lclkrs - Mathbox for Norm Megill

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Theorem lclkrs 41578

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 41572 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 41572 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 4025	. . . 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 2731	. . . 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 41149	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
12	5, 6, 7, 11	ldualvbase 39165	. . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹)
13	2, 4, 12	3sstr4d 3985	. 2 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐷))
14		eqid 2731	. . . . . 6 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
15		eqid 2731	. . . . . 6 ⊢ (0_g‘(Scalar‘𝑈)) = (0_g‘(Scalar‘𝑈))
16		eqid 2731	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
17	14, 15, 16, 5	lfl0f 39108	. . . . 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 41400	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
21		eqidd 2732	. . . . . . . 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 39133	. . . . . . . . 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 6821	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}))) = ( ⊥ ‘(Base‘𝑈)))
27	26	fveq2d 6821	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
28	20, 27, 25	3eqtr4d 2776	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))) = (𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))
29		eqid 2731	. . . . . . . 8 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
30	8, 9, 19, 16, 29	doch1 41398	. . . . . . 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 2766	. . . . 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 20877	. . . . . 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 3964	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}))) ⊆ 𝑅)
38	3	lcfls1lem 41573	. . . 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 4287	. 2 ⊢ (𝜑 → 𝐶 ≠ ∅)
41		eqid 2731	. . . 4 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
42		eqid 2731	. . . 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 2731	. . . 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 2731	. . . . . . . 8 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷)
50		eqid 2731	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷))
51	14, 41, 6, 49, 50, 11	ldualsbase 39172	. . . . . . 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 2833	. . . . 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 41576	. . . 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 41577	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → ((𝑥( ·_𝑠 ‘𝐷)𝑎)(+_g‘𝐷)𝑏) ∈ 𝐶)
56	55	ralrimivvva 3178	. 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 20862	. 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 2111 ≠ wne 2928 ∀wral 3047 {crab 3395 ⊆ wss 3897 ∅c0 4278 {csn 4571 × cxp 5609 ‘cfv 6476 (class class class)co 7341 Basecbs 17115 +_gcplusg 17156 Scalarcsca 17159 ·_𝑠 cvsca 17160 0_gc0g 17338 LModclmod 20788 LSubSpclss 20859 LFnlclfn 39096 LKerclk 39124 LDualcld 39162 HLchlt 39389 LHypclh 40023 DVecHcdvh 41117 ocHcoch 41386

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-riotaBAD 38992

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-tpos 8151 df-undef 8198 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-n0 12377 df-z 12464 df-uz 12728 df-fz 13403 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-sca 17172 df-vsca 17173 df-0g 17340 df-mre 17483 df-mrc 17484 df-acs 17486 df-proset 18195 df-poset 18214 df-plt 18229 df-lub 18245 df-glb 18246 df-join 18247 df-meet 18248 df-p0 18324 df-p1 18325 df-lat 18333 df-clat 18400 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-grp 18844 df-minusg 18845 df-sbg 18846 df-subg 19031 df-cntz 19224 df-oppg 19253 df-lsm 19543 df-cmn 19689 df-abl 19690 df-mgp 20054 df-rng 20066 df-ur 20095 df-ring 20148 df-oppr 20250 df-dvdsr 20270 df-unit 20271 df-invr 20301 df-dvr 20314 df-nzr 20423 df-rlreg 20604 df-domn 20605 df-drng 20641 df-lmod 20790 df-lss 20860 df-lsp 20900 df-lvec 21032 df-lsatoms 39015 df-lshyp 39016 df-lcv 39058 df-lfl 39097 df-lkr 39125 df-ldual 39163 df-oposet 39215 df-ol 39217 df-oml 39218 df-covers 39305 df-ats 39306 df-atl 39337 df-cvlat 39361 df-hlat 39390 df-llines 39537 df-lplanes 39538 df-lvols 39539 df-lines 39540 df-psubsp 39542 df-pmap 39543 df-padd 39835 df-lhyp 40027 df-laut 40028 df-ldil 40143 df-ltrn 40144 df-trl 40198 df-tgrp 40782 df-tendo 40794 df-edring 40796 df-dveca 41042 df-disoa 41068 df-dvech 41118 df-dib 41178 df-dic 41212 df-dih 41268 df-doch 41387 df-djh 41434

This theorem is referenced by: lclkrs2 41579 mapddlssN 41679

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