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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrs | Structured version Visualization version GIF version |
Description: The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace 𝑅 is a subspace of the dual space. TODO: This proof repeats large parts of the lclkr 41234 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 41234 a special case of this? (Contributed by NM, 29-Jan-2015.) |
Ref | Expression |
---|---|
lclkrs.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lclkrs.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lclkrs.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lclkrs.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
lclkrs.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lclkrs.l | ⊢ 𝐿 = (LKer‘𝑈) |
lclkrs.d | ⊢ 𝐷 = (LDual‘𝑈) |
lclkrs.t | ⊢ 𝑇 = (LSubSp‘𝐷) |
lclkrs.c | ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑅)} |
lclkrs.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lclkrs.r | ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
Ref | Expression |
---|---|
lclkrs | ⊢ (𝜑 → 𝐶 ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 4076 | . . . 4 ⊢ {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑅)} ⊆ 𝐹 | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑅)} ⊆ 𝐹) |
3 | lclkrs.c | . . . 4 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑅)} | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑅)}) |
5 | lclkrs.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑈) | |
6 | lclkrs.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑈) | |
7 | eqid 2726 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
8 | lclkrs.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | lclkrs.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | lclkrs.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
11 | 8, 9, 10 | dvhlmod 40811 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
12 | 5, 6, 7, 11 | ldualvbase 38826 | . . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹) |
13 | 2, 4, 12 | 3sstr4d 4027 | . 2 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐷)) |
14 | eqid 2726 | . . . . . 6 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
15 | eqid 2726 | . . . . . 6 ⊢ (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈)) | |
16 | eqid 2726 | . . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
17 | 14, 15, 16, 5 | lfl0f 38769 | . . . . 5 ⊢ (𝑈 ∈ LMod → ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐹) |
18 | 11, 17 | syl 17 | . . . 4 ⊢ (𝜑 → ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐹) |
19 | lclkrs.o | . . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
20 | 8, 9, 19, 16, 10 | dochoc1 41062 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈)) |
21 | eqidd 2727 | . . . . . . . 8 ⊢ (𝜑 → ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})) | |
22 | lclkrs.l | . . . . . . . . . 10 ⊢ 𝐿 = (LKer‘𝑈) | |
23 | 14, 15, 16, 5, 22 | lkr0f 38794 | . . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐹) → ((𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})) = (Base‘𝑈) ↔ ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) |
24 | 11, 18, 23 | syl2anc 582 | . . . . . . . 8 ⊢ (𝜑 → ((𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})) = (Base‘𝑈) ↔ ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) |
25 | 21, 24 | mpbird 256 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})) = (Base‘𝑈)) |
26 | 25 | fveq2d 6907 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) = ( ⊥ ‘(Base‘𝑈))) |
27 | 26 | fveq2d 6907 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈)))) |
28 | 20, 27, 25 | 3eqtr4d 2776 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))) = (𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) |
29 | eqid 2726 | . . . . . . . 8 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
30 | 8, 9, 19, 16, 29 | doch1 41060 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘(Base‘𝑈)) = {(0g‘𝑈)}) |
31 | 10, 30 | syl 17 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(Base‘𝑈)) = {(0g‘𝑈)}) |
32 | 26, 31 | eqtrd 2766 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) = {(0g‘𝑈)}) |
33 | lclkrs.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑆) | |
34 | lclkrs.s | . . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑈) | |
35 | 29, 34 | lss0ss 20928 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑅 ∈ 𝑆) → {(0g‘𝑈)} ⊆ 𝑅) |
36 | 11, 33, 35 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → {(0g‘𝑈)} ⊆ 𝑅) |
