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Theorem cssmre 21630
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 17574: consider the Hilbert space of sequences β„•βŸΆβ„ 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 17639. (Contributed by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
cssmre.v 𝑉 = (Baseβ€˜π‘Š)
cssmre.c 𝐢 = (ClSubSpβ€˜π‘Š)
Assertion
Ref Expression
cssmre (π‘Š ∈ PreHil β†’ 𝐢 ∈ (Mooreβ€˜π‘‰))

Proof of Theorem cssmre
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cssmre.v . . . . . 6 𝑉 = (Baseβ€˜π‘Š)
2 cssmre.c . . . . . 6 𝐢 = (ClSubSpβ€˜π‘Š)
31, 2cssss 21622 . . . . 5 (π‘₯ ∈ 𝐢 β†’ π‘₯ βŠ† 𝑉)
4 velpw 4609 . . . . 5 (π‘₯ ∈ 𝒫 𝑉 ↔ π‘₯ βŠ† 𝑉)
53, 4sylibr 233 . . . 4 (π‘₯ ∈ 𝐢 β†’ π‘₯ ∈ 𝒫 𝑉)
65a1i 11 . . 3 (π‘Š ∈ PreHil β†’ (π‘₯ ∈ 𝐢 β†’ π‘₯ ∈ 𝒫 𝑉))
76ssrdv 3986 . 2 (π‘Š ∈ PreHil β†’ 𝐢 βŠ† 𝒫 𝑉)
81, 2css1 21627 . 2 (π‘Š ∈ PreHil β†’ 𝑉 ∈ 𝐢)
9 intss1 4968 . . . . . . . . . . . 12 (𝑧 ∈ π‘₯ β†’ ∩ π‘₯ βŠ† 𝑧)
10 eqid 2727 . . . . . . . . . . . . 13 (ocvβ€˜π‘Š) = (ocvβ€˜π‘Š)
1110ocv2ss 21610 . . . . . . . . . . . 12 (∩ π‘₯ βŠ† 𝑧 β†’ ((ocvβ€˜π‘Š)β€˜π‘§) βŠ† ((ocvβ€˜π‘Š)β€˜βˆ© π‘₯))
1210ocv2ss 21610 . . . . . . . . . . . 12 (((ocvβ€˜π‘Š)β€˜π‘§) βŠ† ((ocvβ€˜π‘Š)β€˜βˆ© π‘₯) β†’ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)) βŠ† ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜π‘§)))
139, 11, 123syl 18 . . . . . . . . . . 11 (𝑧 ∈ π‘₯ β†’ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)) βŠ† ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜π‘§)))
1413ad2antll 727 . . . . . . . . . 10 (((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) ∧ (𝑦 ∈ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)) ∧ 𝑧 ∈ π‘₯)) β†’ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)) βŠ† ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜π‘§)))
15 simprl 769 . . . . . . . . . 10 (((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) ∧ (𝑦 ∈ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)) ∧ 𝑧 ∈ π‘₯)) β†’ 𝑦 ∈ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)))
1614, 15sseldd 3981 . . . . . . . . 9 (((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) ∧ (𝑦 ∈ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)) ∧ 𝑧 ∈ π‘₯)) β†’ 𝑦 ∈ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜π‘§)))
17 simpl2 1189 . . . . . . . . . . 11 (((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) ∧ (𝑦 ∈ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)) ∧ 𝑧 ∈ π‘₯)) β†’ π‘₯ βŠ† 𝐢)
18 simprr 771 . . . . . . . . . . 11 (((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) ∧ (𝑦 ∈ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)) ∧ 𝑧 ∈ π‘₯)) β†’ 𝑧 ∈ π‘₯)
1917, 18sseldd 3981 . . . . . . . . . 10 (((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) ∧ (𝑦 ∈ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)) ∧ 𝑧 ∈ π‘₯)) β†’ 𝑧 ∈ 𝐢)
2010, 2cssi 21621 . . . . . . . . . 10 (𝑧 ∈ 𝐢 β†’ 𝑧 = ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜π‘§)))
2119, 20syl 17 . . . . . . . . 9 (((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) ∧ (𝑦 ∈ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)) ∧ 𝑧 ∈ π‘₯)) β†’ 𝑧 = ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜π‘§)))
2216, 21eleqtrrd 2831 . . . . . . . 8 (((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) ∧ (𝑦 ∈ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)) ∧ 𝑧 ∈ π‘₯)) β†’ 𝑦 ∈ 𝑧)
