lcdlvec - Mathbox for Norm Megill

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Theorem lcdlvec 42794

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 2763	. . 3 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊)
3		lcdlmod.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
4		eqid 2763	. . 3 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
5		eqid 2763	. . 3 ⊢ (LFnl‘((DVecH‘𝐾)‘𝑊)) = (LFnl‘((DVecH‘𝐾)‘𝑊))
6		eqid 2763	. . 3 ⊢ (LKer‘((DVecH‘𝐾)‘𝑊)) = (LKer‘((DVecH‘𝐾)‘𝑊))
7		eqid 2763	. . 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 42792	. 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 42312	. . . 4 ⊢ (𝜑 → ((DVecH‘𝐾)‘𝑊) ∈ LVec)
11	7, 10	lduallvec 40357	. . 3 ⊢ (𝜑 → (LDual‘((DVecH‘𝐾)‘𝑊)) ∈ LVec)
12		eqid 2763	. . . 4 ⊢ (LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊))) = (LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊)))
13		eqid 2763	. . . 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 42736	. . 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 2763	. . . 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 21399	. . 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 2864	1 ⊢ (𝜑 → 𝐶 ∈ LVec)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2146 {crab 3416 ‘cfv 6547 (class class class)co 7430 ↾_s cress 17423 LSubSpclss 21221 LVecclvec 21392 LFnlclfn 40260 LKerclk 40288 LDualcld 40326 HLchlt 40553 LHypclh 41187 DVecHcdvh 42281 ocHcoch 42550 LCDualclcd 42789

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290 ax-riotaBAD 40156

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7703 df-om 7888 df-1st 8011 df-2nd 8012 df-tpos 8257 df-undef 8304 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-2o 8491 df-er 8731 df-map 8863 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-nn 12349 df-2 12418 df-3 12419 df-4 12420 df-5 12421 df-6 12422 df-n0 12620 df-z 12707 df-uz 12979 df-fz 13653 df-struct 17340 df-sets 17357 df-slot 17375 df-ndx 17387 df-base 17403 df-ress 17424 df-plusg 17456 df-mulr 17457 df-sca 17459 df-vsca 17460 df-0g 17627 df-mre 17771 df-mrc 17772 df-acs 17774 df-proset 18483 df-poset 18502 df-plt 18517 df-lub 18533 df-glb 18534 df-join 18535 df-meet 18536 df-p0 18612 df-p1 18613 df-lat 18621 df-clat 18688 df-mgm 18831 df-sgrp 18923 df-mnd 18939 df-submnd 18994 df-grp 19162 df-minusg 19163 df-sbg 19164 df-subg 19348 df-cntz 19546 df-oppg 19575 df-lsm 19865 df-cmn 20011 df-abl 20012 df-mgp 20376 df-rng 20390 df-ur 20423 df-ring 20476 df-oppr 20582 df-dvdsr 20602 df-unit 20603 df-invr 20633 df-dvr 20646 df-nzr 20778 df-rlreg 20961 df-domn 20962 df-drng 20997 df-lmod 21152 df-lss 21222 df-lsp 21262 df-lvec 21393 df-lsatoms 40179 df-lshyp 40180 df-lcv 40222 df-lfl 40261 df-lkr 40289 df-ldual 40327 df-oposet 40379 df-ol 40381 df-oml 40382 df-covers 40469 df-ats 40470 df-atl 40501 df-cvlat 40525 df-hlat 40554 df-llines 40701 df-lplanes 40702 df-lvols 40703 df-lines 40704 df-psubsp 40706 df-pmap 40707 df-padd 40999 df-lhyp 41191 df-laut 41192 df-ldil 41307 df-ltrn 41308 df-trl 41362 df-tgrp 41946 df-tendo 41958 df-edring 41960 df-dveca 42206 df-disoa 42232 df-dvech 42282 df-dib 42342 df-dic 42376 df-dih 42432 df-doch 42551 df-djh 42598 df-lcdual 42790

This theorem is used by: lcdlmod 42795 mapdcnvatN 42869 mapdat 42870 mapdpglem18 42892 mapdpglem20 42894 mapdpglem22 42896 mapdpglem26 42901 mapdpglem27 42902 mapdpglem30 42905 mapdheq4lem 42934 mapdh6lem1N 42936 mapdh6lem2N 42937 hdmap1l6lem1 43010 hdmap1l6lem2 43011 hdmaprnlem3N 43053 hdmaprnlem3uN 43054 hdmaprnlem9N 43060 hdmap14lem2a 43070 hdmap14lem2N 43072 hdmap14lem3 43073 hdmap14lem6 43076 hdmap14lem9 43079 hgmapval0 43095 hgmapval1 43096 hgmapadd 43097 hgmapmul 43098 hgmaprnlem1N 43099

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