Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrslem1 | Structured version Visualization version GIF version |
Description: The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace 𝑄 is closed under scalar product. (Contributed by NM, 27-Jan-2015.) |
Ref | Expression |
---|---|
lclkrslem1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lclkrslem1.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lclkrslem1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lclkrslem1.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
lclkrslem1.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lclkrslem1.l | ⊢ 𝐿 = (LKer‘𝑈) |
lclkrslem1.d | ⊢ 𝐷 = (LDual‘𝑈) |
lclkrslem1.r | ⊢ 𝑅 = (Scalar‘𝑈) |
lclkrslem1.b | ⊢ 𝐵 = (Base‘𝑅) |
lclkrslem1.t | ⊢ · = ( ·𝑠 ‘𝐷) |
lclkrslem1.c | ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑄)} |
lclkrslem1.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lclkrslem1.q | ⊢ (𝜑 → 𝑄 ∈ 𝑆) |
lclkrslem1.g | ⊢ (𝜑 → 𝐺 ∈ 𝐶) |
lclkrslem1.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
lclkrslem1 | ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lclkrslem1.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lclkrslem1.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
3 | lclkrslem1.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | lclkrslem1.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
5 | lclkrslem1.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
6 | lclkrslem1.d | . . 3 ⊢ 𝐷 = (LDual‘𝑈) | |
7 | lclkrslem1.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑈) | |
8 | lclkrslem1.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
9 | lclkrslem1.t | . . 3 ⊢ · = ( ·𝑠 ‘𝐷) | |
10 | eqid 2821 | . . 3 ⊢ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
11 | lclkrslem1.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
12 | lclkrslem1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
13 | lclkrslem1.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐶) | |
14 | lclkrslem1.c | . . . . . 6 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑄)} | |
15 | 14, 10 | lcfls1c 38671 | . . . . 5 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄)) |
16 | 15 | simplbi 500 | . . . 4 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}) |
17 | 13, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}) |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17 | lclkrlem1 38641 | . 2 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}) |
19 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
20 | 1, 3, 11 | dvhlmod 38245 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
21 | 14 | lcfls1lem 38669 | . . . . . . . 8 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄)) |
22 | 13, 21 | sylib 220 | . . . . . . 7 ⊢ (𝜑 → (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄)) |
23 | 22 | simp1d 1138 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
24 | 4, 7, 8, 6, 9, 20, 12, 23 | ldualvscl 36274 | . . . . 5 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐹) |
25 | 19, 4, 5, 20, 24 | lkrssv 36231 | . . . 4 ⊢ (𝜑 → (𝐿‘(𝑋 · 𝐺)) ⊆ (Base‘𝑈)) |
26 | 1, 3, 11 | dvhlvec 38244 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec) |
27 | 7, 8, 4, 5, 6, 9, 26, 23, 12 | lkrss 36303 | . . . 4 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝑋 · 𝐺))) |
28 | 1, 3, 19, 2 | dochss 38500 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘(𝑋 · 𝐺)) ⊆ (Base‘𝑈) ∧ (𝐿‘𝐺) ⊆ (𝐿‘(𝑋 · 𝐺))) → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) ⊆ ( ⊥ ‘(𝐿‘𝐺))) |
29 | 11, 25, 27, 28 | syl3anc 1367 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) ⊆ ( ⊥ ‘(𝐿‘𝐺))) |
30 | 22 | simp3d 1140 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄) |
31 | 29, 30 | sstrd 3976 | . 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) ⊆ 𝑄) |
32 | 14, 10 | lcfls1c 38671 | . 2 ⊢ ((𝑋 · 𝐺) ∈ 𝐶 ↔ ((𝑋 · 𝐺) ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) ⊆ 𝑄)) |
33 | 18, 31, 32 | sylanbrc 585 | 1 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {crab 3142 ⊆ wss 3935 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 Scalarcsca 16567 ·𝑠 cvsca 16568 LSubSpclss 19702 LFnlclfn 36192 LKerclk 36220 LDualcld 36258 HLchlt 36485 LHypclh 37119 DVecHcdvh 38213 ocHcoch 38482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-riotaBAD 36088 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-tpos 7891 df-undef 7938 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-0g 16714 df-proset 17537 df-poset 17555 df-plt 17567 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-p0 17648 df-p1 17649 df-lat 17655 df-clat 17717 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-subg 18275 df-cntz 18446 df-lsm 18760 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-dvr 19432 df-drng 19503 df-lmod 19635 df-lss 19703 df-lsp 19743 df-lvec 19874 df-lfl 36193 df-lkr 36221 df-ldual 36259 df-oposet 36311 df-ol 36313 df-oml 36314 df-covers 36401 df-ats 36402 df-atl 36433 df-cvlat 36457 df-hlat 36486 df-llines 36633 df-lplanes 36634 df-lvols 36635 df-lines 36636 df-psubsp 36638 df-pmap 36639 df-padd 36931 df-lhyp 37123 df-laut 37124 df-ldil 37239 df-ltrn 37240 df-trl 37294 df-tendo 37890 df-edring 37892 df-disoa 38164 df-dvech 38214 df-dib 38274 df-dic 38308 df-dih 38364 df-doch 38483 |
This theorem is referenced by: lclkrs 38674 |
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