hdmapoc - Mathbox for Norm Megill

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Theorem hdmapoc 42228

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 41652	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑂‘𝑋) ⊆ 𝑉)
8	1, 2, 7	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝑋) ⊆ 𝑉)
9	8	sseld 3933	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (𝑂‘𝑋) → 𝑦 ∈ 𝑉))
10	9	pm4.71rd 562	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝑂‘𝑋) ↔ (𝑦 ∈ 𝑉 ∧ 𝑦 ∈ (𝑂‘𝑋))))
11		eqid 2737	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
12		eqid 2737	. . . . . . . . 9 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
13	3, 4, 1	dvhlmod 41407	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LMod)
14	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑈 ∈ LMod)
15	3, 4, 5, 11, 6	dochlss 41651	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑂‘𝑋) ∈ (LSubSp‘𝑈))
16	1, 2, 15	syl2anc 585	. . . . . . . . . 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 20950	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑦 ∈ (𝑂‘𝑋) ↔ ((LSpan‘𝑈)‘{𝑦}) ⊆ (𝑂‘𝑋)))
20		eqid 2737	. . . . . . . . 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 41628	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
23	21, 18, 22	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
24	3, 20, 4, 5, 6	dochcl 41650	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑂‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
25	1, 2, 24	syl2anc 585	. . . . . . . . . 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 41667	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (((LSpan‘𝑈)‘{𝑦}) ⊆ (𝑂‘𝑋) ↔ (𝑂‘(𝑂‘𝑋)) ⊆ (𝑂‘((LSpan‘𝑈)‘{𝑦}))))
28	18	snssd 4766	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → {𝑦} ⊆ 𝑉)
29	3, 4, 6, 5, 12, 21, 28	dochocsp 41676	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘((LSpan‘𝑈)‘{𝑦})) = (𝑂‘{𝑦}))
30	29	sseq2d 3967	. . . . . . . . 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 41650	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑦} ⊆ 𝑉) → (𝑂‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
33	21, 28, 32	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (𝑂‘{𝑦}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
34	3, 4, 5, 20, 6, 21, 31, 33	dochsscl 41665	. . . . . . . . 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 3923	. . . . . . 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 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
43	31	sselda 3934	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑉)
44		simplr 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → 𝑦 ∈ 𝑉)
45	3, 6, 4, 5, 39, 40, 41, 42, 43, 44	hdmapellkr 42211	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (((𝑆‘𝑧)‘𝑦) = 0 ↔ 𝑦 ∈ (𝑂‘{𝑧})))
46	3, 6, 4, 5, 42, 44, 43	dochsncom 41679	. . . . . . . 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 3158	. . . . . 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 2870	. 2 ⊢ (𝜑 → (𝑂‘𝑋) = {𝑦 ∣ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )})
53		df-rab 3401	. 2 ⊢ {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 } = {𝑦 ∣ (𝑦 ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 )}
54	52, 53	eqtr4di 2790	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 1542 ∈ wcel 2114 {cab 2715 ∀wral 3052 {crab 3400 ⊆ wss 3902 {csn 4581 ran crn 5626 ‘cfv 6493 Basecbs 17140 Scalarcsca 17184 0_gc0g 17363 LModclmod 20815 LSubSpclss 20886 LSpanclspn 20926 HLchlt 39647 LHypclh 40281 DVecHcdvh 41375 DIsoHcdih 41525 ocHcoch 41644 HDMapchdma 42089

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-riotaBAD 39250

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-ot 4590 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8170 df-undef 8217 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-n0 12406 df-z 12493 df-uz 12756 df-fz 13428 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-sca 17197 df-vsca 17198 df-0g 17365 df-mre 17509 df-mrc 17510 df-acs 17512 df-proset 18221 df-poset 18240 df-plt 18255 df-lub 18271 df-glb 18272 df-join 18273 df-meet 18274 df-p0 18350 df-p1 18351 df-lat 18359 df-clat 18426 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-grp 18870 df-minusg 18871 df-sbg 18872 df-subg 19057 df-cntz 19250 df-oppg 19279 df-lsm 19569 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-ring 20174 df-oppr 20277 df-dvdsr 20297 df-unit 20298 df-invr 20328 df-dvr 20341 df-nzr 20450 df-rlreg 20631 df-domn 20632 df-drng 20668 df-lmod 20817 df-lss 20887 df-lsp 20927 df-lvec 21059 df-lsatoms 39273 df-lshyp 39274 df-lcv 39316 df-lfl 39355 df-lkr 39383 df-ldual 39421 df-oposet 39473 df-ol 39475 df-oml 39476 df-covers 39563 df-ats 39564 df-atl 39595 df-cvlat 39619 df-hlat 39648 df-llines 39795 df-lplanes 39796 df-lvols 39797 df-lines 39798 df-psubsp 39800 df-pmap 39801 df-padd 40093 df-lhyp 40285 df-laut 40286 df-ldil 40401 df-ltrn 40402 df-trl 40456 df-tgrp 41040 df-tendo 41052 df-edring 41054 df-dveca 41300 df-disoa 41326 df-dvech 41376 df-dib 41436 df-dic 41470 df-dih 41526 df-doch 41645 df-djh 41692 df-lcdual 41884 df-mapd 41922 df-hvmap 42054 df-hdmap1 42090 df-hdmap 42091

This theorem is referenced by: hlhilocv 42254

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