lclkrs - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrs		Structured version Visualization version GIF version

Theorem lclkrs 42031

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 42025 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 42025 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 4011	. . . 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 2739	. . . 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 41602	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
12	5, 6, 7, 11	ldualvbase 39618	. . 3 ⊢ (𝜑 → (Base‘𝐷) = 𝐹)
13	2, 4, 12	3sstr4d 3970	. 2 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐷))
14		eqid 2739	. . . . . 6 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
15		eqid 2739	. . . . . 6 ⊢ (0_g‘(Scalar‘𝑈)) = (0_g‘(Scalar‘𝑈))
16		eqid 2739	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
17	14, 15, 16, 5	lfl0f 39561	. . . . 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 41853	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
21		eqidd 2740	. . . . . . . 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 39586	. . . . . . . . 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 590	. . . . . . . 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 258	. . . . . . 7 ⊢ (𝜑 → (𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})) = (Base‘𝑈))
26	25	fveq2d 6831	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}))) = ( ⊥ ‘(Base‘𝑈)))
27	26	fveq2d 6831	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
28	20, 27, 25	3eqtr4d 2784	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))) = (𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))})))
29		eqid 2739	. . . . . . . 8 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
30	8, 9, 19, 16, 29	doch1 41851	. . . . . . 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 2774	. . . . 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 20939	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑅 ∈ 𝑆) → {(0_g‘𝑈)} ⊆ 𝑅)
36	11, 33, 35	syl2anc 590	. . . . 5 ⊢ (𝜑 → {(0_g‘𝑈)} ⊆ 𝑅)
37	32, 36	eqsstrd 3949	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}))) ⊆ 𝑅)
38	3	lcfls1lem 42026	. . . 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 1350	. . 3 ⊢ (𝜑 → ((Base‘𝑈) × {(0_g‘(Scalar‘𝑈))}) ∈ 𝐶)
40	39	ne0d 4270	. 2 ⊢ (𝜑 → 𝐶 ≠ ∅)
41		eqid 2739	. . . 4 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
42		eqid 2739	. . . 4 ⊢ ( ·_𝑠 ‘𝐷) = ( ·_𝑠 ‘𝐷)
43	10	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
44	33	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑅 ∈ 𝑆)
45		simpr3 1203	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑏 ∈ 𝐶)
46		eqid 2739	. . . 4 ⊢ (+_g‘𝐷) = (+_g‘𝐷)
47		simpr2 1202	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑎 ∈ 𝐶)
48		simpr1 1201	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → 𝑥 ∈ (Base‘(Scalar‘𝐷)))
49		eqid 2739	. . . . . . . 8 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷)
50		eqid 2739	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝐷))
51	14, 41, 6, 49, 50, 11	ldualsbase 39625	. . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈)))
52	51	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → (Base‘(Scalar‘𝐷)) = (Base‘(Scalar‘𝑈)))
53	48, 52	eleqtrd 2841	. . . . 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 42029	. . . 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 42030	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐷)) ∧ 𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶)) → ((𝑥( ·_𝑠 ‘𝐷)𝑎)(+_g‘𝐷)𝑏) ∈ 𝐶)
56	55	ralrimivvva 3185	. 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 20924	. 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 1350	1 ⊢ (𝜑 → 𝐶 ∈ 𝑇)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 {crab 3391 ⊆ wss 3883 ∅c0 4261 {csn 4555 × cxp 5616 ‘cfv 6485 (class class class)co 7356 Basecbs 17170 +_gcplusg 17211 Scalarcsca 17214 ·_𝑠 cvsca 17215 0_gc0g 17393 LModclmod 20850 LSubSpclss 20921 LFnlclfn 39549 LKerclk 39577 LDualcld 39615 HLchlt 39842 LHypclh 40476 DVecHcdvh 41570 ocHcoch 41839

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-riotaBAD 39445

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-undef 8213 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-0g 17395 df-mre 17539 df-mrc 17540 df-acs 17542 df-proset 18251 df-poset 18270 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18389 df-clat 18456 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-cntz 19283 df-oppg 19312 df-lsm 19602 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-nzr 20485 df-rlreg 20666 df-domn 20667 df-drng 20703 df-lmod 20852 df-lss 20922 df-lsp 20962 df-lvec 21093 df-lsatoms 39468 df-lshyp 39469 df-lcv 39511 df-lfl 39550 df-lkr 39578 df-ldual 39616 df-oposet 39668 df-ol 39670 df-oml 39671 df-covers 39758 df-ats 39759 df-atl 39790 df-cvlat 39814 df-hlat 39843 df-llines 39990 df-lplanes 39991 df-lvols 39992 df-lines 39993 df-psubsp 39995 df-pmap 39996 df-padd 40288 df-lhyp 40480 df-laut 40481 df-ldil 40596 df-ltrn 40597 df-trl 40651 df-tgrp 41235 df-tendo 41247 df-edring 41249 df-dveca 41495 df-disoa 41521 df-dvech 41571 df-dib 41631 df-dic 41665 df-dih 41721 df-doch 41840 df-djh 41887

This theorem is referenced by: lclkrs2 42032 mapddlssN 42132

W3C validator