hdmapoc - Mathbox for Norm Megill

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Theorem hdmapoc 41910

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 41334	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑂‘𝑋) ⊆ 𝑉)
8	1, 2, 7	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑋) ⊆ 𝑉)
9	8	sseld 3936	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝑂‘𝑋) → 𝑦 ∈ 𝑉))
10	9	pm4.71rd 562	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝑂‘𝑋) ↔ (𝑦 ∈ 𝑉 ∧ 𝑦 ∈ (𝑂‘𝑋))))
11		eqid 2729	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
12		eqid 2729	. . . . . . . . 9 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
13	3, 4, 1	dvhlmod 41089	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LMod)
14	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑈 ∈ LMod)
15	3, 4, 5, 11, 6	dochlss 41333	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑂‘𝑋) ∈ (LSubSp‘𝑈))
16	1, 2, 15	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘𝑋) ∈ (LSubSp‘𝑈))
17	16	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘𝑋) ∈ (LSubSp‘𝑈))
18		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
19	5, 11, 12, 14, 17, 18	ellspsn5b 20916	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 ∈ (𝑂‘𝑋) ↔ ((LSpan‘𝑈)‘{𝑦}) ⊆ (𝑂‘𝑋)))
20		eqid 2729	. . . . . . . . 9 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
21	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
22	3, 4, 5, 12, 20	dihlsprn 41310	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
23	21, 18, 22	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
24	3, 20, 4, 5, 6	dochcl 41332	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑂‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
25	1, 2, 24	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
26	25	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
27	3, 20, 6, 21, 23, 26	dochord 41349	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((LSpan‘𝑈)‘{𝑦}) ⊆ (𝑂‘𝑋) ↔ (𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘((LSpan‘𝑈)‘{𝑦}))))
28	18	snssd 4763	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → {𝑦} ⊆ 𝑉)
29	3, 4, 6, 5, 12, 21, 28	dochocsp 41358	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘((LSpan‘𝑈)‘{𝑦})) = (𝑂‘{𝑦}))
30	29	sseq2d 3970	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘((LSpan‘𝑈)‘{𝑦})) ↔ (𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘{𝑦})))
31	2	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑋 ⊆ 𝑉)
32	3, 20, 4, 5, 6	dochcl 41332	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑦} ⊆ 𝑉) → (𝑂‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	21, 28, 32	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
34	3, 4, 5, 20, 6, 21, 31, 33	dochsscl 41347	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑋 ⊆ (𝑂‘{𝑦}) ↔ (𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘{𝑦})))
35	30, 34	bitr4d 282	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘((LSpan‘𝑈)‘{𝑦})) ↔ 𝑋 ⊆ (𝑂‘{𝑦})))
36	19, 27, 35	3bitrd 305	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 ∈ (𝑂‘𝑋) ↔ 𝑋 ⊆ (𝑂‘{𝑦})))
37		dfss3 3926	. . . . . . 7 ⊢ (𝑋 ⊆ (𝑂‘{𝑦}) ↔ ∀𝑧 ∈ 𝑋 𝑧 ∈ (𝑂‘{𝑦}))
38	36, 37	bitrdi 287	. . . . . 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 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
43	31	sselda 3937	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑉)
44		simplr 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → 𝑦 ∈ 𝑉)
45	3, 6, 4, 5, 39, 40, 41, 42, 43, 44	hdmapellkr 41893	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (((𝑆‘𝑧)‘𝑦) = 0 ↔ 𝑦 ∈ (𝑂‘{𝑧})))
46	3, 6, 4, 5, 42, 44, 43	dochsncom 41361	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (𝑦 ∈ (𝑂‘{𝑧}) ↔ 𝑧 ∈ (𝑂‘{𝑦})))
47	45, 46	bitrd 279	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (((𝑆‘𝑧)‘𝑦) = 0 ↔ 𝑧 ∈ (𝑂‘{𝑦})))
48	47	ralbidva 3150	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 ↔ ∀𝑧 ∈ 𝑋 𝑧 ∈ (𝑂‘{𝑦})))
49	38, 48	bitr4d 282	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 ∈ (𝑂‘𝑋) ↔ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 ))
50	49	pm5.32da 579	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝑉 ∧ 𝑦 ∈ (𝑂‘𝑋)) ↔ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )))
51	10, 50	bitrd 279	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝑂‘𝑋) ↔ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )))
52	51	eqabdv 2861	. 2 ⊢ (𝜑 → (𝑂‘𝑋) = {𝑦 ∣ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )})
53		df-rab 3397	. 2 ⊢ {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 } = {𝑦 ∣ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )}
54	52, 53	eqtr4di 2782	1 ⊢ (𝜑 → (𝑂‘𝑋) = {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 })

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {cab 2707 ∀wral 3044 {crab 3396 ⊆ wss 3905 {csn 4579 ran crn 5624 ‘cfv 6486 Basecbs 17138 Scalarcsca 17182 0_gc0g 17361 LModclmod 20781 LSubSpclss 20852 LSpanclspn 20892 HLchlt 39328 LHypclh 39963 DVecHcdvh 41057 DIsoHcdih 41207 ocHcoch 41326 HDMapchdma 41771

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-riotaBAD 38931

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-ot 4588 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 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 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 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 12754 df-fz 13429 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-0g 17363 df-mre 17506 df-mrc 17507 df-acs 17509 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 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-grp 18833 df-minusg 18834 df-sbg 18835 df-subg 19020 df-cntz 19214 df-oppg 19243 df-lsm 19533 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-oppr 20240 df-dvdsr 20260 df-unit 20261 df-invr 20291 df-dvr 20304 df-nzr 20416 df-rlreg 20597 df-domn 20598 df-drng 20634 df-lmod 20783 df-lss 20853 df-lsp 20893 df-lvec 21025 df-lsatoms 38954 df-lshyp 38955 df-lcv 38997 df-lfl 39036 df-lkr 39064 df-ldual 39102 df-oposet 39154 df-ol 39156 df-oml 39157 df-covers 39244 df-ats 39245 df-atl 39276 df-cvlat 39300 df-hlat 39329 df-llines 39477 df-lplanes 39478 df-lvols 39479 df-lines 39480 df-psubsp 39482 df-pmap 39483 df-padd 39775 df-lhyp 39967 df-laut 39968 df-ldil 40083 df-ltrn 40084 df-trl 40138 df-tgrp 40722 df-tendo 40734 df-edring 40736 df-dveca 40982 df-disoa 41008 df-dvech 41058 df-dib 41118 df-dic 41152 df-dih 41208 df-doch 41327 df-djh 41374 df-lcdual 41566 df-mapd 41604 df-hvmap 41736 df-hdmap1 41772 df-hdmap 41773

This theorem is referenced by: hlhilocv 41936

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