587  ax-addcl 10589
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  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-iun 4912  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-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-ov 7151  df-om 7573  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-nn 11631  df-hlim 28741
This theorem is referenced by:  hlimseqi  28958  hlimveci  28959  hlimconvi  28960  hlim2  28961
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