37 | 32, 36 | eqsstrd 4018 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) ⊆ 𝑅) |
38 | 3 | lcfls1lem 41235 | . . . 4 ⊢ (((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐶 ↔ (((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))) = (𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})) ∧ ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) ⊆ 𝑅)) |
39 | 18, 28, 37, 38 | syl3anbrc 1340 | . . 3 ⊢ (𝜑 → ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐶) |
40 | 39 | ne0d 4338 | . 2 ⊢ (𝜑 → 𝐶 ≠ ∅) |
41 | eqid 2726 | . . . 4 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) | |
42 | eqid 2726 | . . . 4 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷) | |
43 | 10 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
44 | 33 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑅 ∈ 𝑆) |
45 | simpr3 1193 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑏 ∈ 𝐶) | |
46 | eqid 2726 | . . . 4 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
47 | simpr2 1192 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑎 ∈ 𝐶) | |
48 | simpr1 1191 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝐷))) | |
49 | eqid 2726 | . . . . . . . 8 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
50 | eqid 2726 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷)) | |
51 | 14, 41, 6, 49, 50, 11 | ldualsbase 38833 | . . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈))) |
52 | 51 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈))) |
53 | 48, 52 | eleqtrd 2828 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝑈))) |
54 | 8, 19, 9, 34, 5, 22, 6, 14, 41, 42, 3, 43, 44, 47, 53 | lclkrslem1 41238 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑥( ·𝑠 ‘𝐷)𝑎) ∈ 𝐶) |
55 | 8, 19, 9, 34, 5, 22, 6, 14, 41, 42, 3, 43, 44, 45, 46, 54 | lclkrslem2 41239 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → ((𝑥( ·𝑠 ‘𝐷)𝑎)(+g‘𝐷)𝑏) ∈ 𝐶) |
56 | 55 | ralrimivvva 3194 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝐷))∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 ((𝑥( ·𝑠 ‘𝐷)𝑎)(+g‘𝐷)𝑏) ∈ 𝐶) |
57 | lclkrs.t | . . 3 ⊢ 𝑇 = (LSubSp‘𝐷) | |
58 | 49, 50, 7, 46, 42, 57 | islss 20913 | . 2 ⊢ (𝐶 ∈ 𝑇 ↔ (𝐶 ⊆ (Base‘𝐷) ∧ 𝐶 ≠ ∅ ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝐷))∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 ((𝑥( ·𝑠 ‘𝐷)𝑎)(+g‘𝐷)𝑏) ∈ 𝐶)) |
59 | 13, 40, 56, 58 | syl3anbrc 1340 | 1 ⊢ (𝜑 → 𝐶 ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 {crab 3419 ⊆ wss 3947 ∅c0 4325 {csn 4633 × cxp 5682 ‘cfv 6556 (class class class)co 7426 Basecbs 17215 +gcplusg 17268 Scalarcsca 17271 ·𝑠 cvsca 17272 0gc0g 17456 LModclmod 20838 LSubSpclss 20910 LFnlclfn 38757 LKerclk 38785 LDualcld 38823 HLchlt 39050 LHypclh 39685 DVecHcdvh 40779 ocHcoch 41048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 ax-riotaBAD 38653 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-iin 5006 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8005 df-2nd 8006 df-tpos 8243 df-undef 8290 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-2o 8499 df-er 8736 df-map 8859 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12613 df-uz 12877 df-fz 13541 df-struct 17151 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-ress 17245 df-plusg 17281 df-mulr 17282 df-sca 17284 df-vsca 17285 df-0g 17458 df-mre 17601 df-mrc 17602 df-acs 17604 df-proset 18322 df-poset 18340 df-plt 18357 df-lub 18373 df-glb 18374 df-join 18375 df-meet 18376 df-p0 18452 df-p1 18453 df-lat 18459 df-clat 18526 df-mgm 18635 df-sgrp 18714 df-mnd 18730 df-submnd 18776 df-grp 18933 df-minusg 18934 df-sbg 18935 df-subg 19119 df-cntz 19313 df-oppg 19342 df-lsm 19636 df-cmn 19782 df-abl 19783 df-mgp 20120 df-rng 20138 df-ur 20167 df-ring 20220 df-oppr 20318 df-dvdsr 20341 df-unit 20342 df-invr 20372 df-dvr 20385 df-nzr 20497 df-rlreg 20674 df-domn 20675 df-drng 20711 df-lmod 20840 df-lss 20911 df-lsp 20951 df-lvec 21083 df-lsatoms 38676 df-lshyp 38677 df-lcv 38719 df-lfl 38758 df-lkr 38786 df-ldual 38824 df-oposet 38876 df-ol 38878 df-oml 38879 df-covers 38966 df-ats 38967 df-atl 38998 df-cvlat 39022 df-hlat 39051 df-llines 39199 df-lplanes 39200 df-lvols 39201 df-lines 39202 df-psubsp 39204 df-pmap 39205 df-padd 39497 df-lhyp 39689 df-laut 39690 df-ldil 39805 df-ltrn 39806 df-trl 39860 df-tgrp 40444 df-tendo 40456 df-edring 40458 df-dveca 40704 df-disoa 40730 df-dvech 40780 df-dib 40840 df-dic 40874 df-dih 40930 df-doch 41049 df-djh 41096 |
This theorem is referenced by: lclkrs2 41241 mapddlssN 41341 |
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