hdmapoc - Mathbox for Norm Megill

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Theorem hdmapoc 42673

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 42097	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑂‘𝑋) ⊆ 𝑉)
8	1, 2, 7	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑋) ⊆ 𝑉)
9	8	sseld 3935	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝑂‘𝑋) → 𝑦 ∈ 𝑉))
10	9	pm4.71rd 571	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝑂‘𝑋) ↔ (𝑦 ∈ 𝑉 ∧ 𝑦 ∈ (𝑂‘𝑋))))
11		eqid 2761	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
12		eqid 2761	. . . . . . . . 9 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
13	3, 4, 1	dvhlmod 41852	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LMod)
14	13	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑈 ∈ LMod)
15	3, 4, 5, 11, 6	dochlss 42096	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑂‘𝑋) ∈ (LSubSp‘𝑈))
16	1, 2, 15	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘𝑋) ∈ (LSubSp‘𝑈))
17	16	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘𝑋) ∈ (LSubSp‘𝑈))
18		simpr 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
19	5, 11, 12, 14, 17, 18	ellspsn5b 21095	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 ∈ (𝑂‘𝑋) ↔ ((LSpan‘𝑈)‘{𝑦}) ⊆ (𝑂‘𝑋)))
20		eqid 2761	. . . . . . . . 9 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
21	1	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
22	3, 4, 5, 12, 20	dihlsprn 42073	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
23	21, 18, 22	syl2anc 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
24	3, 20, 4, 5, 6	dochcl 42095	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑂‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
25	1, 2, 24	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
26	25	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
27	3, 20, 6, 21, 23, 26	dochord 42112	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((LSpan‘𝑈)‘{𝑦}) ⊆ (𝑂‘𝑋) ↔ (𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘((LSpan‘𝑈)‘{𝑦}))))
28	18	snssd 4751	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → {𝑦} ⊆ 𝑉)
29	3, 4, 6, 5, 12, 21, 28	dochocsp 42121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘((LSpan‘𝑈)‘{𝑦})) = (𝑂‘{𝑦}))
30	29	sseq2d 3968	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘((LSpan‘𝑈)‘{𝑦})) ↔ (𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘{𝑦})))
31	2	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑋 ⊆ 𝑉)
32	3, 20, 4, 5, 6	dochcl 42095	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑦} ⊆ 𝑉) → (𝑂‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	21, 28, 32	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
34	3, 4, 5, 20, 6, 21, 31, 33	dochsscl 42110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑋 ⊆ (𝑂‘{𝑦}) ↔ (𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘{𝑦})))
35	30, 34	bitr4d 285	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘((LSpan‘𝑈)‘{𝑦})) ↔ 𝑋 ⊆ (𝑂‘{𝑦})))
36	19, 27, 35	3bitrd 308	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 ∈ (𝑂‘𝑋) ↔ 𝑋 ⊆ (𝑂‘{𝑦})))
37		dfss3 3925	. . . . . . 7 ⊢ (𝑋 ⊆ (𝑂‘{𝑦}) ↔ ∀𝑧 ∈ 𝑋 𝑧 ∈ (𝑂‘{𝑦}))
38	36, 37	bitrdi 290	. . . . . 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 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
43	31	sselda 3936	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑉)
44		simplr 780	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → 𝑦 ∈ 𝑉)
45	3, 6, 4, 5, 39, 40, 41, 42, 43, 44	hdmapellkr 42656	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (((𝑆‘𝑧)‘𝑦) = 0 ↔ 𝑦 ∈ (𝑂‘{𝑧})))
46	3, 6, 4, 5, 42, 44, 43	dochsncom 42124	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (𝑦 ∈ (𝑂‘{𝑧}) ↔ 𝑧 ∈ (𝑂‘{𝑦})))
47	45, 46	bitrd 282	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (((𝑆‘𝑧)‘𝑦) = 0 ↔ 𝑧 ∈ (𝑂‘{𝑦})))
48	47	ralbidva 3184	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 ↔ ∀𝑧 ∈ 𝑋 𝑧 ∈ (𝑂‘{𝑦})))
49	38, 48	bitr4d 285	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 ∈ (𝑂‘𝑋) ↔ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 ))
50	49	pm5.32da 589	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝑉 ∧ 𝑦 ∈ (𝑂‘𝑋)) ↔ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )))
51	10, 50	bitrd 282	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝑂‘𝑋) ↔ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )))
52	51	eqabdv 2894	. 2 ⊢ (𝜑 → (𝑂‘𝑋) = {𝑦 ∣ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )})
53		df-rab 3415	. 2 ⊢ {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 } = {𝑦 ∣ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )}
54	52, 53	eqtr4di 2814	1 ⊢ (𝜑 → (𝑂‘𝑋) = {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 })

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1568 ∈ wcel 2141 {cab 2739 ∀wral 3077 {crab 3414 ⊆ wss 3904 {csn 4588 ran crn 5662 ‘cfv 6536 Basecbs 17268 Scalarcsca 17312 0_gc0g 17491 LModclmod 20960 LSubSpclss 21031 LSpanclspn 21071 HLchlt 40092 LHypclh 40726 DVecHcdvh 41820 DIsoHcdih 41970 ocHcoch 42089 HDMapchdma 42534

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-riotaBAD 39695

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-ot 4597 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7862 df-1st 7985 df-2nd 7986 df-tpos 8221 df-undef 8268 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-n0 12504 df-z 12591 df-uz 12862 df-fz 13535 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-sca 17325 df-vsca 17326 df-0g 17493 df-mre 17637 df-mrc 17638 df-acs 17640 df-proset 18349 df-poset 18368 df-plt 18383 df-lub 18399 df-glb 18400 df-join 18401 df-meet 18402 df-p0 18478 df-p1 18479 df-lat 18487 df-clat 18554 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-submnd 18841 df-grp 19002 df-minusg 19003 df-sbg 19004 df-subg 19188 df-cntz 19386 df-oppg 19415 df-lsm 19705 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-oppr 20418 df-dvdsr 20438 df-unit 20439 df-invr 20469 df-dvr 20482 df-nzr 20595 df-rlreg 20778 df-domn 20779 df-drng 20814 df-lmod 20962 df-lss 21032 df-lsp 21072 df-lvec 21203 df-lsatoms 39718 df-lshyp 39719 df-lcv 39761 df-lfl 39800 df-lkr 39828 df-ldual 39866 df-oposet 39918 df-ol 39920 df-oml 39921 df-covers 40008 df-ats 40009 df-atl 40040 df-cvlat 40064 df-hlat 40093 df-llines 40240 df-lplanes 40241 df-lvols 40242 df-lines 40243 df-psubsp 40245 df-pmap 40246 df-padd 40538 df-lhyp 40730 df-laut 40731 df-ldil 40846 df-ltrn 40847 df-trl 40901 df-tgrp 41485 df-tendo 41497 df-edring 41499 df-dveca 41745 df-disoa 41771 df-dvech 41821 df-dib 41881 df-dic 41915 df-dih 41971 df-doch 42090 df-djh 42137 df-lcdual 42329 df-mapd 42367 df-hvmap 42499 df-hdmap1 42535 df-hdmap 42536

This theorem is referenced by: hlhilocv 42699

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