Theorem axhcompl-zf 27855

Description: Derive axiom ax-hcompl 28059 from Hilbert space under ZF set theory. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
axhil.1	|- U = <. <. +h , .h >. , normh >.
axhil.2	|- U e. CHilOLD

Assertion

Ref	Expression
axhcompl-zf	|- ( F e. Cauchy -> E. x e. ~H F ~~>v x )

Distinct variable groups:   x, F   x, U

Proof of Theorem axhcompl-zf

Step	Hyp	Ref	Expression
1		axhil.2	. . . . . 6 |- U e. CHilOLD
2		simpl 473	. . . . . 6 |- ( ( F e. ( Cau ` ( IndMet ` U ) ) /\ F e. ( ~H ^m NN ) ) -> F e. ( Cau ` ( IndMet ` U ) ) )
3		eqid 2622	. . . . . . 7 |- ( IndMet ` U ) = ( IndMet ` U )
4		eqid 2622	. . . . . . 7 |- ( MetOpen ` ( IndMet ` U ) ) = ( MetOpen ` ( IndMet ` U ) )
5	3, 4	hlcompl 27771	. . . . . 6 |- ( ( U e. CHilOLD /\ F e. ( Cau ` ( IndMet ` U ) ) ) -> F e. dom ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) )
6	1, 2, 5	sylancr 695	. . . . 5 |- ( ( F e. ( Cau ` ( IndMet ` U ) ) /\ F e. ( ~H ^m NN ) ) -> F e. dom ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) )
7		eldm2g 5320	. . . . . 6 |- ( F e. ( Cau ` ( IndMet ` U ) ) -> ( F e. dom ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) <-> E. x <. F , x >. e. ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) ) )
8	7	adantr 481	. . . . 5 |- ( ( F e. ( Cau ` ( IndMet ` U ) ) /\ F e. ( ~H ^m NN ) ) -> ( F e. dom ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) <-> E. x <. F , x >. e. ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) ) )
9	6, 8	mpbid 222	. . . 4 |- ( ( F e. ( Cau ` ( IndMet ` U ) ) /\ F e. ( ~H ^m NN ) ) -> E. x <. F , x >. e. ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) )
10		df-br 4654	. . . . . 6 |- ( F ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) x <-> <. F , x >. e. ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) )
11	1	hlnvi 27748	. . . . . . . . . 10 |- U e. NrmCVec
12		df-hba 27826	. . . . . . . . . . . 12 |- ~H = ( BaseSet ` <. <. +h , .h >. , normh >. )
13		axhil.1	. . . . . . . . . . . . 13 |- U = <. <. +h , .h >. , normh >.
14	13	fveq2i 6194	. . . . . . . . . . . 12 |- ( BaseSet ` U ) = ( BaseSet ` <. <. +h , .h >. , normh >. )
15	12, 14	eqtr4i 2647	. . . . . . . . . . 11 |- ~H = ( BaseSet ` U )
16	15, 3	imsxmet 27547	. . . . . . . . . 10 |- ( U e. NrmCVec -> ( IndMet ` U ) e. ( *Met ` ~H ) )
17	4	mopntopon 22244	. . . . . . . . . 10 |- ( ( IndMet ` U ) e. ( *Met ` ~H ) -> ( MetOpen ` ( IndMet ` U ) ) e. (TopOn ` ~H ) )
18	11, 16, 17	mp2b 10	. . . . . . . . 9 |- ( MetOpen ` ( IndMet ` U ) ) e. (TopOn ` ~H )
19		lmcl 21101	. . . . . . . . 9 |- ( ( ( MetOpen ` ( IndMet ` U ) ) e. (TopOn ` ~H ) /\ F ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) x ) -> x e. ~H )
20	18, 19	mpan 706	. . . . . . . 8 |- ( F ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) x -> x e. ~H )
21	20	a1i 11	. . . . . . 7 |- ( ( F e. ( Cau ` ( IndMet ` U ) ) /\ F e. ( ~H ^m NN ) ) -> ( F ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) x -> x e. ~H ) )
22	13, 11, 15, 3, 4	h2hlm 27837	. . . . . . . . . . . 12 |- ~~>v = ( ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) |` ( ~H ^m NN ) )
23	22	breqi 4659	. . . . . . . . . . 11 |- ( F ~~>v x <-> F ( ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) |` ( ~H ^m NN ) ) x )
