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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 39583 | . . . . . . 7 ⊢ (𝜑 → 𝐶 = (𝐷 ↾s 𝐵)) |
12 | 11 | fveq2d 6772 | . . . . . 6 ⊢ (𝜑 → (LSubSp‘𝐶) = (LSubSp‘(𝐷 ↾s 𝐵))) |
13 | 1, 12 | eqtrid 2791 | . . . . 5 ⊢ (𝜑 → 𝑆 = (LSubSp‘(𝐷 ↾s 𝐵))) |
14 | 13 | eleq2d 2825 | . . . 4 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ 𝑢 ∈ (LSubSp‘(𝐷 ↾s 𝐵)))) |
15 | 2, 5, 9 | dvhlmod 39103 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
16 | 8, 15 | lduallmod 37146 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ LMod) |
17 | lcdlss.t | . . . . . 6 ⊢ 𝑇 = (LSubSp‘𝐷) | |
18 | 2, 5, 3, 6, 7, 8, 17, 10, 9 | lclkr 39526 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑇) |
19 | eqid 2739 | . . . . . 6 ⊢ (𝐷 ↾s 𝐵) = (𝐷 ↾s 𝐵) | |
20 | eqid 2739 | . . . . . 6 ⊢ (LSubSp‘(𝐷 ↾s 𝐵)) = (LSubSp‘(𝐷 ↾s 𝐵)) | |
21 | 19, 17, 20 | lsslss 20204 | . . . . 5 ⊢ ((𝐷 ∈ LMod ∧ 𝐵 ∈ 𝑇) → (𝑢 ∈ (LSubSp‘(𝐷 ↾s 𝐵)) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵))) |
22 | 16, 18, 21 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝑢 ∈ (LSubSp‘(𝐷 ↾s 𝐵)) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵))) |
23 | 14, 22 | bitrd 278 | . . 3 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵))) |
24 | elin 3907 | . . . 4 ⊢ (𝑢 ∈ (𝑇 ∩ 𝒫 𝐵) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ∈ 𝒫 𝐵)) | |
25 | velpw 4543 | . . . . 5 ⊢ (𝑢 ∈ 𝒫 𝐵 ↔ 𝑢 ⊆ 𝐵) | |
26 | 25 | anbi2i 622 | . . . 4 ⊢ ((𝑢 ∈ 𝑇 ∧ 𝑢 ∈ 𝒫 𝐵) ↔ (𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵)) |
27 | 24, 26 | bitr2i 275 | . . 3 ⊢ ((𝑢 ∈ 𝑇 ∧ 𝑢 ⊆ 𝐵) ↔ 𝑢 ∈ (𝑇 ∩ 𝒫 𝐵)) |
28 | 23, 27 | bitrdi 286 | . 2 ⊢ (𝜑 → (𝑢 ∈ 𝑆 ↔ 𝑢 ∈ (𝑇 ∩ 𝒫 𝐵))) |
29 | 28 | eqrdv 2737 | 1 ⊢ (𝜑 → 𝑆 = (𝑇 ∩ 𝒫 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1541 ∈ wcel 2109 {crab 3069 ∩ cin 3890 ⊆ wss 3891 𝒫 cpw 4538 ‘cfv 6430 (class class class)co 7268 ↾s cress 16922 LModclmod 20104 LSubSpclss 20174 LFnlclfn 37050 LKerclk 37078 LDualcld 37116 HLchlt 37343 LHypclh 37977 DVecHcdvh 39071 ocHcoch 39340 LCDualclcd 39579 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-riotaBAD 36946 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-tpos 8026 df-undef 8073 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-n0 12217 df-z 12303 df-uz 12565 df-fz 13222 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-sca 16959 df-vsca 16960 df-0g 17133 df-mre 17276 df-mrc 17277 df-acs 17279 df-proset 17994 df-poset 18012 df-plt 18029 df-lub 18045 df-glb 18046 df-join 18047 df-meet 18048 df-p0 18124 df-p1 18125 df-lat 18131 df-clat 18198 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-submnd 18412 df-grp 18561 df-minusg 18562 df-sbg 18563 df-subg 18733 df-cntz 18904 df-oppg 18931 df-lsm 19222 df-cmn 19369 df-abl 19370 df-mgp 19702 df-ur 19719 df-ring 19766 df-oppr 19843 df-dvdsr 19864 df-unit 19865 df-invr 19895 df-dvr 19906 df-drng 19974 df-lmod 20106 df-lss 20175 df-lsp 20215 df-lvec 20346 df-lsatoms 36969 df-lshyp 36970 df-lcv 37012 df-lfl 37051 df-lkr 37079 df-ldual 37117 df-oposet 37169 df-ol 37171 df-oml 37172 df-covers 37259 df-ats 37260 df-atl 37291 df-cvlat 37315 df-hlat 37344 df-llines 37491 df-lplanes 37492 df-lvols 37493 df-lines 37494 df-psubsp 37496 df-pmap 37497 df-padd 37789 df-lhyp 37981 df-laut 37982 df-ldil 38097 df-ltrn 38098 df-trl 38152 df-tgrp 38736 df-tendo 38748 df-edring 38750 df-dveca 38996 df-disoa 39022 df-dvech 39072 df-dib 39132 df-dic 39166 df-dih 39222 df-doch 39341 df-djh 39388 df-lcdual 39580 |
This theorem is referenced by: lcdlss2N 39613 mapdrn2 39644 |
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