hdmapoc - Mathbox for Norm Megill

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Theorem hdmapoc 42908

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 42332	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑂‘𝑋) ⊆ 𝑉)
8	1, 2, 7	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑋) ⊆ 𝑉)
9	8	sseld 3929	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝑂‘𝑋) → 𝑦 ∈ 𝑉))
10	9	pm4.71rd 572	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝑂‘𝑋) ↔ (𝑦 ∈ 𝑉 ∧ 𝑦 ∈ (𝑂‘𝑋))))
11		eqid 2760	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
12		eqid 2760	. . . . . . . . 9 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
13	3, 4, 1	dvhlmod 42087	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LMod)
14	13	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑈 ∈ LMod)
15	3, 4, 5, 11, 6	dochlss 42331	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑂‘𝑋) ∈ (LSubSp‘𝑈))
16	1, 2, 15	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘𝑋) ∈ (LSubSp‘𝑈))
17	16	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘𝑋) ∈ (LSubSp‘𝑈))
18		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
19	5, 11, 12, 14, 17, 18	ellspsn5b 21232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 ∈ (𝑂‘𝑋) ↔ ((LSpan‘𝑈)‘{𝑦}) ⊆ (𝑂‘𝑋)))
20		eqid 2760	. . . . . . . . 9 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
21	1	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
22	3, 4, 5, 12, 20	dihlsprn 42308	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
23	21, 18, 22	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
24	3, 20, 4, 5, 6	dochcl 42330	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑂‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
25	1, 2, 24	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
26	25	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
27	3, 20, 6, 21, 23, 26	dochord 42347	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((LSpan‘𝑈)‘{𝑦}) ⊆ (𝑂‘𝑋) ↔ (𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘((LSpan‘𝑈)‘{𝑦}))))
28	18	snssd 4746	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → {𝑦} ⊆ 𝑉)
29	3, 4, 6, 5, 12, 21, 28	dochocsp 42356	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘((LSpan‘𝑈)‘{𝑦})) = (𝑂‘{𝑦}))
30	29	sseq2d 3962	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘((LSpan‘𝑈)‘{𝑦})) ↔ (𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘{𝑦})))
31	2	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑋 ⊆ 𝑉)
32	3, 20, 4, 5, 6	dochcl 42330	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑦} ⊆ 𝑉) → (𝑂‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	21, 28, 32	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
34	3, 4, 5, 20, 6, 21, 31, 33	dochsscl 42345	. . . . . . . . 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 3919	. . . . . . 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 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
43	31	sselda 3930	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑉)
44		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → 𝑦 ∈ 𝑉)
45	3, 6, 4, 5, 39, 40, 41, 42, 43, 44	hdmapellkr 42891	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (((𝑆‘𝑧)‘𝑦) = 0 ↔ 𝑦 ∈ (𝑂‘{𝑧})))
46	3, 6, 4, 5, 42, 44, 43	dochsncom 42359	. . . . . . . 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 3183	. . . . . 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 590	. . . 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 2893	. 2 ⊢ (𝜑 → (𝑂‘𝑋) = {𝑦 ∣ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )})
53		df-rab 3413	. 2 ⊢ {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 } = {𝑦 ∣ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )}
54	52, 53	eqtr4di 2813	1 ⊢ (𝜑 → (𝑂‘𝑋) = {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 })

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {cab 2738 ∀wral 3076 {crab 3412 ⊆ wss 3898 {csn 4583 ran crn 5648 ‘cfv 6527 Basecbs 17349 Scalarcsca 17393 0_gc0g 17572 LModclmod 21097 LSubSpclss 21168 LSpanclspn 21208 HLchlt 40327 LHypclh 40961 DVecHcdvh 42055 DIsoHcdih 42205 ocHcoch 42324 HDMapchdma 42769

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11228 ax-resscn 11229 ax-1cn 11230 ax-icn 11231 ax-addcl 11232 ax-addrcl 11233 ax-mulcl 11234 ax-mulrcl 11235 ax-mulcom 11236 ax-addass 11237 ax-mulass 11238 ax-distr 11239 ax-i2m1 11240 ax-1ne0 11241 ax-1rid 11242 ax-rnegex 11243 ax-rrecex 11244 ax-cnre 11245 ax-pre-lttri 11246 ax-pre-lttrn 11247 ax-pre-ltadd 11248 ax-pre-mulgt0 11249 ax-riotaBAD 39930

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-ot 4592 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-tpos 8221 df-undef 8268 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-pnf 11317 df-mnf 11318 df-xr 11319 df-ltxr 11320 df-le 11321 df-sub 11515 df-neg 11516 df-nn 12306 df-2 12375 df-3 12376 df-4 12377 df-5 12378 df-6 12379 df-n0 12577 df-z 12664 df-uz 12936 df-fz 13610 df-struct 17287 df-sets 17304 df-slot 17322 df-ndx 17334 df-base 17350 df-ress 17371 df-plusg 17403 df-mulr 17404 df-sca 17406 df-vsca 17407 df-0g 17574 df-mre 17718 df-mrc 17719 df-acs 17721 df-proset 18430 df-poset 18449 df-plt 18464 df-lub 18480 df-glb 18481 df-join 18482 df-meet 18483 df-p0 18559 df-p1 18560 df-lat 18568 df-clat 18635 df-mgm 18778 df-sgrp 18870 df-mnd 18886 df-submnd 18941 df-grp 19109 df-minusg 19110 df-sbg 19111 df-subg 19295 df-cntz 19493 df-oppg 19522 df-lsm 19812 df-cmn 19958 df-abl 19959 df-mgp 20323 df-rng 20337 df-ur 20370 df-ring 20423 df-oppr 20529 df-dvdsr 20549 df-unit 20550 df-invr 20580 df-dvr 20593 df-nzr 20725 df-rlreg 20908 df-domn 20909 df-drng 20944 df-lmod 21099 df-lss 21169 df-lsp 21209 df-lvec 21340 df-lsatoms 39953 df-lshyp 39954 df-lcv 39996 df-lfl 40035 df-lkr 40063 df-ldual 40101 df-oposet 40153 df-ol 40155 df-oml 40156 df-covers 40243 df-ats 40244 df-atl 40275 df-cvlat 40299 df-hlat 40328 df-llines 40475 df-lplanes 40476 df-lvols 40477 df-lines 40478 df-psubsp 40480 df-pmap 40481 df-padd 40773 df-lhyp 40965 df-laut 40966 df-ldil 41081 df-ltrn 41082 df-trl 41136 df-tgrp 41720 df-tendo 41732 df-edring 41734 df-dveca 41980 df-disoa 42006 df-dvech 42056 df-dib 42116 df-dic 42150 df-dih 42206 df-doch 42325 df-djh 42372 df-lcdual 42564 df-mapd 42602 df-hvmap 42734 df-hdmap1 42770 df-hdmap 42771

This theorem is used by: hlhilocv 42934

W3C validator