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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 39996 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 39996 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 4037 | . . . 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 2736 | . . . 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 39573 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
12 | 5, 6, 7, 11 | ldualvbase 37588 | . . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹) |
13 | 2, 4, 12 | 3sstr4d 3991 | . 2 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐷)) |
14 | eqid 2736 | . . . . . 6 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
15 | eqid 2736 | . . . . . 6 ⊢ (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈)) | |
16 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
17 | 14, 15, 16, 5 | lfl0f 37531 | . . . . 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 39824 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈)) |
21 | eqidd 2737 | . . . . . . . 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 37556 | . . . . . . . . 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 584 | . . . . . . . 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 6846 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) = ( ⊥ ‘(Base‘𝑈))) |
27 | 26 | fveq2d 6846 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈)))) |
28 | 20, 27, 25 | 3eqtr4d 2786 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))) = (𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) |
29 | eqid 2736 | . . . . . . . 8 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
30 | 8, 9, 19, 16, 29 | doch1 39822 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘(Base‘𝑈)) = {(0g‘𝑈)}) |
31 | 10, 30 | syl 17 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(Base‘𝑈)) = {(0g‘𝑈)}) |
32 | 26, 31 | eqtrd 2776 | . . . . 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 20409 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑅 ∈ 𝑆) → {(0g‘𝑈)} ⊆ 𝑅) |
36 | 11, 33, 35 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → {(0g‘𝑈)} ⊆ 𝑅) |
37 | 32, 36 | eqsstrd 3982 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))) ⊆ 𝑅) |
38 | 3 | lcfls1lem 39997 | . . . 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 1343 | . . 3 ⊢ (𝜑 → ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}) ∈ 𝐶) |
40 | 39 | ne0d 4295 | . 2 ⊢ (𝜑 → 𝐶 ≠ ∅) |
41 | eqid 2736 | . . . 4 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) | |
42 | eqid 2736 | . . . 4 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷) | |
43 | 10 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
44 | 33 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑅 ∈ 𝑆) |
45 | simpr3 1196 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑏 ∈ 𝐶) | |
46 | eqid 2736 | . . . 4 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
47 | simpr2 1195 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑎 ∈ 𝐶) | |
48 | simpr1 1194 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝐷))) | |
49 | eqid 2736 | . . . . . . . 8 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
50 | eqid 2736 | . . . . . . . 8 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷)) | |
51 | 14, 41, 6, 49, 50, 11 | ldualsbase 37595 | . . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈))) |
52 | 51 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈))) |
53 | 48, 52 | eleqtrd 2840 | . . . . 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 40000 | . . . 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 40001 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → ((𝑥( ·𝑠 ‘𝐷)𝑎)(+g‘𝐷)𝑏) ∈ 𝐶) |
56 | 55 | ralrimivvva 3200 | . 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 20395 | . 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 1343 | 1 ⊢ (𝜑 → 𝐶 ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 {crab 3407 ⊆ wss 3910 ∅c0 4282 {csn 4586 × cxp 5631 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 +gcplusg 17133 Scalarcsca 17136 ·𝑠 cvsca 17137 0gc0g 17321 LModclmod 20322 LSubSpclss 20392 LFnlclfn 37519 LKerclk 37547 LDualcld 37585 HLchlt 37812 LHypclh 38447 DVecHcdvh 39541 ocHcoch 39810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-riotaBAD 37415 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-undef 8204 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-0g 17323 df-mre 17466 df-mrc 17467 df-acs 17469 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-p1 18315 df-lat 18321 df-clat 18388 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-cntz 19097 df-oppg 19124 df-lsm 19418 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-drng 20187 df-lmod 20324 df-lss 20393 df-lsp 20433 df-lvec 20564 df-lsatoms 37438 df-lshyp 37439 df-lcv 37481 df-lfl 37520 df-lkr 37548 df-ldual 37586 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 df-llines 37961 df-lplanes 37962 df-lvols 37963 df-lines 37964 df-psubsp 37966 df-pmap 37967 df-padd 38259 df-lhyp 38451 df-laut 38452 df-ldil 38567 df-ltrn 38568 df-trl 38622 df-tgrp 39206 df-tendo 39218 df-edring 39220 df-dveca 39466 df-disoa 39492 df-dvech 39542 df-dib 39602 df-dic 39636 df-dih 39692 df-doch 39811 df-djh 39858 |
This theorem is referenced by: lclkrs2 40003 mapddlssN 40103 |
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