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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 2740 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
4 | eqid 2740 | . . 3 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
5 | eqid 2740 | . . 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 41588 | . 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 41087 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
13 | 3, 4, 5, 10, 11, 12 | ldual0v 39126 | . 2 ⊢ (𝜑 → 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})) |
14 | 9, 13 | eqtr4d 2783 | 1 ⊢ (𝜑 → 𝑂 = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 {csn 4648 × cxp 5699 ‘cfv 6576 Basecbs 17278 Scalarcsca 17334 0gc0g 17519 LDualcld 39099 HLchlt 39326 LHypclh 39961 DVecHcdvh 41055 LCDualclcd 41563 |
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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5304 ax-sep 5318 ax-nul 5325 ax-pow 5384 ax-pr 5448 ax-un 7773 ax-cnex 11243 ax-resscn 11244 ax-1cn 11245 ax-icn 11246 ax-addcl 11247 ax-addrcl 11248 ax-mulcl 11249 ax-mulrcl 11250 ax-mulcom 11251 ax-addass 11252 ax-mulass 11253 ax-distr 11254 ax-i2m1 11255 ax-1ne0 11256 ax-1rid 11257 ax-rnegex 11258 ax-rrecex 11259 ax-cnre 11260 ax-pre-lttri 11261 ax-pre-lttrn 11262 ax-pre-ltadd 11263 ax-pre-mulgt0 11264 ax-riotaBAD 38929 |
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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4933 df-int 4972 df-iun 5018 df-iin 5019 df-br 5168 df-opab 5230 df-mpt 5251 df-tr 5285 df-id 5594 df-eprel 5600 df-po 5608 df-so 5609 df-fr 5653 df-we 5655 df-xp 5707 df-rel 5708 df-cnv 5709 df-co 5710 df-dm 5711 df-rn 5712 df-res 5713 df-ima 5714 df-pred 6335 df-ord 6401 df-on 6402 df-lim 6403 df-suc 6404 df-iota 6528 df-fun 6578 df-fn 6579 df-f 6580 df-f1 6581 df-fo 6582 df-f1o 6583 df-fv 6584 df-riota 7407 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7717 df-om 7907 df-1st 8033 df-2nd 8034 df-tpos 8270 df-undef 8317 df-frecs 8325 df-wrecs 8356 df-recs 8430 df-rdg 8469 df-1o 8525 df-2o 8526 df-er 8766 df-map 8889 df-en 9007 df-dom 9008 df-sdom 9009 df-fin 9010 df-pnf 11329 df-mnf 11330 df-xr 11331 df-ltxr 11332 df-le 11333 df-sub 11526 df-neg 11527 df-nn 12299 df-2 12361 df-3 12362 df-4 12363 df-5 12364 df-6 12365 df-n0 12559 df-z 12646 df-uz 12911 df-fz 13579 df-struct 17214 df-sets 17231 df-slot 17249 df-ndx 17261 df-base 17279 df-ress 17308 df-plusg 17344 df-mulr 17345 df-sca 17347 df-vsca 17348 df-0g 17521 df-mre 17664 df-mrc 17665 df-acs 17667 df-proset 18385 df-poset 18403 df-plt 18420 df-lub 18436 df-glb 18437 df-join 18438 df-meet 18439 df-p0 18515 df-p1 18516 df-lat 18522 df-clat 18589 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18839 df-grp 18996 df-minusg 18997 df-sbg 18998 df-subg 19183 df-cntz 19377 df-oppg 19406 df-lsm 19698 df-cmn 19844 df-abl 19845 df-mgp 20182 df-rng 20200 df-ur 20229 df-ring 20282 df-oppr 20380 df-dvdsr 20403 df-unit 20404 df-invr 20434 df-dvr 20447 df-nzr 20559 df-rlreg 20736 df-domn 20737 df-drng 20773 df-lmod 20902 df-lss 20973 df-lsp 21013 df-lvec 21145 df-lsatoms 38952 df-lshyp 38953 df-lcv 38995 df-lfl 39034 df-lkr 39062 df-ldual 39100 df-oposet 39152 df-ol 39154 df-oml 39155 df-covers 39242 df-ats 39243 df-atl 39274 df-cvlat 39298 df-hlat 39327 df-llines 39475 df-lplanes 39476 df-lvols 39477 df-lines 39478 df-psubsp 39480 df-pmap 39481 df-padd 39773 df-lhyp 39965 df-laut 39966 df-ldil 40081 df-ltrn 40082 df-trl 40136 df-tgrp 40720 df-tendo 40732 df-edring 40734 df-dveca 40980 df-disoa 41006 df-dvech 41056 df-dib 41116 df-dic 41150 df-dih 41206 df-doch 41325 df-djh 41372 df-lcdual 41564 |
This theorem is referenced by: lcdlkreqN 41599 hvmap1o2 41742 |
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