Theorem chlim 9092

Description: The limit property of a closed subspace of a Hilbert space.

Hypothesis

Ref	Expression
chlim.1	|- A e. V

Assertion

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

Proof of Theorem chlim

Step	Hyp	Ref	Expression
1		closedsub 9081	. . . 4 |- (H e. CH <-> (H e. SH /\ A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H)))
2	1	pm3.27bi 326	. . 3 |- (H e. CH -> A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H))
3		nnex 5895	. . . . . . 7 |- NN e. V
4		fex 3649	. . . . . . 7 |- ((F:NN-->H /\ NN e. V) -> F e. V)
5	3, 4	mpan2 695	. . . . . 6 |- (F:NN-->H -> F e. V)
6	5	adantr 389	. . . . 5 |- ((F:NN-->H /\ F ~~>v A) -> F e. V)
7		feq1 3617	. . . . . . . . . 10 |- (f = F -> (f:NN-->H <-> F:NN-->H))
8		breq1 2619	. . . . . . . . . 10 |- (f = F -> (f ~~>v x <-> F ~~>v x))
9	7, 8	anbi12d 627	. . . . . . . . 9 |- (f = F -> ((f:NN-->H /\ f ~~>v x) <-> (F:NN-->H /\ F ~~>v x)))
10	9	imbi1d 612	. . . . . . . 8 |- (f = F -> (((f:NN-->H /\ f ~~>v x) -> x e. H) <-> ((F:NN-->H /\ F ~~>v x) -> x e. H)))
11	10	albidv 1278	. . . . . . 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	cla4gv 1860	. . . . . 6 |- (F e. V -> (A.fA.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 2620	. . . . . . . . 9 |- (x = A -> (F ~~>v x <-> F ~~>v A))
15	14	anbi2d 615	. . . . . . . 8 |- (x = A -> ((F:NN-->H /\ F ~~>v x) <-> (F:NN-->H /\ F ~~>v A)))
16		eleq1 1533	. . . . . . . 8 |- (x = A -> (x e. H <-> A e. H))
17	15, 16	imbi12d 625	. . . . . . 7 |- (x = A -> (((F:NN-->H /\ F ~~>v x) -> x e. H) <-> ((F:NN-->H /\ F ~~>v A) -> A e. H)))
18	13, 17	cla4v 1866	. . . . . 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 22	. . . . 5 |- (F e. V -> (A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H) -> ((F:NN-->H /\ F ~~>v A) -> A e. H)))
20	6, 19	syl 10	. . . 4 |- ((F:NN-->H /\ F ~~>v A) -> (A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H) -> ((F:NN-->H /\ F ~~>v A) -> A e. H)))
21	20	pm2.43b 67	. . 3 |- (A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H) -> ((F:NN-->H /\ F ~~>v A) -> A e. H))
22	2, 21	syl 10	. 2 |- (H e. CH -> ((F:NN-->H /\ F ~~>v A) -> A e. H))
23	22	3impib 830	1 |- ((H e. CH /\ F:NN-->H /\ F ~~>v A) -> A e. H)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774  A.wal 953   = wceq 955   e. wcel 957  Vcvv 1809   class class class wbr 2616  -->wf 3175  NNcn 5283   ~~>v chli 8780  SHcsh 8781  CHcch 8782

This theorem is referenced by:  chintcl 9283  osumlem6 9573

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-inf2 4612

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-pss 2053  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2501  df-int 2531  df-iun 2565  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949  df-lim 2950  df-suc 2951  df-om 3129  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fv 3195  df-rdg 3929  df-opr 3962  df-oprab 3963  df-1st 4076  df-2nd 4077  df-1o 4130  df-oadd 4132  df-omul 4133  df-er 4258  df-ec 4260  df-qs 4263  df-ni 4987  df-pli 4988  df-mi 4989  df-lti 4990  df-plpq 5022  df-mpq 5023  df-enq 5024  df-nq 5025  df-plq 5026  df-mq 5027  df-rq 5028  df-ltq 5029  df-1q 5030  df-np 5073  df-1p 5074  df-plp 5075  df-mp 5076  df-ltp 5077  df-plpr 5151  df-mpr 5152  df-enr 5153  df-nr 5154  df-plr 5155  df-mr 5156  df-ltr 5157  df-0r 5158  df-1r 5159  df-m1r 5160  df-c 5227  df-0 5228  df-1 5229  df-i 5230  df-r 5231  df-plus 5232  df-mul 5233  df-sub 5343  df-neg 5345  df-n 5887  df-ch 9080

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