2322expr 455 . . . . . . 7 (((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) ∧ 𝑦 ∈ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯))) β†’ (𝑧 ∈ π‘₯ β†’ 𝑦 ∈ 𝑧))
2423alrimiv 1922 . . . . . 6 (((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) ∧ 𝑦 ∈ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯))) β†’ βˆ€π‘§(𝑧 ∈ π‘₯ β†’ 𝑦 ∈ 𝑧))
25 vex 3475 . . . . . . 7 𝑦 ∈ V
2625elint 4957 . . . . . 6 (𝑦 ∈ ∩ π‘₯ ↔ βˆ€π‘§(𝑧 ∈ π‘₯ β†’ 𝑦 ∈ 𝑧))
2724, 26sylibr 233 . . . . 5 (((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) ∧ 𝑦 ∈ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯))) β†’ 𝑦 ∈ ∩ π‘₯)
2827ex 411 . . . 4 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ (𝑦 ∈ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)) β†’ 𝑦 ∈ ∩ π‘₯))
2928ssrdv 3986 . . 3 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)) βŠ† ∩ π‘₯)
30 simp1 1133 . . . 4 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ π‘Š ∈ PreHil)
31 intssuni 4975 . . . . . 6 (π‘₯ β‰  βˆ… β†’ ∩ π‘₯ βŠ† βˆͺ π‘₯)
32313ad2ant3 1132 . . . . 5 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ ∩ π‘₯ βŠ† βˆͺ π‘₯)
33 simp2 1134 . . . . . . 7 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ π‘₯ βŠ† 𝐢)
3473ad2ant1 1130 . . . . . . 7 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ 𝐢 βŠ† 𝒫 𝑉)
3533, 34sstrd 3990 . . . . . 6 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ π‘₯ βŠ† 𝒫 𝑉)
36 sspwuni 5105 . . . . . 6 (π‘₯ βŠ† 𝒫 𝑉 ↔ βˆͺ π‘₯ βŠ† 𝑉)
3735, 36sylib 217 . . . . 5 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ βˆͺ π‘₯ βŠ† 𝑉)
3832, 37sstrd 3990 . . . 4 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ ∩ π‘₯ βŠ† 𝑉)
391, 2, 10iscss2 21623 . . . 4 ((π‘Š ∈ PreHil ∧ ∩ π‘₯ βŠ† 𝑉) β†’ (∩ π‘₯ ∈ 𝐢 ↔ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)) βŠ† ∩ π‘₯))
4030, 38, 39syl2anc 582 . . 3 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ (∩ π‘₯ ∈ 𝐢 ↔ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜βˆ© π‘₯)) βŠ† ∩ π‘₯))
4129, 40mpbird 256 . 2 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ ∩ π‘₯ ∈ 𝐢)
427, 8, 41ismred 17587 1 (π‘Š ∈ PreHil β†’ 𝐢 ∈ (Mooreβ€˜π‘‰))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084  βˆ€wal 1531   = wceq 1533   ∈ wcel 2098   β‰  wne 2936   βŠ† wss 3947  βˆ…c0 4324  π’« cpw 4604  βˆͺ cuni 4910  βˆ© cint 4951  β€˜cfv 6551  Basecbs 17185  Moorecmre 17567  PreHilcphl 21561  ocvcocv 21597  ClSubSpccss 21598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-2nd 7998  df-tpos 8236  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-er 8729  df-map 8851  df-en 8969  df-dom 8970  df-sdom 8971  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-sets 17138  df-slot 17156  df-ndx 17168  df-base 17186  df-plusg 17251  df-mulr 17252  df-sca 17254  df-vsca 17255  df-ip 17256  df-0g 17428  df-mre 17571  df-mgm 18605  df-sgrp 18684  df-mnd 18700  df-mhm 18745  df-grp 18898  df-ghm 19173  df-mgp 20080  df-ur 20127  df-ring 20180  df-oppr 20278  df-rhm 20416  df-staf 20730  df-srng 20731  df-lmod 20750  df-lmhm 20912  df-lvec 20993  df-sra 21063  df-rgmod 21064  df-phl 21563  df-ocv 21600  df-css 21601
This theorem is referenced by:  mrccss  21631
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