![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdcl2 | Structured version Visualization version GIF version |
Description: The mapping of a subspace of vector space H is a subspace in the dual space of functionals with closed kernels. (Contributed by NM, 31-Jan-2015.) |
Ref | Expression |
---|---|
mapdlss.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdlss.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdlss.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdlss.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
mapdlss.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdlss.t | ⊢ 𝑇 = (LSubSp‘𝐶) |
mapdlss.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdlss.r | ⊢ (𝜑 → 𝑅 ∈ 𝑆) |
Ref | Expression |
---|---|
mapdcl2 | ⊢ (𝜑 → (𝑀‘𝑅) ∈ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdlss.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdlss.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
3 | mapdlss.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | mapdlss.s | . . 3 ⊢ 𝑆 = (LSubSp‘𝑈) | |
5 | mapdlss.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | mapdlss.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑆) | |
7 | 1, 2, 3, 4, 5, 6 | mapdcl 37812 | . 2 ⊢ (𝜑 → (𝑀‘𝑅) ∈ ran 𝑀) |
8 | mapdlss.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
9 | mapdlss.t | . . 3 ⊢ 𝑇 = (LSubSp‘𝐶) | |
10 | 1, 2, 8, 9, 5 | mapdrn2 37810 | . 2 ⊢ (𝜑 → ran 𝑀 = 𝑇) |
11 | 7, 10 | eleqtrd 2861 | 1 ⊢ (𝜑 → (𝑀‘𝑅) ∈ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ran crn 5358 ‘cfv 6137 LSubSpclss 19328 HLchlt 35509 LHypclh 36143 DVecHcdvh 37237 LCDualclcd 37745 mapdcmpd 37783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-riotaBAD 35112 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-tpos 7636 df-undef 7683 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-n0 11647 df-z 11733 df-uz 11997 df-fz 12648 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-sca 16358 df-vsca 16359 df-0g 16492 df-mre 16636 df-mrc 16637 df-acs 16639 df-proset 17318 df-poset 17336 df-plt 17348 df-lub 17364 df-glb 17365 df-join 17366 df-meet 17367 df-p0 17429 df-p1 17430 df-lat 17436 df-clat 17498 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-submnd 17726 df-grp 17816 df-minusg 17817 df-sbg 17818 df-subg 17979 df-cntz 18137 df-oppg 18163 df-lsm 18439 df-cmn 18585 df-abl 18586 df-mgp 18881 df-ur 18893 df-ring 18940 df-oppr 19014 df-dvdsr 19032 df-unit 19033 df-invr 19063 df-dvr 19074 df-drng 19145 df-lmod 19261 df-lss 19329 df-lsp 19371 df-lvec 19502 df-lsatoms 35135 df-lshyp 35136 df-lcv 35178 df-lfl 35217 df-lkr 35245 df-ldual 35283 df-oposet 35335 df-ol 35337 df-oml 35338 df-covers 35425 df-ats 35426 df-atl 35457 df-cvlat 35481 df-hlat 35510 df-llines 35657 df-lplanes 35658 df-lvols 35659 df-lines 35660 df-psubsp 35662 df-pmap 35663 df-padd 35955 df-lhyp 36147 df-laut 36148 df-ldil 36263 df-ltrn 36264 df-trl 36318 df-tgrp 36902 df-tendo 36914 df-edring 36916 df-dveca 37162 df-disoa 37188 df-dvech 37238 df-dib 37298 df-dic 37332 df-dih 37388 df-doch 37507 df-djh 37554 df-lcdual 37746 df-mapd 37784 |
This theorem is referenced by: mapdcv 37819 mapdin 37821 mapdlsm 37823 mapdat 37826 mapdpglem2a 37833 mapdpglem3 37834 mapdpglem6 37837 mapdpglem8 37838 mapdpglem12 37842 mapdpglem13 37843 mapdpglem17N 37847 mapdpglem18 37848 mapdpglem19 37849 mapdpglem21 37851 mapdpglem23 37853 hdmaprnlem8N 38015 hdmaprnlem3eN 38017 |
Copyright terms: Public domain | W3C validator |