lclkrs - Mathbox for Norm Megill

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Theorem lclkrs 41799

Description: The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace 𝑅 is a subspace of the dual space. TODO: This proof repeats large parts of the lclkr 41793 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 41793 a special case of this? (Contributed by NM, 29-Jan-2015.)

**Hypotheses**
Ref	Expression
lclkrs.h	⊢ 𝐻 = (LHyp‘𝐾)
lclkrs.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lclkrs.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lclkrs.s	⊢ 𝑆 = (LSubSp‘𝑈)
lclkrs.f	⊢ 𝐹 = (LFnl‘𝑈)
lclkrs.l	⊢ 𝐿 = (LKer‘𝑈)
lclkrs.d	⊢ 𝐷 = (LDual‘𝑈)
lclkrs.t	⊢ 𝑇 = (LSubSp‘𝐷)
lclkrs.c	⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑅)}
lclkrs.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lclkrs.r	⊢ (𝜑 → 𝑅 ∈ 𝑆)

**Assertion**
Ref	Expression
lclkrs	⊢ (𝜑 → 𝐶 ∈ 𝑇)

Distinct variable groups: 𝐷,𝑓 𝑓,𝐹 𝑓,𝐿 𝑅,𝑓 𝑈,𝑓 ⊥ ,𝑓 Allowed substitution hints: 𝜑(𝑓) 𝐶(𝑓) 𝑆(𝑓) 𝑇(𝑓) 𝐻(𝑓) 𝐾(𝑓) 𝑊(𝑓)

Proof of Theorem lclkrs Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4032	. . . 4 ⊢ {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑅)} ⊆ 𝐹
2	1	a1i 11	. . 3 ⊢ (𝜑 → {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑅)} ⊆ 𝐹)
3		lclkrs.c	. . . 4 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑅)}
4	3	a1i 11	. . 3 ⊢ (𝜑 → 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑅)})
5		lclkrs.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑈)
6		lclkrs.d	. . . 4 ⊢ 𝐷 = (LDual‘𝑈)
7		eqid 2736	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
8		lclkrs.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
9		lclkrs.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
10		lclkrs.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11	8, 9, 10	dvhlmod 41370	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
12	5, 6, 7, 11	ldualvbase 39386	. . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹)
13	2, 4, 12	3sstr4d 3989	. 2 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐷))
14		eqid 2736	. . . . . 6 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
15		eqid 2736	. . . . . 6 ⊢ (0_g‘(Scalar‘𝑈)) = (0_g‘(Scalar‘𝑈))
16		eqid 2736	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
17	14, 15, 16, 5	lfl0f 39329	. . . . 5 ⊢ (𝑈 ∈ LMod → ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) ∈ 𝐹)
18	11, 17	syl 17	. . . 4 ⊢ (𝜑 → ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) ∈ 𝐹)
19		lclkrs.o	. . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
20	8, 9, 19, 16, 10	dochoc1 41621	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
21		eqidd 2737	. . . . . . . 8 ⊢ (𝜑 → ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) = ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}))
22		lclkrs.l	. . . . . . . . . 10 ⊢ 𝐿 = (LKer‘𝑈)
23	14, 15, 16, 5, 22	lkr0f 39354	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) ∈ 𝐹) → ((𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})) = (Base‘𝑈) ↔ ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) = ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))
24	11, 18, 23	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})) = (Base‘𝑈) ↔ ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) = ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))
25	21, 24	mpbird 257	. . . . . . 7 ⊢ (𝜑 → (𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})) = (Base‘𝑈))
26	25	fveq2d 6838	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}))) = ( ⊥ ‘(Base‘𝑈)))
27	26	fveq2d 6838	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
28	20, 27, 25	3eqtr4d 2781	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))) = (𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))
29		eqid 2736	. . . . . . . 8 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
30	8, 9, 19, 16, 29	doch1 41619	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘(Base‘𝑈)) = {(0_g‘𝑈)})
31	10, 30	syl 17	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(Base‘𝑈)) = {(0_g‘𝑈)})
32	26, 31	eqtrd 2771	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}))) = {(0_g‘𝑈)})
33		lclkrs.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
34		lclkrs.s	. . . . . . 7 ⊢ 𝑆 = (LSubSp‘𝑈)
35	29, 34	lss0ss 20900	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑅 ∈ 𝑆) → {(0_g‘𝑈)} ⊆ 𝑅)
36	11, 33, 35	syl2anc 584	. . . . 5 ⊢ (𝜑 → {(0_g‘𝑈)} ⊆ 𝑅)
37	32, 36	eqsstrd 3968	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}))) ⊆ 𝑅)
38	3	lcfls1lem 41794	. . . 4 ⊢ (((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) ∈ 𝐶 ↔ (((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))) = (𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})) ∧ ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}))) ⊆ 𝑅))
39	18, 28, 37, 38	syl3anbrc 1344	. . 3 ⊢ (𝜑 → ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) ∈ 𝐶)
40	39	ne0d 4294	. 2 ⊢ (𝜑 → 𝐶 ≠ ∅)
41		eqid 2736	. . . 4 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
42		eqid 2736	. . . 4 ⊢ ( ·_𝑠 ‘𝐷) = ( ·_𝑠 ‘𝐷)
43	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
44	33	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑅 ∈ 𝑆)
45		simpr3 1197	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑏 ∈ 𝐶)
46		eqid 2736	. . . 4 ⊢ (+_g‘𝐷) = (+_g‘𝐷)
47		simpr2 1196	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑎 ∈ 𝐶)
48		simpr1 1195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝐷)))
49		eqid 2736	. . . . . . . 8 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷)
50		eqid 2736	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷))
51	14, 41, 6, 49, 50, 11	ldualsbase 39393	. . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈)))
52	51	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈)))
53	48, 52	eleqtrd 2838	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝑈)))
54	8, 19, 9, 34, 5, 22, 6, 14, 41, 42, 3, 43, 44, 47, 53	lclkrslem1 41797	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝑥( ·_𝑠 ‘𝐷)𝑎) ∈ 𝐶)
55	8, 19, 9, 34, 5, 22, 6, 14, 41, 42, 3, 43, 44, 45, 46, 54	lclkrslem2 41798	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → ((𝑥( ·_𝑠 ‘𝐷)𝑎)(+_g‘𝐷)𝑏) ∈ 𝐶)
56	55	ralrimivvva 3182	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝐷))∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 ((𝑥( ·_𝑠 ‘𝐷)𝑎)(+_g‘𝐷)𝑏) ∈ 𝐶)
57		lclkrs.t	. . 3 ⊢ 𝑇 = (LSubSp‘𝐷)
58	49, 50, 7, 46, 42, 57	islss 20885	. 2 ⊢ (𝐶 ∈ 𝑇 ↔ (𝐶 ⊆ (Base‘𝐷) ∧ 𝐶 ≠ ∅ ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝐷))∀𝑎 ∈ 𝐶 ∀𝑏 ∈ 𝐶 ((𝑥( ·_𝑠 ‘𝐷)𝑎)(+_g‘𝐷)𝑏) ∈ 𝐶))
59	13, 40, 56, 58	syl3anbrc 1344	1 ⊢ (𝜑 → 𝐶 ∈ 𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∀wral 3051 {crab 3399 ⊆ wss 3901 ∅c0 4285 {csn 4580 × cxp 5622 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 +_gcplusg 17177 Scalarcsca 17180 ·_𝑠 cvsca 17181 0_gc0g 17359 LModclmod 20811 LSubSpclss 20882 LFnlclfn 39317 LKerclk 39345 LDualcld 39383 HLchlt 39610 LHypclh 40244 DVecHcdvh 41338 ocHcoch 41607

