hdmapoc - Mathbox for Norm Megill

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Theorem hdmapoc 39066

Description: Express our constructed orthocomplement (polarity) in terms of the Hilbert space definition of orthocomplement. Lines 24 and 25 in [Holland95] p. 14. (Contributed by NM, 17-Jun-2015.)

**Hypotheses**
Ref	Expression
hdmapoc.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmapoc.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapoc.v	⊢ 𝑉 = (Base‘𝑈)
hdmapoc.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmapoc.z	⊢ 0 = (0_g‘𝑅)
hdmapoc.o	⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
hdmapoc.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapoc.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmapoc.x	⊢ (𝜑 → 𝑋 ⊆ 𝑉)

**Assertion**
Ref	Expression
hdmapoc	⊢ (𝜑 → (𝑂‘𝑋) = {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 })

Distinct variable groups: 𝑦,𝑧,𝑂 𝑧,𝑉 𝑦,𝑋,𝑧 𝜑,𝑦,𝑧 Allowed substitution hints: 𝑅(𝑦,𝑧) 𝑆(𝑦,𝑧) 𝑈(𝑦,𝑧) 𝐻(𝑦,𝑧) 𝐾(𝑦,𝑧) 𝑉(𝑦) 𝑊(𝑦,𝑧) 0 (𝑦,𝑧)

**Proof of Theorem hdmapoc**
Step	Hyp	Ref	Expression
1		hdmapoc.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		hdmapoc.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑉)
3		hdmapoc.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
4		hdmapoc.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		hdmapoc.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈)
6		hdmapoc.o	. . . . . . . 8 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
7	3, 4, 5, 6	dochssv 38490	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑂‘𝑋) ⊆ 𝑉)
8	1, 2, 7	syl2anc 586	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑋) ⊆ 𝑉)
9	8	sseld 3965	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝑂‘𝑋) → 𝑦 ∈ 𝑉))
10	9	pm4.71rd 565	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝑂‘𝑋) ↔ (𝑦 ∈ 𝑉 ∧ 𝑦 ∈ (𝑂‘𝑋))))
11		eqid 2821	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
12		eqid 2821	. . . . . . . . 9 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
13	3, 4, 1	dvhlmod 38245	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LMod)
14	13	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑈 ∈ LMod)
15	3, 4, 5, 11, 6	dochlss 38489	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑂‘𝑋) ∈ (LSubSp‘𝑈))
16	1, 2, 15	syl2anc 586	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘𝑋) ∈ (LSubSp‘𝑈))
17	16	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘𝑋) ∈ (LSubSp‘𝑈))
18		simpr 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
19	5, 11, 12, 14, 17, 18	lspsnel5 19766	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 ∈ (𝑂‘𝑋) ↔ ((LSpan‘𝑈)‘{𝑦}) ⊆ (𝑂‘𝑋)))
20		eqid 2821	. . . . . . . . 9 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
21	1	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
22	3, 4, 5, 12, 20	dihlsprn 38466	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
23	21, 18, 22	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
24	3, 20, 4, 5, 6	dochcl 38488	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑂‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
25	1, 2, 24	syl2anc 586	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
26	25	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
27	3, 20, 6, 21, 23, 26	dochord 38505	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((LSpan‘𝑈)‘{𝑦}) ⊆ (𝑂‘𝑋) ↔ (𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘((LSpan‘𝑈)‘{𝑦}))))
28	18	snssd 4741	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → {𝑦} ⊆ 𝑉)
29	3, 4, 6, 5, 12, 21, 28	dochocsp 38514	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘((LSpan‘𝑈)‘{𝑦})) = (𝑂‘{𝑦}))
30	29	sseq2d 3998	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘((LSpan‘𝑈)‘{𝑦})) ↔ (𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘{𝑦})))
31	2	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑋 ⊆ 𝑉)
32	3, 20, 4, 5, 6	dochcl 38488	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑦} ⊆ 𝑉) → (𝑂‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	21, 28, 32	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
34	3, 4, 5, 20, 6, 21, 31, 33	dochsscl 38503	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑋 ⊆ (𝑂‘{𝑦}) ↔ (𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘{𝑦})))
35	30, 34	bitr4d 284	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘((LSpan‘𝑈)‘{𝑦})) ↔ 𝑋 ⊆ (𝑂‘{𝑦})))
36	19, 27, 35	3bitrd 307	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 ∈ (𝑂‘𝑋) ↔ 𝑋 ⊆ (𝑂‘{𝑦})))
37		dfss3 3955	. . . . . . 7 ⊢ (𝑋 ⊆ (𝑂‘{𝑦}) ↔ ∀𝑧 ∈ 𝑋 𝑧 ∈ (𝑂‘{𝑦}))
38	36, 37	syl6bb 289	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 ∈ (𝑂‘𝑋) ↔ ∀𝑧 ∈ 𝑋 𝑧 ∈ (𝑂‘{𝑦})))
39		hdmapoc.r	. . . . . . . . 9 ⊢ 𝑅 = (Scalar‘𝑈)
40		hdmapoc.z	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑅)
41		hdmapoc.s	. . . . . . . . 9 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
42	1	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
43	31	sselda 3966	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑉)
44		simplr 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → 𝑦 ∈ 𝑉)
45	3, 6, 4, 5, 39, 40, 41, 42, 43, 44	hdmapellkr 39049	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (((𝑆‘𝑧)‘𝑦) = 0 ↔ 𝑦 ∈ (𝑂‘{𝑧})))
46	3, 6, 4, 5, 42, 44, 43	dochsncom 38517	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (𝑦 ∈ (𝑂‘{𝑧}) ↔ 𝑧 ∈ (𝑂‘{𝑦})))
47	45, 46	bitrd 281	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (((𝑆‘𝑧)‘𝑦) = 0 ↔ 𝑧 ∈ (𝑂‘{𝑦})))
48	47	ralbidva 3196	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 ↔ ∀𝑧 ∈ 𝑋 𝑧 ∈ (𝑂‘{𝑦})))
49	38, 48	bitr4d 284	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 ∈ (𝑂‘𝑋) ↔ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 ))
50	49	pm5.32da 581	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝑉 ∧ 𝑦 ∈ (𝑂‘𝑋)) ↔ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )))
51	10, 50	bitrd 281	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝑂‘𝑋) ↔ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )))
52	51	abbi2dv 2950	. 2 ⊢ (𝜑 → (𝑂‘𝑋) = {𝑦 ∣ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )})
53		df-rab 3147	. 2 ⊢ {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 } = {𝑦 ∣ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )}
54	52, 53	syl6eqr 2874	1 ⊢ (𝜑 → (𝑂‘𝑋) = {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 })

