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Theorem hlimi 22678
 Description: Express the predicate: The limit of vector sequence in a Hilbert space is , i.e. converges to . This means that for any real , no matter how small, there always exists an integer such that the norm of any later vector in the sequence minus the limit is less than . 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
Assertion
Ref Expression
hlimi
Distinct variable groups:   ,,,   ,,,

Proof of Theorem hlimi
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-hlim 22463 . . . 4
21relopabi 4991 . . 3
32brrelexi 4909 . 2
4 nnex 9995 . . . 4
5 fex 5960 . . . 4
64, 5mpan2 653 . . 3
8 hlim.1 . . 3
9 feq1 5567 . . . . . 6
10 eleq1 2495 . . . . . 6
119, 10bi2anan9 844 . . . . 5
12 fveq1 5718 . . . . . . . . . 10
13 oveq12 6081 . . . . . . . . . 10
1412, 13sylan 458 . . . . . . . . 9
1514fveq2d 5723 . . . . . . . 8
1615breq1d 4214 . . . . . . 7
1716rexralbidv 2741 . . . . . 6
1817ralbidv 2717 . . . . 5
1911, 18anbi12d 692 . . . 4
2019, 1brabga 4461 . . 3
218, 20mpan2 653 . 2
223, 7, 21pm5.21nii 343 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725  wral 2697  wrex 2698  cvv 2948   class class class wbr 4204  wf 5441  cfv 5445  (class class class)co 6072   clt 9109  cn 9989  cuz 10477  crp 10601  chil 22410  cno 22414   cmv 22416   chli 22418 This theorem is referenced by:  hlimseqi  22679  hlimveci  22680  hlimconvi  22681  hlim2  22682 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-i2m1 9047  ax-1ne0 9048  ax-rrecex 9051  ax-cnre 9052 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-recs 6624  df-rdg 6659  df-nn 9990  df-hlim 22463
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