lclkrslem1 - Mathbox for Norm Megill

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Theorem lclkrslem1 41834

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 41833	. . . . 5 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∧ ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑄))
16	15	simplbi 497	. . . 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 41803	. 2 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
19		eqid 2737	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
20	1, 3, 11	dvhlmod 41407	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
21	14	lcfls1lem 41831	. . . . . . . 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 39436	. . . . 5 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐹)
25	19, 4, 5, 20, 24	lkrssv 39393	. . . 4 ⊢ (𝜑 → (𝐿‘(𝑋 · 𝐺)) ⊆ (Base‘𝑈))
26	1, 3, 11	dvhlvec 41406	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec)
27	7, 8, 4, 5, 6, 9, 26, 23, 12	lkrss 39465	. . . 4 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (𝐿‘(𝑋 · 𝐺)))
28	1, 3, 19, 2	dochss 41662	. . . 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 3945	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) ⊆ 𝑄)
32	14, 10	lcfls1c 41833	. 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 3400 ⊆ wss 3902 ‘cfv 6493 (class class class)co 7360 Basecbs 17140 Scalarcsca 17184 ·_𝑠 cvsca 17185 LSubSpclss 20886 LFnlclfn 39354 LKerclk 39382 LDualcld 39420 HLchlt 39647 LHypclh 40281 DVecHcdvh 41375 ocHcoch 41644

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 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-riotaBAD 39250

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 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8170 df-undef 8217 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-n0 12406 df-z 12493 df-uz 12756 df-fz 13428 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-sca 17197 df-vsca 17198 df-0g 17365 df-proset 18221 df-poset 18240 df-plt 18255 df-lub 18271 df-glb 18272 df-join 18273 df-meet 18274 df-p0 18350 df-p1 18351 df-lat 18359 df-clat 18426 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 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 20277 df-dvdsr 20297 df-unit 20298 df-invr 20328 df-dvr 20341 df-nzr 20450 df-rlreg 20631 df-domn 20632 df-drng 20668 df-lmod 20817 df-lss 20887 df-lsp 20927 df-lvec 21059 df-lfl 39355 df-lkr 39383 df-ldual 39421 df-oposet 39473 df-ol 39475 df-oml 39476 df-covers 39563 df-ats 39564 df-atl 39595 df-cvlat 39619 df-hlat 39648 df-llines 39795 df-lplanes 39796 df-lvols 39797 df-lines 39798 df-psubsp 39800 df-pmap 39801 df-padd 40093 df-lhyp 40285 df-laut 40286 df-ldil 40401 df-ltrn 40402 df-trl 40456 df-tendo 41052 df-edring 41054 df-disoa 41326 df-dvech 41376 df-dib 41436 df-dic 41470 df-dih 41526 df-doch 41645

This theorem is referenced by: lclkrs 41836

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