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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcd0v2 | Structured version Visualization version GIF version |
Description: The zero functional in the set of functionals with closed kernels. (Contributed by NM, 27-Mar-2015.) |
Ref | Expression |
---|---|
lcd0v2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcd0v2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcd0v2.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcd0v2.z | ⊢ 0 = (0g‘𝐷) |
lcd0v2.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
lcd0v2.o | ⊢ 𝑂 = (0g‘𝐶) |
lcd0v2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
lcd0v2 | ⊢ (𝜑 → 𝑂 = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcd0v2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcd0v2.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | eqid 2728 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
4 | eqid 2728 | . . 3 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
5 | eqid 2728 | . . 3 ⊢ (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈)) | |
6 | lcd0v2.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
7 | lcd0v2.o | . . 3 ⊢ 𝑂 = (0g‘𝐶) | |
8 | lcd0v2.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lcd0v 41079 | . 2 ⊢ (𝜑 → 𝑂 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})) |
10 | lcd0v2.d | . . 3 ⊢ 𝐷 = (LDual‘𝑈) | |
11 | lcd0v2.z | . . 3 ⊢ 0 = (0g‘𝐷) | |
12 | 1, 2, 8 | dvhlmod 40578 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
13 | 3, 4, 5, 10, 11, 12 | ldual0v 38617 | . 2 ⊢ (𝜑 → 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})) |
14 | 9, 13 | eqtr4d 2771 | 1 ⊢ (𝜑 → 𝑂 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {csn 4625 × cxp 5671 ‘cfv 6543 Basecbs 17174 Scalarcsca 17230 0gc0g 17415 LDualcld 38590 HLchlt 38817 LHypclh 39452 DVecHcdvh 40546 LCDualclcd 41054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-riotaBAD 38420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 df-om 7866 df-1st 7988 df-2nd 7989 df-tpos 8226 df-undef 8273 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-0g 17417 df-mre 17560 df-mrc 17561 df-acs 17563 df-proset 18281 df-poset 18299 df-plt 18316 df-lub 18332 df-glb 18333 df-join 18334 df-meet 18335 df-p0 18411 df-p1 18412 df-lat 18418 df-clat 18485 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-grp 18887 df-minusg 18888 df-sbg 18889 df-subg 19072 df-cntz 19262 df-oppg 19291 df-lsm 19585 df-cmn 19731 df-abl 19732 df-mgp 20069 df-rng 20087 df-ur 20116 df-ring 20169 df-oppr 20267 df-dvdsr 20290 df-unit 20291 df-invr 20321 df-dvr 20334 df-drng 20620 df-lmod 20739 df-lss 20810 df-lsp 20850 df-lvec 20982 df-lsatoms 38443 df-lshyp 38444 df-lcv 38486 df-lfl 38525 df-lkr 38553 df-ldual 38591 df-oposet 38643 df-ol 38645 df-oml 38646 df-covers 38733 df-ats 38734 df-atl 38765 df-cvlat 38789 df-hlat 38818 df-llines 38966 df-lplanes 38967 df-lvols 38968 df-lines 38969 df-psubsp 38971 df-pmap 38972 df-padd 39264 df-lhyp 39456 df-laut 39457 df-ldil 39572 df-ltrn 39573 df-trl 39627 df-tgrp 40211 df-tendo 40223 df-edring 40225 df-dveca 40471 df-disoa 40497 df-dvech 40547 df-dib 40607 df-dic 40641 df-dih 40697 df-doch 40816 df-djh 40863 df-lcdual 41055 |
This theorem is referenced by: lcdlkreqN 41090 hvmap1o2 41233 |
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