lcdlvec - Mathbox for Norm Megill

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Theorem lcdlvec 42427

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 2765	. . 3 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊)
3		lcdlmod.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
4		eqid 2765	. . 3 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
5		eqid 2765	. . 3 ⊢ (LFnl‘((DVecH‘𝐾)‘𝑊)) = (LFnl‘((DVecH‘𝐾)‘𝑊))
6		eqid 2765	. . 3 ⊢ (LKer‘((DVecH‘𝐾)‘𝑊)) = (LKer‘((DVecH‘𝐾)‘𝑊))
7		eqid 2765	. . 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 42425	. 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 41945	. . . 4 ⊢ (𝜑 → ((DVecH‘𝐾)‘𝑊) ∈ LVec)
11	7, 10	lduallvec 39990	. . 3 ⊢ (𝜑 → (LDual‘((DVecH‘𝐾)‘𝑊)) ∈ LVec)
12		eqid 2765	. . . 4 ⊢ (LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊))) = (LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊)))
13		eqid 2765	. . . 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 42369	. . 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 2765	. . . 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 21284	. . 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 596	. 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 2865	1 ⊢ (𝜑 → 𝐶 ∈ LVec)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 {crab 3418 ‘cfv 6540 (class class class)co 7419 ↾_s cress 17316 LSubSpclss 21106 LVecclvec 21277 LFnlclfn 39893 LKerclk 39921 LDualcld 39959 HLchlt 40186 LHypclh 40820 DVecHcdvh 41914 ocHcoch 42183 LCDualclcd 42422

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11175 ax-resscn 11176 ax-1cn 11177 ax-icn 11178 ax-addcl 11179 ax-addrcl 11180 ax-mulcl 11181 ax-mulrcl 11182 ax-mulcom 11183 ax-addass 11184 ax-mulass 11185 ax-distr 11186 ax-i2m1 11187 ax-1ne0 11188 ax-1rid 11189 ax-rnegex 11190 ax-rrecex 11191 ax-cnre 11192 ax-pre-lttri 11193 ax-pre-lttrn 11194 ax-pre-ltadd 11195 ax-pre-mulgt0 11196 ax-riotaBAD 39789

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-tpos 8228 df-undef 8275 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11462 df-neg 11463 df-nn 12253 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-n0 12524 df-z 12611 df-uz 12883 df-fz 13556 df-struct 17233 df-sets 17250 df-slot 17268 df-ndx 17280 df-base 17296 df-ress 17317 df-plusg 17349 df-mulr 17350 df-sca 17352 df-vsca 17353 df-0g 17520 df-mre 17664 df-mrc 17665 df-acs 17667 df-proset 18376 df-poset 18395 df-plt 18410 df-lub 18426 df-glb 18427 df-join 18428 df-meet 18429 df-p0 18505 df-p1 18506 df-lat 18514 df-clat 18581 df-mgm 18724 df-sgrp 18813 df-mnd 18829 df-submnd 18883 df-grp 19051 df-minusg 19052 df-sbg 19053 df-subg 19237 df-cntz 19435 df-oppg 19464 df-lsm 19754 df-cmn 19900 df-abl 19901 df-mgp 20265 df-rng 20279 df-ur 20312 df-ring 20365 df-oppr 20469 df-dvdsr 20489 df-unit 20490 df-invr 20520 df-dvr 20533 df-nzr 20664 df-rlreg 20847 df-domn 20848 df-drng 20883 df-lmod 21037 df-lss 21107 df-lsp 21147 df-lvec 21278 df-lsatoms 39812 df-lshyp 39813 df-lcv 39855 df-lfl 39894 df-lkr 39922 df-ldual 39960 df-oposet 40012 df-ol 40014 df-oml 40015 df-covers 40102 df-ats 40103 df-atl 40134 df-cvlat 40158 df-hlat 40187 df-llines 40334 df-lplanes 40335 df-lvols 40336 df-lines 40337 df-psubsp 40339 df-pmap 40340 df-padd 40632 df-lhyp 40824 df-laut 40825 df-ldil 40940 df-ltrn 40941 df-trl 40995 df-tgrp 41579 df-tendo 41591 df-edring 41593 df-dveca 41839 df-disoa 41865 df-dvech 41915 df-dib 41975 df-dic 42009 df-dih 42065 df-doch 42184 df-djh 42231 df-lcdual 42423

This theorem is used by: lcdlmod 42428 mapdcnvatN 42502 mapdat 42503 mapdpglem18 42525 mapdpglem20 42527 mapdpglem22 42529 mapdpglem26 42534 mapdpglem27 42535 mapdpglem30 42538 mapdheq4lem 42567 mapdh6lem1N 42569 mapdh6lem2N 42570 hdmap1l6lem1 42643 hdmap1l6lem2 42644 hdmaprnlem3N 42686 hdmaprnlem3uN 42687 hdmaprnlem9N 42693 hdmap14lem2a 42703 hdmap14lem2N 42705 hdmap14lem3 42706 hdmap14lem6 42709 hdmap14lem9 42712 hgmapval0 42728 hgmapval1 42729 hgmapadd 42730 hgmapmul 42731 hgmaprnlem1N 42732

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