Theorem cssmre 20037

Description: The closed subspaces of a pre-Hilbert space are a Moore system. Unlike many of our other examples of closure systems, this one is not usually an algebraic closure system df-acs 16249: consider the Hilbert space of sequences NN --> RR with convergent sum; the subspace of all sequences with finite support is the classic example of a non-closed subspace, but for every finite set of sequences of finite support, there is a finite-dimensional (and hence closed) subspace containing all of the sequences, so if closed subspaces were an algebraic closure system this would violate acsfiel 16315. (Contributed by Mario Carneiro, 13-Oct-2015.)

Hypotheses

Ref	Expression
cssmre.v	|- V = ( Base ` W )
cssmre.c	|- C = ( CSubSp ` W )

Assertion

Ref	Expression
cssmre	|- ( W e. PreHil -> C e. (Moore ` V ) )

Proof of Theorem cssmre

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cssmre.v	. . . . . 6 |- V = ( Base ` W )
2		cssmre.c	. . . . . 6 |- C = ( CSubSp ` W )
3	1, 2	cssss 20029	. . . . 5 |- ( x e. C -> x C_ V )
4		selpw 4165	. . . . 5 |- ( x e. ~P V <-> x C_ V )
5	3, 4	sylibr 224	. . . 4 |- ( x e. C -> x e. ~P V )
6	5	a1i 11	. . 3 |- ( W e. PreHil -> ( x e. C -> x e. ~P V ) )
7	6	ssrdv 3609	. 2 |- ( W e. PreHil -> C C_ ~P V )
8	1, 2	css1 20034	. 2 |- ( W e. PreHil -> V e. C )
9		intss1 4492	. . . . . . . . . . . 12 |- ( z e. x -> |^| x C_ z )
10		eqid 2622	. . . . . . . . . . . . 13 |- ( ocv ` W ) = ( ocv ` W )
11	10	ocv2ss 20017	. . . . . . . . . . . 12 |- ( |^| x C_ z -> ( ( ocv ` W ) ` z ) C_ ( ( ocv ` W ) ` |^| x ) )
12	10	ocv2ss 20017	. . . . . . . . . . . 12 |- ( ( ( ocv ` W ) ` z ) C_ ( ( ocv ` W ) ` |^| x ) -> ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) C_ ( ( ocv ` W ) ` ( ( ocv ` W ) ` z ) ) )
13	9, 11, 12	3syl 18	. . . . . . . . . . 11 |- ( z e. x -> ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) C_ ( ( ocv ` W ) ` ( ( ocv ` W ) ` z ) ) )
14	13	ad2antll 765	. . . . . . . . . 10 |- ( ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) /\ ( y e. ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) /\ z e. x ) ) -> ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) C_ ( ( ocv ` W ) ` ( ( ocv ` W ) ` z ) ) )
15		simprl 794	. . . . . . . . . 10 |- ( ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) /\ ( y e. ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) /\ z e. x ) ) -> y e. ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) )
16	14, 15	sseldd 3604	. . . . . . . . 9 |- ( ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) /\ ( y e. ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) /\ z e. x ) ) -> y e. ( ( ocv ` W ) ` ( ( ocv ` W ) ` z ) ) )
17		simpl2 1065	. . . . . . . . . . 11 |- ( ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) /\ ( y e. ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) /\ z e. x ) ) -> x C_ C )
18		simprr 796	. . . . . . . . . . 11 |- ( ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) /\ ( y e. ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) /\ z e. x ) ) -> z e. x )
19	17, 18	sseldd 3604	. . . . . . . . . 10 |- ( ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) /\ ( y e. ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) /\ z e. x ) ) -> z e. C )
20	10, 2	cssi 20028	. . . . . . . . . 10 |- ( z e. C -> z = ( ( ocv ` W ) ` ( ( ocv ` W ) ` z ) ) )
21	19, 20	syl 17	. . . . . . . . 9 |- ( ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) /\ ( y e. ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) /\ z e. x ) ) -> z = ( ( ocv ` W ) ` ( ( ocv ` W ) ` z ) ) )
22	16, 21	eleqtrrd 2704	. . . . . . . 8 |- ( ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) /\ ( y e. ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) /\ z e. x ) ) -> y e. z )
23	22	expr 643	. . . . . . 7 |- ( ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) /\ y e. ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) ) -> ( z e. x -> y e. z ) )
24	23	alrimiv 1855	. . . . . 6 |- ( ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) /\ y e. ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) ) -> A. z ( z e. x -> y e. z ) )
25		vex 3203	. . . . . . 7 |- y e. _V
26	25	elint 4481	. . . . . 6 |- ( y e. |^| x <-> A. z ( z e. x -> y e. z ) )
27	24, 26	sylibr 224	. . . . 5 |- ( ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) /\ y e. ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) ) -> y e. |^| x )
28	27	ex 450	. . . 4 |- ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) -> ( y e. ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) -> y e. |^| x ) )
29	28	ssrdv 3609	. . 3 |- ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) -> ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) C_ |^| x )
30		simp1 1061	. . . 4 |- ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) -> W e. PreHil )
31		intssuni 4499	. . . . . 6 |- ( x =/= (/) -> |^| x C_ U. x )
32	31	3ad2ant3 1084	. . . . 5 |- ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) -> |^| x C_ U. x )
33		simp2 1062	. . . . . . 7 |- ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) -> x C_ C )
34	7	3ad2ant1 1082	. . . . . . 7 |- ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) -> C C_ ~P V )
35	33, 34	sstrd 3613	. . . . . 6 |- ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) -> x C_ ~P V )
36		sspwuni 4611	. . . . . 6 |- ( x C_ ~P V <-> U. x C_ V )
37	35, 36	sylib 208	. . . . 5 |- ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) -> U. x C_ V )
38	32, 37	sstrd 3613	. . . 4 |- ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) -> |^| x C_ V )
39	1, 2, 10	iscss2 20030	. . . 4 |- ( ( W e. PreHil /\ |^| x C_ V ) -> ( |^| x e. C <-> ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) C_ |^| x ) )
40	30, 38, 39	syl2anc 693	. . 3 |- ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) -> ( |^| x e. C <-> ( ( ocv ` W ) ` ( ( ocv ` W ) ` |^| x ) ) C_ |^| x ) )
41	29, 40	mpbird 247	. 2 |- ( ( W e. PreHil /\ x C_ C /\ x =/= (/) ) -> |^| x e. C )
42	7, 8, 41	ismred 16262	1 |- ( W e. PreHil -> C e. (Moore ` V ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   ` cfv 5888   Basecbs 15857  Moorecmre 16242   PreHilcphl 19969   ocvcocv 20004   CSubSpccss 20005

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

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-int 4476  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-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-0g 16102  df-mre 16246  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-rnghom 18715  df-staf 18845  df-srng 18846  df-lmod 18865  df-lmhm 19022  df-lvec 19103  df-sra 19172  df-rgmod 19173  df-phl 19971  df-ocv 20007  df-css 20008

This theorem is referenced by:  mrccss  20038