Theorem hlimi 28045

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 e. _V

Assertion

Ref	Expression
hlimi	|- ( F ~~>v A <-> ( ( F : NN --> ~H /\ A e. ~H ) /\ A. x e. RR+ E. y e. NN A. z e. ( ZZ>= ` y ) ( normh ` ( ( F ` z ) -h 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.

Step	Hyp	Ref	Expression
1		df-hlim 27829	. . . 4 |- ~~>v = { <. f , w >. | ( ( f : NN --> ~H /\ w e. ~H ) /\ A. x e. RR+ E. y e. NN A. z e. ( ZZ>= ` y ) ( normh ` ( ( f ` z ) -h w ) ) < x ) }
2	1	relopabi 5245	. . 3 |- Rel ~~>v
3	2	brrelexi 5158	. 2 |- ( F ~~>v A -> F e. _V )
4		nnex 11026	. . . 4 |- NN e. _V
5		fex 6490	. . . 4 |- ( ( F : NN --> ~H /\ NN e. _V ) -> F e. _V )
6	4, 5	mpan2 707	. . 3 |- ( F : NN --> ~H -> F e. _V )
7	6	ad2antrr 762	. 2 |- ( ( ( F : NN --> ~H /\ A e. ~H ) /\ A. x e. RR+ E. y e. NN A. z e. ( ZZ>= ` y ) ( normh ` ( ( F ` z ) -h A ) ) < x ) -> F e. _V )
8		hlim.1	. . 3 |- A e. _V
9		feq1 6026	. . . . . 6 |- ( f = F -> ( f : NN --> ~H <-> F : NN --> ~H ) )
10		eleq1 2689	. . . . . 6 |- ( w = A -> ( w e. ~H <-> A e. ~H ) )
11	9, 10	bi2anan9 917	. . . . 5 |- ( ( f = F /\ w = A ) -> ( ( f : NN --> ~H /\ w e. ~H ) <-> ( F : NN --> ~H /\ A e. ~H ) ) )
12		fveq1 6190	. . . . . . . . . 10 |- ( f = F -> ( f ` z ) = ( F ` z ) )
13		oveq12 6659	. . . . . . . . . 10 |- ( ( ( f ` z ) = ( F ` z ) /\ w = A ) -> ( ( f ` z ) -h w ) = ( ( F ` z ) -h A ) )
14	12, 13	sylan 488	. . . . . . . . 9 |- ( ( f = F /\ w = A ) -> ( ( f ` z ) -h w ) = ( ( F ` z ) -h A ) )
15	14	fveq2d 6195	. . . . . . . 8 |- ( ( f = F /\ w = A ) -> ( normh ` ( ( f ` z ) -h w ) ) = ( normh ` ( ( F ` z ) -h A ) ) )
16	15	breq1d 4663	. . . . . . 7 |- ( ( f = F /\ w = A ) -> ( ( normh ` ( ( f ` z ) -h w ) ) < x <-> ( normh ` ( ( F ` z ) -h A ) ) < x ) )
17	16	rexralbidv 3058	. . . . . 6 |- ( ( f = F /\ w = A ) -> ( E. y e. NN A. z e. ( ZZ>= ` y ) ( normh ` ( ( f ` z ) -h w ) ) < x <-> E. y e. NN A. z e. ( ZZ>= ` y ) ( normh ` ( ( F ` z ) -h A ) ) < x ) )
18	17	ralbidv 2986	. . . . 5 |- ( ( f = F /\ w = A ) -> ( A. x e. RR+ E. y e. NN A. z e. ( ZZ>= ` y ) ( normh ` ( ( f ` z ) -h w ) ) < x <-> A. x e. RR+ E. y e. NN A. z e. ( ZZ>= ` y ) ( normh ` ( ( F ` z ) -h A ) ) < x ) )
19	11, 18	anbi12d 747	. . . 4 |- ( ( f = F /\ w = A ) -> ( ( ( f : NN --> ~H /\ w e. ~H ) /\ A. x e. RR+ E. y e. NN A. z e. ( ZZ>= ` y ) ( normh ` ( ( f ` z ) -h w ) ) < x ) <-> ( ( F : NN --> ~H /\ A e. ~H ) /\ A. x e. RR+ E. y e. NN A. z e. ( ZZ>= ` y ) ( normh ` ( ( F ` z ) -h A ) ) < x ) ) )
20	19, 1	brabga 4989	. . 3 |- ( ( F e. _V /\ A e. _V ) -> ( F ~~>v A <-> ( ( F : NN --> ~H /\ A e. ~H ) /\ A. x e. RR+ E. y e. NN A. z e. ( ZZ>= ` y ) ( normh ` ( ( F ` z ) -h A ) ) < x ) ) )
21	8, 20	mpan2 707	. 2 |- ( F e. _V -> ( F ~~>v A <-> ( ( F : NN --> ~H /\ A e. ~H ) /\ A. x e. RR+ E. y e. NN A. z e. ( ZZ>= ` y ) ( normh ` ( ( F ` z ) -h A ) ) < x ) ) )
22	3, 7, 21	pm5.21nii 368	1 |- ( F ~~>v A <-> ( ( F : NN --> ~H /\ A e. ~H ) /\ A. x e. RR+ E. y e. NN A. z e. ( ZZ>= ` y ) ( normh ` ( ( F ` z ) -h A ) ) < x ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   < clt 10074   NNcn 11020   ZZ>=cuz 11687   RR+crp 11832   ~Hchil 27776   normhcno 27780   -h cmv 27782   ~~>v chli 27784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-nn 11021  df-hlim 27829

This theorem is referenced by:  hlimseqi  28046  hlimveci  28047  hlimconvi  28048  hlim2  28049