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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcd0vcl | Structured version Visualization version GIF version |
Description: Closure of the zero functional in the set of functionals with closed kernels. (Contributed by NM, 15-Mar-2015.) |
Ref | Expression |
---|---|
lcdv0cl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcdv0cl.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
lcdv0cl.v | ⊢ 𝑉 = (Base‘𝐶) |
lcdv0cl.o | ⊢ 𝑂 = (0g‘𝐶) |
lcdv0cl.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
lcd0vcl | ⊢ (𝜑 → 𝑂 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcdv0cl.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcdv0cl.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
3 | lcdv0cl.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | lcdlmod 41497 | . 2 ⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | lcdv0cl.v | . . 3 ⊢ 𝑉 = (Base‘𝐶) | |
6 | lcdv0cl.o | . . 3 ⊢ 𝑂 = (0g‘𝐶) | |
7 | 5, 6 | lmod0vcl 20906 | . 2 ⊢ (𝐶 ∈ LMod → 𝑂 ∈ 𝑉) |
8 | 4, 7 | syl 17 | 1 ⊢ (𝜑 → 𝑂 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ‘cfv 6572 Basecbs 17253 0gc0g 17494 LModclmod 20875 HLchlt 39254 LHypclh 39889 LCDualclcd 41491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-riotaBAD 38857 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-iin 5022 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-om 7900 df-1st 8026 df-2nd 8027 df-tpos 8263 df-undef 8310 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-n0 12550 df-z 12636 df-uz 12900 df-fz 13564 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-sca 17322 df-vsca 17323 df-0g 17496 df-mre 17639 df-mrc 17640 df-acs 17642 df-proset 18360 df-poset 18378 df-plt 18395 df-lub 18411 df-glb 18412 df-join 18413 df-meet 18414 df-p0 18490 df-p1 18491 df-lat 18497 df-clat 18564 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-submnd 18814 df-grp 18971 df-minusg 18972 df-sbg 18973 df-subg 19158 df-cntz 19352 df-oppg 19381 df-lsm 19673 df-cmn 19819 df-abl 19820 df-mgp 20157 df-rng 20175 df-ur 20204 df-ring 20257 df-oppr 20355 df-dvdsr 20378 df-unit 20379 df-invr 20409 df-dvr 20422 df-nzr 20534 df-rlreg 20711 df-domn 20712 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38880 df-lshyp 38881 df-lcv 38923 df-lfl 38962 df-lkr 38990 df-ldual 39028 df-oposet 39080 df-ol 39082 df-oml 39083 df-covers 39170 df-ats 39171 df-atl 39202 df-cvlat 39226 df-hlat 39255 df-llines 39403 df-lplanes 39404 df-lvols 39405 df-lines 39406 df-psubsp 39408 df-pmap 39409 df-padd 39701 df-lhyp 39893 df-laut 39894 df-ldil 40009 df-ltrn 40010 df-trl 40064 df-tgrp 40648 df-tendo 40660 df-edring 40662 df-dveca 40908 df-disoa 40934 df-dvech 40984 df-dib 41044 df-dic 41078 df-dih 41134 df-doch 41253 df-djh 41300 df-lcdual 41492 |
This theorem is referenced by: mapd0 41570 mapdspex 41573 mapdhcl 41632 |
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