24		vex 3203	. . . . . . . . . . . 12 |- x e. _V
25	24	brres 5402	. . . . . . . . . . 11 |- ( F ( ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) |` ( ~H ^m NN ) ) x <-> ( F ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) x /\ F e. ( ~H ^m NN ) ) )
26		ancom 466	. . . . . . . . . . 11 |- ( ( F ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) x /\ F e. ( ~H ^m NN ) ) <-> ( F e. ( ~H ^m NN ) /\ F ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) x ) )
27	23, 25, 26	3bitri 286	. . . . . . . . . 10 |- ( F ~~>v x <-> ( F e. ( ~H ^m NN ) /\ F ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) x ) )
28	27	baib 944	. . . . . . . . 9 |- ( F e. ( ~H ^m NN ) -> ( F ~~>v x <-> F ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) x ) )
29	28	adantl 482	. . . . . . . 8 |- ( ( F e. ( Cau ` ( IndMet ` U ) ) /\ F e. ( ~H ^m NN ) ) -> ( F ~~>v x <-> F ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) x ) )
30	29	biimprd 238	. . . . . . 7 |- ( ( F e. ( Cau ` ( IndMet ` U ) ) /\ F e. ( ~H ^m NN ) ) -> ( F ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) x -> F ~~>v x ) )
31	21, 30	jcad 555	. . . . . 6 |- ( ( F e. ( Cau ` ( IndMet ` U ) ) /\ F e. ( ~H ^m NN ) ) -> ( F ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) x -> ( x e. ~H /\ F ~~>v x ) ) )
32	10, 31	syl5bir 233	. . . . 5 |- ( ( F e. ( Cau ` ( IndMet ` U ) ) /\ F e. ( ~H ^m NN ) ) -> ( <. F , x >. e. ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) -> ( x e. ~H /\ F ~~>v x ) ) )
33	32	eximdv 1846	. . . 4 |- ( ( F e. ( Cau ` ( IndMet ` U ) ) /\ F e. ( ~H ^m NN ) ) -> ( E. x <. F , x >. e. ( ~~> t ` ( MetOpen ` ( IndMet ` U ) ) ) -> E. x ( x e. ~H /\ F ~~>v x ) ) )
34	9, 33	mpd 15	. . 3 |- ( ( F e. ( Cau ` ( IndMet ` U ) ) /\ F e. ( ~H ^m NN ) ) -> E. x ( x e. ~H /\ F ~~>v x ) )
35		elin 3796	. . 3 |- ( F e. ( ( Cau ` ( IndMet ` U ) ) i^i ( ~H ^m NN ) ) <-> ( F e. ( Cau ` ( IndMet ` U ) ) /\ F e. ( ~H ^m NN ) ) )
36		df-rex 2918	. . 3 |- ( E. x e. ~H F ~~>v x <-> E. x ( x e. ~H /\ F ~~>v x ) )
37	34, 35, 36	3imtr4i 281	. 2 |- ( F e. ( ( Cau ` ( IndMet ` U ) ) i^i ( ~H ^m NN ) ) -> E. x e. ~H F ~~>v x )
38	13, 11, 15, 3	h2hcau 27836	. 2 |- Cauchy = ( ( Cau ` ( IndMet ` U ) ) i^i ( ~H ^m NN ) )
39	37, 38	eleq2s 2719	1 |- ( F e. Cauchy -> E. x e. ~H F ~~>v x )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   i^i cin 3573   <.cop 4183   class class class wbr 4653   dom cdm 5114   |` cres 5116   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   NNcn 11020   *Metcxmt 19731   MetOpencmopn 19736  TopOnctopon 20715   ~~> tclm 21030   Caucca 23051   NrmCVeccnv 27439   BaseSetcba 27441   IndMetcims 27446   CHilOLDchlo 27741   ~Hchil 27776   +h cva 27777   .h csm 27778   normhcno 27780   Cauchyccau 27783   ~~>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-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-bases 20750  df-ntr 20824  df-nei 20902  df-lm 21033  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-cfil 23053  df-cau 23054  df-cmet 23055  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-vs 27454  df-nmcv 27455  df-ims 27456  df-cbn 27719  df-hlo 27742  df-hba 27826  df-hvsub 27828  df-hlim 27829  df-hcau 27830

This theorem is referenced by: (None)