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Theorem cssmre 21113
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 17476: 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 17541. (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 21105 . . . . 5 (π‘₯ ∈ 𝐢 β†’ π‘₯ βŠ† 𝑉)
4 velpw 4570 . . . . 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 3955 . 2 (π‘Š ∈ PreHil β†’ 𝐢 βŠ† 𝒫 𝑉)
81, 2css1 21110 . 2 (π‘Š ∈ PreHil β†’ 𝑉 ∈ 𝐢)
9 intss1 4929 . . . . . . . . . . . 12 (𝑧 ∈ π‘₯ β†’ ∩ π‘₯ βŠ† 𝑧)
10 eqid 2737 . . . . . . . . . . . . 13 (ocvβ€˜π‘Š) = (ocvβ€˜π‘Š)
1110ocv2ss 21093 . . . . . . . . . . . 12 (∩ π‘₯ βŠ† 𝑧 β†’ ((ocvβ€˜π‘Š)β€˜π‘§) βŠ† ((ocvβ€˜π‘Š)β€˜βˆ© π‘₯))
1210ocv2ss 21093 . . . . . . . . . . . 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 728 . . . . . . . . . 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 770 . . . . . . . . . 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 3950 . . . . . . . . 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 1193 . . . . . . . . . . 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 772 . . . . . . . . . . 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 3950 . . . . . . . . . 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 21104 . . . . . . . . . 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 2841 . . . . . . . 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 458 . . . . . . 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 1931 . . . . . 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 3452 . . . . . . 7 𝑦 ∈ V
2625elint 4918 . . . . . 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 414 . . . 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 3955 . . 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 1137 . . . 4 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ π‘Š ∈ PreHil)
31 intssuni 4936 . . . . . 6 (π‘₯ β‰  βˆ… β†’ ∩ π‘₯ βŠ† βˆͺ π‘₯)
32313ad2ant3 1136 . . . . 5 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ ∩ π‘₯ βŠ† βˆͺ π‘₯)
33 simp2 1138 . . . . . . 7 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ π‘₯ βŠ† 𝐢)
3473ad2ant1 1134 . . . . . . 7 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ 𝐢 βŠ† 𝒫 𝑉)
3533, 34sstrd 3959 . . . . . 6 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ π‘₯ βŠ† 𝒫 𝑉)
36 sspwuni 5065 . . . . . 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 3959 . . . 4 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ ∩ π‘₯ βŠ† 𝑉)
391, 2, 10iscss2 21106 . . . 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 585 . . 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 257 . 2 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† 𝐢 ∧ π‘₯ β‰  βˆ…) β†’ ∩ π‘₯ ∈ 𝐢)
427, 8, 41ismred 17489 1 (π‘Š ∈ PreHil β†’ 𝐢 ∈ (Mooreβ€˜π‘‰))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107   β‰  wne 2944   βŠ† wss 3915  βˆ…c0 4287  π’« cpw 4565  βˆͺ cuni 4870  βˆ© cint 4912  β€˜cfv 6501  Basecbs 17090  Moorecmre 17469  PreHilcphl 21044  ocvcocv 21080  ClSubSpccss 21081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-tpos 8162  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-plusg 17153  df-mulr 17154  df-sca 17156  df-vsca 17157  df-ip 17158  df-0g 17330  df-mre 17473  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-mhm 18608  df-grp 18758  df-ghm 19013  df-mgp 19904  df-ur 19921  df-ring 19973  df-oppr 20056  df-rnghom 20155  df-staf 20320  df-srng 20321  df-lmod 20340  df-lmhm 20499  df-lvec 20580  df-sra 20649  df-rgmod 20650  df-phl 21046  df-ocv 21083  df-css 21084
This theorem is referenced by:  mrccss  21114
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