lclkrslem1 - Mathbox for Norm Megill

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Theorem lclkrslem1 41974

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 2737	. . 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 41973	. . . . 5 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
16	15	simplbi 496	. . . 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 41943	. 2 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
19		eqid 2737	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
20	1, 3, 11	dvhlmod 41547	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
21	14	lcfls1lem 41971	. . . . . . . 8 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
22	13, 21	sylib 218	. . . . . . 7 ⊢ (𝜑 → (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
23	22	simp1d 1143	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
24	4, 7, 8, 6, 9, 20, 12, 23	ldualvscl 39576	. . . . 5 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐹)
25	19, 4, 5, 20, 24	lkrssv 39533	. . . 4 ⊢ (𝜑 → (𝐿‘(𝑋 · 𝐺)) ⊆ (Base‘𝑈))
26	1, 3, 11	dvhlvec 41546	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec)
27	7, 8, 4, 5, 6, 9, 26, 23, 12	lkrss 39605	. . . 4 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝑋 · 𝐺)))
28	1, 3, 19, 2	dochss 41802	. . . 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 1374	. . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) ⊆ ( ⊥ ‘(𝐿‘𝐺)))
30	22	simp3d 1145	. . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄)
31	29, 30	sstrd 3933	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) ⊆ 𝑄)
32	14, 10	lcfls1c 41973	. 2 ⊢ ((𝑋 · 𝐺) ∈ 𝐶 ↔ ((𝑋 · 𝐺) ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) ⊆ 𝑄))
33	18, 31, 32	sylanbrc 584	1 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {crab 3390 ⊆ wss 3890 ‘cfv 6490 (class class class)co 7358 Basecbs 17137 Scalarcsca 17181 ·_𝑠 cvsca 17182 LSubSpclss 20884 LFnlclfn 39494 LKerclk 39522 LDualcld 39560 HLchlt 39787 LHypclh 40421 DVecHcdvh 41515 ocHcoch 41784

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-riotaBAD 39390

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 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 8167 df-undef 8214 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-er 8634 df-map 8766 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-n0 12403 df-z 12490 df-uz 12753 df-fz 13425 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17138 df-ress 17159 df-plusg 17191 df-mulr 17192 df-sca 17194 df-vsca 17195 df-0g 17362 df-proset 18218 df-poset 18237 df-plt 18252 df-lub 18268 df-glb 18269 df-join 18270 df-meet 18271 df-p0 18347 df-p1 18348 df-lat 18356 df-clat 18423 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18710 df-grp 18870 df-minusg 18871 df-sbg 18872 df-subg 19057 df-cntz 19250 df-lsm 19569 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-ring 20174 df-oppr 20275 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-nzr 20448 df-rlreg 20629 df-domn 20630 df-drng 20666 df-lmod 20815 df-lss 20885 df-lsp 20925 df-lvec 21057 df-lfl 39495 df-lkr 39523 df-ldual 39561 df-oposet 39613 df-ol 39615 df-oml 39616 df-covers 39703 df-ats 39704 df-atl 39735 df-cvlat 39759 df-hlat 39788 df-llines 39935 df-lplanes 39936 df-lvols 39937 df-lines 39938 df-psubsp 39940 df-pmap 39941 df-padd 40233 df-lhyp 40425 df-laut 40426 df-ldil 40541 df-ltrn 40542 df-trl 40596 df-tendo 41192 df-edring 41194 df-disoa 41466 df-dvech 41516 df-dib 41576 df-dic 41610 df-dih 41666 df-doch 41785

This theorem is referenced by: lclkrs 41976

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