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Mirrors > Home > MPE Home > Th. List > cssmre | Structured version Visualization version GIF version |
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 16072: consider the Hilbert space of sequences ℕ⟶ℝ 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 16138. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
cssmre.v | ⊢ 𝑉 = (Base‘𝑊) |
cssmre.c | ⊢ 𝐶 = (CSubSp‘𝑊) |
Ref | Expression |
---|---|
cssmre | ⊢ (𝑊 ∈ PreHil → 𝐶 ∈ (Moore‘𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cssmre.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
2 | cssmre.c | . . . . . 6 ⊢ 𝐶 = (CSubSp‘𝑊) | |
3 | 1, 2 | cssss 19848 | . . . . 5 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ⊆ 𝑉) |
4 | selpw 4115 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝑉 ↔ 𝑥 ⊆ 𝑉) | |
5 | 3, 4 | sylibr 223 | . . . 4 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝒫 𝑉) |
6 | 5 | a1i 11 | . . 3 ⊢ (𝑊 ∈ PreHil → (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝒫 𝑉)) |
7 | 6 | ssrdv 3574 | . 2 ⊢ (𝑊 ∈ PreHil → 𝐶 ⊆ 𝒫 𝑉) |
8 | 1, 2 | css1 19853 | . 2 ⊢ (𝑊 ∈ PreHil → 𝑉 ∈ 𝐶) |
9 | intss1 4427 | . . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑥 → ∩ 𝑥 ⊆ 𝑧) | |
10 | eqid 2610 | . . . . . . . . . . . . 13 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
11 | 10 | ocv2ss 19836 | . . . . . . . . . . . 12 ⊢ (∩ 𝑥 ⊆ 𝑧 → ((ocv‘𝑊)‘𝑧) ⊆ ((ocv‘𝑊)‘∩ 𝑥)) |
12 | 10 | ocv2ss 19836 | . . . . . . . . . . . 12 ⊢ (((ocv‘𝑊)‘𝑧) ⊆ ((ocv‘𝑊)‘∩ 𝑥) → ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥)) ⊆ ((ocv‘𝑊)‘((ocv‘𝑊)‘𝑧))) |
13 | 9, 11, 12 | 3syl 18 | . . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥)) ⊆ ((ocv‘𝑊)‘((ocv‘𝑊)‘𝑧))) |
14 | 13 | ad2antll 761 | . . . . . . . . . 10 ⊢ (((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) ∧ (𝑦 ∈ ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥)) ∧ 𝑧 ∈ 𝑥)) → ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥)) ⊆ ((ocv‘𝑊)‘((ocv‘𝑊)‘𝑧))) |
15 | simprl 790 | . . . . . . . . . 10 ⊢ (((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) ∧ (𝑦 ∈ ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥)) ∧ 𝑧 ∈ 𝑥)) → 𝑦 ∈ ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥))) | |
16 | 14, 15 | sseldd 3569 | . . . . . . . . 9 ⊢ (((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) ∧ (𝑦 ∈ ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥)) ∧ 𝑧 ∈ 𝑥)) → 𝑦 ∈ ((ocv‘𝑊)‘((ocv‘𝑊)‘𝑧))) |
17 | simpl2 1058 | . . . . . . . . . . 11 ⊢ (((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) ∧ (𝑦 ∈ ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥)) ∧ 𝑧 ∈ 𝑥)) → 𝑥 ⊆ 𝐶) | |
18 | simprr 792 | . . . . . . . . . . 11 ⊢ (((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) ∧ (𝑦 ∈ ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥)) ∧ 𝑧 ∈ 𝑥)) → 𝑧 ∈ 𝑥) | |
19 | 17, 18 | sseldd 3569 | . . . . . . . . . 10 ⊢ (((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) ∧ (𝑦 ∈ ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥)) ∧ 𝑧 ∈ 𝑥)) → 𝑧 ∈ 𝐶) |
20 | 10, 2 | cssi 19847 | . . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐶 → 𝑧 = ((ocv‘𝑊)‘((ocv‘𝑊)‘𝑧))) |
21 | 19, 20 | syl 17 | . . . . . . . . 9 ⊢ (((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) ∧ (𝑦 ∈ ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥)) ∧ 𝑧 ∈ 𝑥)) → 𝑧 = ((ocv‘𝑊)‘((ocv‘𝑊)‘𝑧))) |
22 | 16, 21 | eleqtrrd 2691 | . . . . . . . 8 ⊢ (((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) ∧ (𝑦 ∈ ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥)) ∧ 𝑧 ∈ 𝑥)) → 𝑦 ∈ 𝑧) |
23 | 22 | expr 641 | . . . . . . 7 ⊢ (((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥))) → (𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)) |
24 | 23 | alrimiv 1842 | . . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥))) → ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)) |
25 | vex 3176 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
26 | 25 | elint 4416 | . . . . . 6 ⊢ (𝑦 ∈ ∩ 𝑥 ↔ ∀𝑧(𝑧 ∈ 𝑥 → 𝑦 ∈ 𝑧)) |
27 | 24, 26 | sylibr 223 | . . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥))) → 𝑦 ∈ ∩ 𝑥) |
28 | 27 | ex 449 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → (𝑦 ∈ ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥)) → 𝑦 ∈ ∩ 𝑥)) |
29 | 28 | ssrdv 3574 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥)) ⊆ ∩ 𝑥) |
30 | simp1 1054 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → 𝑊 ∈ PreHil) | |
31 | intssuni 4434 | . . . . . 6 ⊢ (𝑥 ≠ ∅ → ∩ 𝑥 ⊆ ∪ 𝑥) | |
32 | 31 | 3ad2ant3 1077 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ ∪ 𝑥) |
33 | simp2 1055 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝐶) | |
34 | 7 | 3ad2ant1 1075 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → 𝐶 ⊆ 𝒫 𝑉) |
35 | 33, 34 | sstrd 3578 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝒫 𝑉) |
36 | sspwuni 4547 | . . . . . 6 ⊢ (𝑥 ⊆ 𝒫 𝑉 ↔ ∪ 𝑥 ⊆ 𝑉) | |
37 | 35, 36 | sylib 207 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ∪ 𝑥 ⊆ 𝑉) |
38 | 32, 37 | sstrd 3578 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ⊆ 𝑉) |
39 | 1, 2, 10 | iscss2 19849 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ ∩ 𝑥 ⊆ 𝑉) → (∩ 𝑥 ∈ 𝐶 ↔ ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥)) ⊆ ∩ 𝑥)) |
40 | 30, 38, 39 | syl2anc 691 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → (∩ 𝑥 ∈ 𝐶 ↔ ((ocv‘𝑊)‘((ocv‘𝑊)‘∩ 𝑥)) ⊆ ∩ 𝑥)) |
41 | 29, 40 | mpbird 246 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ⊆ 𝐶 ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ 𝐶) |
42 | 7, 8, 41 | ismred 16085 | 1 ⊢ (𝑊 ∈ PreHil → 𝐶 ∈ (Moore‘𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∀wal 1473 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ∪ cuni 4372 ∩ cint 4410 ‘cfv 5804 Basecbs 15695 Moorecmre 16065 PreHilcphl 19788 ocvcocv 19823 CSubSpccss 19824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-0g 15925 df-mre 16069 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-ghm 17481 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-rnghom 18538 df-staf 18668 df-srng 18669 df-lmod 18688 df-lmhm 18843 df-lvec 18924 df-sra 18993 df-rgmod 18994 df-phl 19790 df-ocv 19826 df-css 19827 |
This theorem is referenced by: mrccss 19857 |
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