Theorem chlimi 28091

Description: The limit property of a closed subspace of a Hilbert space. (Contributed by NM, 14-Sep-1999.) (New usage is discouraged.)

Hypothesis

Ref	Expression
chlim.1	|- A e. _V

Assertion

Ref	Expression
chlimi	|- ( ( H e. CH /\ F : NN --> H /\ F ~~>v A ) -> A e. H )

Proof of Theorem chlimi

Dummy variables x f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isch2 28080	. . . 4 |- ( H e. CH <-> ( H e. SH /\ A. f A. x ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) ) )
2	1	simprbi 480	. . 3 |- ( H e. CH -> A. f A. x ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) )
3		nnex 11026	. . . . . . 7 |- NN e. _V
4		fex 6490	. . . . . . 7 |- ( ( F : NN --> H /\ NN e. _V ) -> F e. _V )
5	3, 4	mpan2 707	. . . . . 6 |- ( F : NN --> H -> F e. _V )
6	5	adantr 481	. . . . 5 |- ( ( F : NN --> H /\ F ~~>v A ) -> F e. _V )
7		feq1 6026	. . . . . . . . . 10 |- ( f = F -> ( f : NN --> H <-> F : NN --> H ) )
8		breq1 4656	. . . . . . . . . 10 |- ( f = F -> ( f ~~>v x <-> F ~~>v x ) )
9	7, 8	anbi12d 747	. . . . . . . . 9 |- ( f = F -> ( ( f : NN --> H /\ f ~~>v x ) <-> ( F : NN --> H /\ F ~~>v x ) ) )
10	9	imbi1d 331	. . . . . . . 8 |- ( f = F -> ( ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) <-> ( ( F : NN --> H /\ F ~~>v x ) -> x e. H ) ) )
11	10	albidv 1849	. . . . . . 7 |- ( f = F -> ( A. x ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) <-> A. x ( ( F : NN --> H /\ F ~~>v x ) -> x e. H ) ) )
12	11	spcgv 3293	. . . . . 6 |- ( F e. _V -> ( A. f A. x ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) -> A. x ( ( F : NN --> H /\ F ~~>v x ) -> x e. H ) ) )
13		chlim.1	. . . . . . 7 |- A e. _V
14		breq2 4657	. . . . . . . . 9 |- ( x = A -> ( F ~~>v x <-> F ~~>v A ) )
15	14	anbi2d 740	. . . . . . . 8 |- ( x = A -> ( ( F : NN --> H /\ F ~~>v x ) <-> ( F : NN --> H /\ F ~~>v A ) ) )
16		eleq1 2689	. . . . . . . 8 |- ( x = A -> ( x e. H <-> A e. H ) )
17	15, 16	imbi12d 334	. . . . . . 7 |- ( x = A -> ( ( ( F : NN --> H /\ F ~~>v x ) -> x e. H ) <-> ( ( F : NN --> H /\ F ~~>v A ) -> A e. H ) ) )
18	13, 17	spcv 3299	. . . . . 6 |- ( A. x ( ( F : NN --> H /\ F ~~>v x ) -> x e. H ) -> ( ( F : NN --> H /\ F ~~>v A ) -> A e. H ) )
19	12, 18	syl6 35	. . . . 5 |- ( F e. _V -> ( A. f A. x ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) -> ( ( F : NN --> H /\ F ~~>v A ) -> A e. H ) ) )
20	6, 19	syl 17	. . . 4 |- ( ( F : NN --> H /\ F ~~>v A ) -> ( A. f A. x ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) -> ( ( F : NN --> H /\ F ~~>v A ) -> A e. H ) ) )
21	20	pm2.43b 55	. . 3 |- ( A. f A. x ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) -> ( ( F : NN --> H /\ F ~~>v A ) -> A e. H ) )
22	2, 21	syl 17	. 2 |- ( H e. CH -> ( ( F : NN --> H /\ F ~~>v A ) -> A e. H ) )
23	22	3impib 1262	1 |- ( ( H e. CH /\ F : NN --> H /\ F ~~>v A ) -> A e. H )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   -->wf 5884   NNcn 11020   ~~>v chli 27784   SHcsh 27785   CHcch 27786

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-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-map 7859  df-nn 11021  df-ch 28078

This theorem is referenced by:  hhsscms  28136  chintcli  28190  chscllem4  28499