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 ∀wral 3138 {crab 3142 ⊆ wss 3935 {csn 4566 ran crn 5555 ‘cfv 6354 Basecbs 16482 Scalarcsca 16567 0_gc0g 16712 LModclmod 19633 LSubSpclss 19702 LSpanclspn 19742 HLchlt 36485 LHypclh 37119 DVecHcdvh 38213 DIsoHcdih 38363 ocHcoch 38482 HDMapchdma 38927

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-riotaBAD 36088

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-ot 4575 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-tpos 7891 df-undef 7938 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-0g 16714 df-mre 16856 df-mrc 16857 df-acs 16859 df-proset 17537 df-poset 17555 df-plt 17567 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-p0 17648 df-p1 17649 df-lat 17655 df-clat 17717 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-subg 18275 df-cntz 18446 df-oppg 18473 df-lsm 18760 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-dvr 19432 df-drng 19503 df-lmod 19635 df-lss 19703 df-lsp 19743 df-lvec 19874 df-lsatoms 36111 df-lshyp 36112 df-lcv 36154 df-lfl 36193 df-lkr 36221 df-ldual 36259 df-oposet 36311 df-ol 36313 df-oml 36314 df-covers 36401 df-ats 36402 df-atl 36433 df-cvlat 36457 df-hlat 36486 df-llines 36633 df-lplanes 36634 df-lvols 36635 df-lines 36636 df-psubsp 36638 df-pmap 36639 df-padd 36931 df-lhyp 37123 df-laut 37124 df-ldil 37239 df-ltrn 37240 df-trl 37294 df-tgrp 37878 df-tendo 37890 df-edring 37892 df-dveca 38138 df-disoa 38164 df-dvech 38214 df-dib 38274 df-dic 38308 df-dih 38364 df-doch 38483 df-djh 38530 df-lcdual 38722 df-mapd 38760 df-hvmap 38892 df-hdmap1 38928 df-hdmap 38929

This theorem is referenced by: hlhilocv 39092

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