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Theorem hlimi 21597
 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
StepHypRef Expression
1 df-hlim 21382 . . . 4
21relopabi 4718 . . 3
32brrelexi 4636 . 2
4 nnex 9632 . . . 4
5 fex 5601 . . . 4
64, 5mpan2 655 . . 3
8 hlim.1 . . 3
9 feq1 5232 . . . . . 6
10 eleq1 2313 . . . . . 6
119, 10bi2anan9 848 . . . . 5
12 fveq1 5376 . . . . . . . . . 10
13 oveq12 5719 . . . . . . . . . 10
1412, 13sylan 459 . . . . . . . . 9
1514fveq2d 5381 . . . . . . . 8
1615breq1d 3930 . . . . . . 7
1716rexralbidv 2549 . . . . . 6
1817ralbidv 2527 . . . . 5
1911, 18anbi12d 694 . . . 4
2019, 1brabga 4172 . . 3
218, 20mpan2 655 . 2
223, 7, 21pm5.21nii 344 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wceq 1619   wcel 1621  wral 2509  wrex 2510  cvv 2727   class class class wbr 3920  wf 4588  cfv 4592  (class class class)co 5710   clt 8747  cn 9626  cuz 10109  crp 10233  chil 21329  cno 21333   cmv 21335   chli 21337 This theorem is referenced by:  hlimseqi  21598  hlimveci  21599  hlimconvi  21600  hlim2  21601 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-i2m1 8685  ax-1ne0 8686  ax-rrecex 8689  ax-cnre 8690 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-recs 6274  df-rdg 6309  df-n 9627  df-hlim 21382
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