lclkrs - Mathbox for Norm Megill

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Theorem lclkrs 41540

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 41534 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 41534 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 4046	. . . 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 2730	. . . 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 41111	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
12	5, 6, 7, 11	ldualvbase 39126	. . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹)
13	2, 4, 12	3sstr4d 4005	. 2 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐷))
14		eqid 2730	. . . . . 6 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
15		eqid 2730	. . . . . 6 ⊢ (0_g‘(Scalar‘𝑈)) = (0_g‘(Scalar‘𝑈))
16		eqid 2730	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
17	14, 15, 16, 5	lfl0f 39069	. . . . 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 41362	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
21		eqidd 2731	. . . . . . . 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 39094	. . . . . . . . 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 6865	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}))) = ( ⊥ ‘(Base‘𝑈)))
27	26	fveq2d 6865	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
28	20, 27, 25	3eqtr4d 2775	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))) = (𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))
29		eqid 2730	. . . . . . . 8 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
30	8, 9, 19, 16, 29	doch1 41360	. . . . . . 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 2765	. . . . 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 20862	. . . . . 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 3984	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}))) ⊆ 𝑅)
38	3	lcfls1lem 41535	. . . 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 4308	. 2 ⊢ (𝜑 → 𝐶 ≠ ∅)
41		eqid 2730	. . . 4 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
42		eqid 2730	. . . 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 2730	. . . 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 2730	. . . . . . . 8 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷)
50		eqid 2730	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷))
51	14, 41, 6, 49, 50, 11	ldualsbase 39133	. . . . . . 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 2831	. . . . 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 41538	. . . 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 41539	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → ((𝑥( ·_𝑠 ‘𝐷)𝑎)(+_g‘𝐷)𝑏) ∈ 𝐶)
56	55	ralrimivvva 3184	. 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 20847	. 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 1540 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 {crab 3408 ⊆ wss 3917 ∅c0 4299 {csn 4592 × cxp 5639 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 +_gcplusg 17227 Scalarcsca 17230 ·_𝑠 cvsca 17231 0_gc0g 17409 LModclmod 20773 LSubSpclss 20844 LFnlclfn 39057 LKerclk 39085 LDualcld 39123 HLchlt 39350 LHypclh 39985 DVecHcdvh 41079 ocHcoch 41348

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-riotaBAD 38953

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-undef 8255 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-0g 17411 df-mre 17554 df-mrc 17555 df-acs 17557 df-proset 18262 df-poset 18281 df-plt 18296 df-lub 18312 df-glb 18313 df-join 18314 df-meet 18315 df-p0 18391 df-p1 18392 df-lat 18398 df-clat 18465 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-cntz 19256 df-oppg 19285 df-lsm 19573 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-nzr 20429 df-rlreg 20610 df-domn 20611 df-drng 20647 df-lmod 20775 df-lss 20845 df-lsp 20885 df-lvec 21017 df-lsatoms 38976 df-lshyp 38977 df-lcv 39019 df-lfl 39058 df-lkr 39086 df-ldual 39124 df-oposet 39176 df-ol 39178 df-oml 39179 df-covers 39266 df-ats 39267 df-atl 39298 df-cvlat 39322 df-hlat 39351 df-llines 39499 df-lplanes 39500 df-lvols 39501 df-lines 39502 df-psubsp 39504 df-pmap 39505 df-padd 39797 df-lhyp 39989 df-laut 39990 df-ldil 40105 df-ltrn 40106 df-trl 40160 df-tgrp 40744 df-tendo 40756 df-edring 40758 df-dveca 41004 df-disoa 41030 df-dvech 41080 df-dib 41140 df-dic 41174 df-dih 41230 df-doch 41349 df-djh 41396

This theorem is referenced by: lclkrs2 41541 mapddlssN 41641

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