lclkrslem1 - Mathbox for Norm Megill

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Theorem lclkrslem1 42166

Description: The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace 𝑄 is closed under scalar product. (Contributed by NM, 27-Jan-2015.)

**Hypotheses**
Ref	Expression
lclkrslem1.h	⊢ 𝐻 = (LHyp‘𝐾)
lclkrslem1.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lclkrslem1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lclkrslem1.s	⊢ 𝑆 = (LSubSp‘𝑈)
lclkrslem1.f	⊢ 𝐹 = (LFnl‘𝑈)
lclkrslem1.l	⊢ 𝐿 = (LKer‘𝑈)
lclkrslem1.d	⊢ 𝐷 = (LDual‘𝑈)
lclkrslem1.r	⊢ 𝑅 = (Scalar‘𝑈)
lclkrslem1.b	⊢ 𝐵 = (Base‘𝑅)
lclkrslem1.t	⊢ · = ( ·_𝑠 ‘𝐷)
lclkrslem1.c	⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑄)}
lclkrslem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lclkrslem1.q	⊢ (𝜑 → 𝑄 ∈ 𝑆)
lclkrslem1.g	⊢ (𝜑 → 𝐺 ∈ 𝐶)
lclkrslem1.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
lclkrslem1	⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐶)

Distinct variable groups: ⊥ ,𝑓 𝑓,𝐹 𝑓,𝐺 𝑓,𝐿 𝑄,𝑓 · ,𝑓 𝑓,𝑋 Allowed substitution hints: 𝜑(𝑓) 𝐵(𝑓) 𝐶(𝑓) 𝐷(𝑓) 𝑅(𝑓) 𝑆(𝑓) 𝑈(𝑓) 𝐻(𝑓) 𝐾(𝑓) 𝑊(𝑓)

**Proof of Theorem lclkrslem1**
Step	Hyp	Ref	Expression
1		lclkrslem1.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		lclkrslem1.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lclkrslem1.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lclkrslem1.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
5		lclkrslem1.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
6		lclkrslem1.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
7		lclkrslem1.r	. . 3 ⊢ 𝑅 = (Scalar‘𝑈)
8		lclkrslem1.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
9		lclkrslem1.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝐷)
10		eqid 2763	. . 3 ⊢ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
11		lclkrslem1.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		lclkrslem1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
13		lclkrslem1.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐶)
14		lclkrslem1.c	. . . . . 6 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ ( ⊥ ‘(𝐿‘𝑓)) ⊆ 𝑄)}
15	14, 10	lcfls1c 42165	. . . . 5 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
16	15	simplbi 500	. . . 4 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
17	13, 16	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17	lclkrlem1 42135	. 2 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
19		eqid 2763	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
20	1, 3, 11	dvhlmod 41739	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
21	14	lcfls1lem 42163	. . . . . . . 8 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
22	13, 21	sylib 220	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
23	22	simp1d 1156	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
24	4, 7, 8, 6, 9, 20, 12, 23	ldualvscl 39768	. . . . 5 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐹)
25	19, 4, 5, 20, 24	lkrssv 39725	. . . 4 ⊢ (𝜑 → (𝐿‘(𝑋 · 𝐺)) ⊆ (Base‘𝑈))
26	1, 3, 11	dvhlvec 41738	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec)
27	7, 8, 4, 5, 6, 9, 26, 23, 12	lkrss 39797	. . . 4 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝑋 · 𝐺)))
28	1, 3, 19, 2	dochss 41994	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘(𝑋 · 𝐺)) ⊆ (Base‘𝑈) ∧ (𝐿‘𝐺) ⊆ (𝐿‘(𝑋 · 𝐺))) → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) ⊆ ( ⊥ ‘(𝐿‘𝐺)))
29	11, 25, 27, 28	syl3anc 1392	. . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) ⊆ ( ⊥ ‘(𝐿‘𝐺)))
30	22	simp3d 1158	. . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄)
31	29, 30	sstrd 3947	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) ⊆ 𝑄)
32	14, 10	lcfls1c 42165	. 2 ⊢ ((𝑋 · 𝐺) ∈ 𝐶 ↔ ((𝑋 · 𝐺) ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) ⊆ 𝑄))
33	18, 31, 32	sylanbrc 592	1 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1099 = wceq 1561 ∈ wcel 2143 {crab 3415 ⊆ wss 3905 ‘cfv 6521 (class class class)co 7396 Basecbs 17255 Scalarcsca 17299 ·_𝑠 cvsca 17300 LSubSpclss 21005 LFnlclfn 39686 LKerclk 39714 LDualcld 39752 HLchlt 39979 LHypclh 40613 DVecHcdvh 41707 ocHcoch 41976

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7718 ax-cnex 11140 ax-resscn 11141 ax-1cn 11142 ax-icn 11143 ax-addcl 11144 ax-addrcl 11145 ax-mulcl 11146 ax-mulrcl 11147 ax-mulcom 11148 ax-addass 11149 ax-mulass 11150 ax-distr 11151 ax-i2m1 11152 ax-1ne0 11153 ax-1rid 11154 ax-rnegex 11155 ax-rrecex 11156 ax-cnre 11157 ax-pre-lttri 11158 ax-pre-lttrn 11159 ax-pre-ltadd 11160 ax-pre-mulgt0 11161 ax-riotaBAD 39582

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-tpos 8206 df-undef 8253 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11427 df-neg 11428 df-nn 12221 df-2 12290 df-3 12291 df-4 12292 df-5 12293 df-6 12294 df-n0 12492 df-z 12579 df-uz 12850 df-fz 13523 df-struct 17193 df-sets 17210 df-slot 17228 df-ndx 17240 df-base 17256 df-ress 17277 df-plusg 17309 df-mulr 17310 df-sca 17312 df-vsca 17313 df-0g 17480 df-proset 18336 df-poset 18355 df-plt 18370 df-lub 18386 df-glb 18387 df-join 18388 df-meet 18389 df-p0 18465 df-p1 18466 df-lat 18474 df-clat 18541 df-mgm 18684 df-sgrp 18763 df-mnd 18779 df-submnd 18828 df-grp 18988 df-minusg 18989 df-sbg 18990 df-subg 19175 df-cntz 19367 df-lsm 19686 df-cmn 19832 df-abl 19833 df-mgp 20197 df-rng 20209 df-ur 20242 df-ring 20295 df-oppr 20396 df-dvdsr 20416 df-unit 20417 df-invr 20447 df-dvr 20460 df-nzr 20573 df-rlreg 20754 df-domn 20755 df-drng 20790 df-lmod 20936 df-lss 21006 df-lsp 21046 df-lvec 21177 df-lfl 39687 df-lkr 39715 df-ldual 39753 df-oposet 39805 df-ol 39807 df-oml 39808 df-covers 39895 df-ats 39896 df-atl 39927 df-cvlat 39951 df-hlat 39980 df-llines 40127 df-lplanes 40128 df-lvols 40129 df-lines 40130 df-psubsp 40132 df-pmap 40133 df-padd 40425 df-lhyp 40617 df-laut 40618 df-ldil 40733 df-ltrn 40734 df-trl 40788 df-tendo 41384 df-edring 41386 df-disoa 41658 df-dvech 41708 df-dib 41768 df-dic 41802 df-dih 41858 df-doch 41977

This theorem is referenced by: lclkrs 42168

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