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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 38120 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 38120 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 3946 | . . . 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 2778 | . . . 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 37697 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
12 | 5, 6, 7, 11 | ldualvbase 35713 | . . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹) |
13 | 2, 4, 12 | 3sstr4d 3904 | . 2 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐷)) |
14 | eqid 2778 | . . . . . 6 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
15 | eqid 2778 | . . . . . 6 ⊢ (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈)) | |
16 | eqid 2778 | . . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
17 | 14, 15, 16, 5 | lfl0f 35656 | . . . . 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 37948 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈)) |
21 | eqidd 2779 | . . . . . . . 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 35681 | . . . . . . . . 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 576 | . . . . . . . 8 ⊢ (𝜑 → ((𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})) = (Base‘𝑈) ↔ ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) |
25 | 21, 24 | mpbird 249 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})) = (Base‘𝑈)) |
26 | 25 | fveq2d 6503 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) = ( ⊥ ‘(Base‘𝑈))) |
27 | 26 | fveq2d 6503 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈)))) |
28 | 20, 27, 25 | 3eqtr4d 2824 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))) = (𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) |
29 | eqid 2778 | . . . . . . . 8 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
30 | 8, 9, 19, 16, 29 | doch1 37946 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘(Base‘𝑈)) = {(0g‘𝑈)}) |
31 | 10, 30 | syl 17 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(Base‘𝑈)) = {(0g‘𝑈)}) |
32 | 26, 31 | eqtrd 2814 | . . . . 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 19442 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑅 ∈ 𝑆) → {(0g‘𝑈)} ⊆ 𝑅) |
36 | 11, 33, 35 | syl2anc 576 | . . . . 5 ⊢ (𝜑 → {(0g‘𝑈)} ⊆ 𝑅) |
37 | 32, 36 | eqsstrd 3895 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) ⊆ 𝑅) |
38 | 3 | lcfls1lem 38121 | . . . 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 1323 | . . 3 ⊢ (𝜑 → ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐶) |
40 | 39 | ne0d 4187 | . 2 ⊢ (𝜑 → 𝐶 ≠ ∅) |
41 | eqid 2778 | . . . 4 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) | |
42 | eqid 2778 | . . . 4 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷) | |
43 | 10 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
44 | 33 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑅 ∈ 𝑆) |
45 | simpr3 1176 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑏 ∈ 𝐶) | |
46 | eqid 2778 | . . . 4 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
47 | simpr2 1175 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑎 ∈ 𝐶) | |
48 | simpr1 1174 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝐷))) | |
49 | eqid 2778 | . . . . . . . 8 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
50 | eqid 2778 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷)) | |
51 | 14, 41, 6, 49, 50, 11 | ldualsbase 35720 | . . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈))) |
52 | 51 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈))) |
53 | 48, 52 | eleqtrd 2868 | . . . . 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 38124 | . . . 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 38125 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → ((𝑥( ·𝑠 ‘𝐷)𝑎)(+g‘𝐷)𝑏) ∈ 𝐶) |
56 | 55 | ralrimivvva 3142 | . 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 19428 | . 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 1323 | 1 ⊢ (𝜑 → 𝐶 ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 ∀wral 3088 {crab 3092 ⊆ wss 3829 ∅c0 4178 {csn 4441 × cxp 5405 ‘cfv 6188 (class class class)co 6976 Basecbs 16339 +gcplusg 16421 Scalarcsca 16424 ·𝑠 cvsca 16425 0gc0g 16569 LModclmod 19356 LSubSpclss 19425 LFnlclfn 35644 LKerclk 35672 LDualcld 35710 HLchlt 35937 LHypclh 36571 DVecHcdvh 37665 ocHcoch 37934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-riotaBAD 35540 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-om 7397 df-1st 7501 df-2nd 7502 df-tpos 7695 df-undef 7742 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-n0 11708 df-z 11794 df-uz 12059 df-fz 12709 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-sca 16437 df-vsca 16438 df-0g 16571 df-mre 16715 df-mrc 16716 df-acs 16718 df-proset 17396 df-poset 17414 df-plt 17426 df-lub 17442 df-glb 17443 df-join 17444 df-meet 17445 df-p0 17507 df-p1 17508 df-lat 17514 df-clat 17576 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-submnd 17804 df-grp 17894 df-minusg 17895 df-sbg 17896 df-subg 18060 df-cntz 18218 df-oppg 18245 df-lsm 18522 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-ring 19022 df-oppr 19096 df-dvdsr 19114 df-unit 19115 df-invr 19145 df-dvr 19156 df-drng 19227 df-lmod 19358 df-lss 19426 df-lsp 19466 df-lvec 19597 df-lsatoms 35563 df-lshyp 35564 df-lcv 35606 df-lfl 35645 df-lkr 35673 df-ldual 35711 df-oposet 35763 df-ol 35765 df-oml 35766 df-covers 35853 df-ats 35854 df-atl 35885 df-cvlat 35909 df-hlat 35938 df-llines 36085 df-lplanes 36086 df-lvols 36087 df-lines 36088 df-psubsp 36090 df-pmap 36091 df-padd 36383 df-lhyp 36575 df-laut 36576 df-ldil 36691 df-ltrn 36692 df-trl 36746 df-tgrp 37330 df-tendo 37342 df-edring 37344 df-dveca 37590 df-disoa 37616 df-dvech 37666 df-dib 37726 df-dic 37760 df-dih 37816 df-doch 37935 df-djh 37982 |
This theorem is referenced by: lclkrs2 38127 mapddlssN 38227 |
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