lcdlvec - Mathbox for Norm Megill

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Theorem lcdlvec 40974

Description: The dual vector space of functionals with closed kernels is a left vector space. (Contributed by NM, 14-Mar-2015.)

**Hypotheses**
Ref	Expression
lcdlmod.h	⊢ 𝐻 = (LHyp‘𝐾)
lcdlmod.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
lcdlmod.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))

**Assertion**
Ref	Expression
lcdlvec	⊢ (𝜑 → 𝐶 ∈ LVec)

Proof of Theorem lcdlvec Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcdlmod.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		eqid 2726	. . 3 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊)
3		lcdlmod.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
4		eqid 2726	. . 3 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
5		eqid 2726	. . 3 ⊢ (LFnl‘((DVecH‘𝐾)‘𝑊)) = (LFnl‘((DVecH‘𝐾)‘𝑊))
6		eqid 2726	. . 3 ⊢ (LKer‘((DVecH‘𝐾)‘𝑊)) = (LKer‘((DVecH‘𝐾)‘𝑊))
7		eqid 2726	. . 3 ⊢ (LDual‘((DVecH‘𝐾)‘𝑊)) = (LDual‘((DVecH‘𝐾)‘𝑊))
8		lcdlmod.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9	1, 2, 3, 4, 5, 6, 7, 8	lcdval 40972	. 2 ⊢ (𝜑 → 𝐶 = ((LDual‘((DVecH‘𝐾)‘𝑊)) ↾_s {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑊)) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓)}))
10	1, 4, 8	dvhlvec 40492	. . . 4 ⊢ (𝜑 → ((DVecH‘𝐾)‘𝑊) ∈ LVec)
11	7, 10	lduallvec 38536	. . 3 ⊢ (𝜑 → (LDual‘((DVecH‘𝐾)‘𝑊)) ∈ LVec)
12		eqid 2726	. . . 4 ⊢ (LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊))) = (LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊)))
13		eqid 2726	. . . 4 ⊢ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑊)) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓)} = {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑊)) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓)}
14	1, 4, 2, 5, 6, 7, 12, 13, 8	lclkr 40916	. . 3 ⊢ (𝜑 → {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑊)) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓)} ∈ (LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊))))
15		eqid 2726	. . . 4 ⊢ ((LDual‘((DVecH‘𝐾)‘𝑊)) ↾_s {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑊)) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓)}) = ((LDual‘((DVecH‘𝐾)‘𝑊)) ↾_s {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑊)) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓)})
16	15, 12	lsslvec 20954	. . 3 ⊢ (((LDual‘((DVecH‘𝐾)‘𝑊)) ∈ LVec ∧ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑊)) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓)} ∈ (LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊)))) → ((LDual‘((DVecH‘𝐾)‘𝑊)) ↾_s {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑊)) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓)}) ∈ LVec)
17	11, 14, 16	syl2anc 583	. 2 ⊢ (𝜑 → ((LDual‘((DVecH‘𝐾)‘𝑊)) ↾_s {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑊)) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑓)}) ∈ LVec)
18	9, 17	eqeltrd 2827	1 ⊢ (𝜑 → 𝐶 ∈ LVec)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3426 ‘cfv 6536 (class class class)co 7404 ↾_s cress 17179 LSubSpclss 20775 LVecclvec 20947 LFnlclfn 38439 LKerclk 38467 LDualcld 38505 HLchlt 38732 LHypclh 39367 DVecHcdvh 40461 ocHcoch 40730 LCDualclcd 40969

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-riotaBAD 38335

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-undef 8256 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-sca 17219 df-vsca 17220 df-0g 17393 df-mre 17536 df-mrc 17537 df-acs 17539 df-proset 18257 df-poset 18275 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18394 df-clat 18461 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-submnd 18711 df-grp 18863 df-minusg 18864 df-sbg 18865 df-subg 19047 df-cntz 19230 df-oppg 19259 df-lsm 19553 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-oppr 20233 df-dvdsr 20256 df-unit 20257 df-invr 20287 df-dvr 20300 df-drng 20586 df-lmod 20705 df-lss 20776 df-lsp 20816 df-lvec 20948 df-lsatoms 38358 df-lshyp 38359 df-lcv 38401 df-lfl 38440 df-lkr 38468 df-ldual 38506 df-oposet 38558 df-ol 38560 df-oml 38561 df-covers 38648 df-ats 38649 df-atl 38680 df-cvlat 38704 df-hlat 38733 df-llines 38881 df-lplanes 38882 df-lvols 38883 df-lines 38884 df-psubsp 38886 df-pmap 38887 df-padd 39179 df-lhyp 39371 df-laut 39372 df-ldil 39487 df-ltrn 39488 df-trl 39542 df-tgrp 40126 df-tendo 40138 df-edring 40140 df-dveca 40386 df-disoa 40412 df-dvech 40462 df-dib 40522 df-dic 40556 df-dih 40612 df-doch 40731 df-djh 40778 df-lcdual 40970

This theorem is referenced by: lcdlmod 40975 mapdcnvatN 41049 mapdat 41050 mapdpglem18 41072 mapdpglem20 41074 mapdpglem22 41076 mapdpglem26 41081 mapdpglem27 41082 mapdpglem30 41085 mapdheq4lem 41114 mapdh6lem1N 41116 mapdh6lem2N 41117 hdmap1l6lem1 41190 hdmap1l6lem2 41191 hdmaprnlem3N 41233 hdmaprnlem3uN 41234 hdmaprnlem9N 41240 hdmap14lem2a 41250 hdmap14lem2N 41252 hdmap14lem3 41253 hdmap14lem6 41256 hdmap14lem9 41259 hgmapval0 41275 hgmapval1 41276 hgmapadd 41277 hgmapmul 41278 hgmaprnlem1N 41279

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