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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2 | Structured version Visualization version GIF version |
Description: The set of functionals having closed kernels is closed under vector (functional) addition. Lemmas lclkrlem2a 37645 through lclkrlem2y 37669 are used for the proof. Here we express lclkrlem2y 37669 in terms of membership in the set 𝐶 of functionals with closed kernels. (Contributed by NM, 18-Jan-2015.) |
Ref | Expression |
---|---|
lclkrlem2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lclkrlem2.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lclkrlem2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lclkrlem2.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lclkrlem2.l | ⊢ 𝐿 = (LKer‘𝑈) |
lclkrlem2.d | ⊢ 𝐷 = (LDual‘𝑈) |
lclkrlem2.p | ⊢ + = (+g‘𝐷) |
lclkrlem2.c | ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
lclkrlem2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lclkrlem2.e | ⊢ (𝜑 → 𝐸 ∈ 𝐶) |
lclkrlem2.g | ⊢ (𝜑 → 𝐺 ∈ 𝐶) |
Ref | Expression |
---|---|
lclkrlem2 | ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lclkrlem2.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
2 | lclkrlem2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | lclkrlem2.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
4 | lclkrlem2.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | lclkrlem2.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
6 | lclkrlem2.d | . . 3 ⊢ 𝐷 = (LDual‘𝑈) | |
7 | lclkrlem2.p | . . 3 ⊢ + = (+g‘𝐷) | |
8 | lclkrlem2.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | lclkrlem2.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐶) | |
10 | lclkrlem2.c | . . . . . 6 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
11 | 10 | lcfl1lem 37629 | . . . . 5 ⊢ (𝐸 ∈ 𝐶 ↔ (𝐸 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸))) |
12 | 11 | simplbi 493 | . . . 4 ⊢ (𝐸 ∈ 𝐶 → 𝐸 ∈ 𝐹) |
13 | 9, 12 | syl 17 | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
14 | lclkrlem2.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐶) | |
15 | 10 | lcfl1lem 37629 | . . . . 5 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
16 | 15 | simplbi 493 | . . . 4 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐹) |
17 | 14, 16 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
18 | 10, 13 | lcfl1 37630 | . . . 4 ⊢ (𝜑 → (𝐸 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸))) |
19 | 9, 18 | mpbid 224 | . . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐸))) = (𝐿‘𝐸)) |
20 | 10, 17 | lcfl1 37630 | . . . 4 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
21 | 14, 20 | mpbid 224 | . . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 13, 17, 19, 21 | lclkrlem2y 37669 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
23 | 2, 4, 8 | dvhlmod 37248 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
24 | 5, 6, 7, 23, 13, 17 | ldualvaddcl 35268 | . . 3 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐹) |
25 | 10, 24 | lcfl1 37630 | . 2 ⊢ (𝜑 → ((𝐸 + 𝐺) ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))) |
26 | 22, 25 | mpbird 249 | 1 ⊢ (𝜑 → (𝐸 + 𝐺) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 {crab 3093 ‘cfv 6135 (class class class)co 6922 +gcplusg 16338 LFnlclfn 35195 LKerclk 35223 LDualcld 35261 HLchlt 35488 LHypclh 36122 DVecHcdvh 37216 ocHcoch 37485 |
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 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-riotaBAD 35091 |
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 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-undef 7681 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-mre 16632 df-mrc 16633 df-acs 16635 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-oppg 18159 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 df-lsatoms 35114 df-lshyp 35115 df-lcv 35157 df-lfl 35196 df-lkr 35224 df-ldual 35262 df-oposet 35314 df-ol 35316 df-oml 35317 df-covers 35404 df-ats 35405 df-atl 35436 df-cvlat 35460 df-hlat 35489 df-llines 35636 df-lplanes 35637 df-lvols 35638 df-lines 35639 df-psubsp 35641 df-pmap 35642 df-padd 35934 df-lhyp 36126 df-laut 36127 df-ldil 36242 df-ltrn 36243 df-trl 36297 df-tgrp 36881 df-tendo 36893 df-edring 36895 df-dveca 37141 df-disoa 37167 df-dvech 37217 df-dib 37277 df-dic 37311 df-dih 37367 df-doch 37486 df-djh 37533 |
This theorem is referenced by: lclkr 37671 lclkrslem2 37676 |
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