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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcdlss | Structured version Visualization version GIF version |
Description: Subspaces of a dual vector space of functionals with closed kernels. (Contributed by NM, 13-Mar-2015.) |
Ref | Expression |
---|---|
lcdlss.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcdlss.o | ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) |
lcdlss.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
lcdlss.s | ⊢ 𝑆 = (LSubSp‘𝐶) |
lcdlss.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcdlss.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcdlss.l | ⊢ 𝐿 = (LKer‘𝑈) |
lcdlss.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcdlss.t | ⊢ 𝑇 = (LSubSp‘𝐷) |
lcdlss.b | ⊢ 𝐵 = {𝑓 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
lcdlss.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
lcdlss | ⊢ (𝜑 → 𝑆 = (𝑇 ∩ 𝒫 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcdlss.s | . . . . . 6 ⊢ 𝑆 = (LSubSp‘𝐶) | |
2 | lcdlss.h | . . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | lcdlss.o | . . . . . . . 8 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) | |
4 | lcdlss.c | . . . . . . . 8 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
5 | lcdlss.u | . . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | lcdlss.f | . . . . . . . 8 ⊢ 𝐹 = (LFnl‘𝑈) | |
7 | lcdlss.l | . . . . . . . 8 ⊢ 𝐿 = (LKer‘𝑈) | |
8 | lcdlss.d | . . . . . . . 8 ⊢ 𝐷 = (LDual‘𝑈) | |
9 | lcdlss.k | . . . . . . . 8 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | lcdlss.b | . . . . . . . 8 ⊢ 𝐵 = {𝑓 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | lcdval2 38720 | . . . . . . 7 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s 𝐵)) |
12 | 11 | fveq2d 6668 | . . . . . 6 ⊢ (𝜑 → (LSubSp‘𝐶) = (LSubSp‘(𝐷 ↾s 𝐵))) |
13 | 1, 12 | syl5eq 2868 | . . . . 5 ⊢ (𝜑 → 𝑆 = (LSubSp‘(𝐷 ↾s 𝐵))) |
14 | 13 | eleq2d 2898 | . . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ 𝑢 ∈ (LSubSp‘(𝐷 ↾s 𝐵)))) |
15 | 2, 5, 9 | dvhlmod 38240 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
16 | 8, 15 | lduallmod 36283 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ LMod) |
17 | lcdlss.t | . . . . . 6 ⊢ 𝑇 = (LSubSp‘𝐷) | |
18 | 2, 5, 3, 6, 7, 8, 17, 10, 9 | lclkr 38663 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑇) |
19 | eqid 2821 | . . . . . 6 ⊢ (𝐷 ↾s 𝐵) = (𝐷 ↾s 𝐵) | |
20 | eqid 2821 | . . . . . 6 ⊢ (LSubSp‘(𝐷 ↾s 𝐵)) = (LSubSp‘(𝐷 ↾s 𝐵)) | |
21 | 19, 17, 20 | lsslss 19727 | . . . . 5 ⊢ ((𝐷 ∈ LMod ∧ 𝐵 ∈ 𝑇) → (𝑢 ∈ (LSubSp‘(𝐷 ↾s 𝐵)) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵))) |
22 | 16, 18, 21 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑢 ∈ (LSubSp‘(𝐷 ↾s 𝐵)) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵))) |
23 | 14, 22 | bitrd 281 | . . 3 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵))) |
24 | elin 4168 | . . . 4 ⊢ (𝑢 ∈ (𝑇 ∩ 𝒫 𝐵) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ∈ 𝒫 𝐵)) | |
25 | velpw 4546 | . . . . 5 ⊢ (𝑢 ∈ 𝒫 𝐵 ↔ 𝑢 ⊆ 𝐵) | |
26 | 25 | anbi2i 624 | . . . 4 ⊢ ((𝑢 ∈ 𝑇 ∧ 𝑢 ∈ 𝒫 𝐵) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵)) |
27 | 24, 26 | bitr2i 278 | . . 3 ⊢ ((𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵) ↔ 𝑢 ∈ (𝑇 ∩ 𝒫 𝐵)) |
28 | 23, 27 | syl6bb 289 | . 2 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ 𝑢 ∈ (𝑇 ∩ 𝒫 𝐵))) |
29 | 28 | eqrdv 2819 | 1 ⊢ (𝜑 → 𝑆 = (𝑇 ∩ 𝒫 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4538 ‘cfv 6349 (class class class)co 7150 ↾s cress 16478 LModclmod 19628 LSubSpclss 19697 LFnlclfn 36187 LKerclk 36215 LDualcld 36253 HLchlt 36480 LHypclh 37114 DVecHcdvh 38208 ocHcoch 38477 LCDualclcd 38716 |
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 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-riotaBAD 36083 |
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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-undef 7933 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-0g 16709 df-mre 16851 df-mrc 16852 df-acs 16854 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-oppg 18468 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-lsatoms 36106 df-lshyp 36107 df-lcv 36149 df-lfl 36188 df-lkr 36216 df-ldual 36254 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 df-lvols 36630 df-lines 36631 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-lhyp 37118 df-laut 37119 df-ldil 37234 df-ltrn 37235 df-trl 37289 df-tgrp 37873 df-tendo 37885 df-edring 37887 df-dveca 38133 df-disoa 38159 df-dvech 38209 df-dib 38269 df-dic 38303 df-dih 38359 df-doch 38478 df-djh 38525 df-lcdual 38717 |
This theorem is referenced by: lcdlss2N 38750 mapdrn2 38781 |
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