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-riotaBAD 39213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8168 df-undef 8215 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-n0 12402 df-z 12489 df-uz 12752 df-fz 13424 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-0g 17361 df-mre 17505 df-mrc 17506 df-acs 17508 df-proset 18217 df-poset 18236 df-plt 18251 df-lub 18267 df-glb 18268 df-join 18269 df-meet 18270 df-p0 18346 df-p1 18347 df-lat 18355 df-clat 18422 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-cntz 19246 df-oppg 19275 df-lsm 19565 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-nzr 20446 df-rlreg 20627 df-domn 20628 df-drng 20664 df-lmod 20813 df-lss 20883 df-lsp 20923 df-lvec 21055 df-lsatoms 39236 df-lshyp 39237 df-lcv 39279 df-lfl 39318 df-lkr 39346 df-ldual 39384 df-oposet 39436 df-ol 39438 df-oml 39439 df-covers 39526 df-ats 39527 df-atl 39558 df-cvlat 39582 df-hlat 39611 df-llines 39758 df-lplanes 39759 df-lvols 39760 df-lines 39761 df-psubsp 39763 df-pmap 39764 df-padd 40056 df-lhyp 40248 df-laut 40249 df-ldil 40364 df-ltrn 40365 df-trl 40419 df-tgrp 41003 df-tendo 41015 df-edring 41017 df-dveca 41263 df-disoa 41289 df-dvech 41339 df-dib 41399 df-dic 41433 df-dih 41489 df-doch 41608 df-djh 41655

This theorem is referenced by: lclkrs2 41800 mapddlssN 41900

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