dochcl - Mathbox for Norm Megill

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Theorem dochcl 40735

Description: Closure of subspace orthocomplement for DVecH vector space. (Contributed by NM, 9-Mar-2014.)

**Hypotheses**
Ref	Expression
dochcl.h	⊢ 𝐻 = (LHyp‘𝐾)
dochcl.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dochcl.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochcl.v	⊢ 𝑉 = (Base‘𝑈)
dochcl.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
dochcl	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ ran 𝐼)

Proof of Theorem dochcl Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2726	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2726	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
3		eqid 2726	. . 3 ⊢ (oc‘𝐾) = (oc‘𝐾)
4		dochcl.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
5		dochcl.i	. . 3 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
6		dochcl.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		dochcl.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
8		dochcl.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
9	1, 2, 3, 4, 5, 6, 7, 8	dochval 40733	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) = (𝐼‘((oc‘𝐾)‘((glb‘𝐾)‘{𝑦 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑦)}))))
10		hlop 38743	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
11	10	ad2antrr 723	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝐾 ∈ OP)
12		hlclat 38739	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
13	12	ad2antrr 723	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝐾 ∈ CLat)
14		ssrab2 4072	. . . . 5 ⊢ {𝑦 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑦)} ⊆ (Base‘𝐾)
15	1, 2	clatglbcl 18468	. . . . 5 ⊢ ((𝐾 ∈ CLat ∧ {𝑦 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑦)} ⊆ (Base‘𝐾)) → ((glb‘𝐾)‘{𝑦 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑦)}) ∈ (Base‘𝐾))
16	13, 14, 15	sylancl 585	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ((glb‘𝐾)‘{𝑦 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑦)}) ∈ (Base‘𝐾))
17	1, 3	opoccl 38575	. . . 4 ⊢ ((𝐾 ∈ OP ∧ ((glb‘𝐾)‘{𝑦 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑦)}) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((glb‘𝐾)‘{𝑦 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑦)})) ∈ (Base‘𝐾))
18	11, 16, 17	syl2anc 583	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ((oc‘𝐾)‘((glb‘𝐾)‘{𝑦 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑦)})) ∈ (Base‘𝐾))
19	1, 4, 5	dihcl 40652	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((oc‘𝐾)‘((glb‘𝐾)‘{𝑦 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑦)})) ∈ (Base‘𝐾)) → (𝐼‘((oc‘𝐾)‘((glb‘𝐾)‘{𝑦 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑦)}))) ∈ ran 𝐼)
20	18, 19	syldan 590	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝐼‘((oc‘𝐾)‘((glb‘𝐾)‘{𝑦 ∈ (Base‘𝐾) ∣ 𝑋 ⊆ (𝐼‘𝑦)}))) ∈ ran 𝐼)
21	9, 20	eqeltrd 2827	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ ran 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3426 ⊆ wss 3943 ran crn 5670 ‘cfv 6536 Basecbs 17151 occoc 17212 glbcglb 18273 CLatccla 18461 OPcops 38553 HLchlt 38731 LHypclh 39366 DVecHcdvh 40460 DIsoHcdih 40610 ocHcoch 40729

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-riotaBAD 38334

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-undef 8256 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-0g 17394 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19048 df-cntz 19231 df-lsm 19554 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-ring 20138 df-oppr 20234 df-dvdsr 20257 df-unit 20258 df-invr 20288 df-dvr 20301 df-drng 20587 df-lmod 20706 df-lss 20777 df-lsp 20817 df-lvec 20949 df-oposet 38557 df-ol 38559 df-oml 38560 df-covers 38647 df-ats 38648 df-atl 38679 df-cvlat 38703 df-hlat 38732 df-llines 38880 df-lplanes 38881 df-lvols 38882 df-lines 38883 df-psubsp 38885 df-pmap 38886 df-padd 39178 df-lhyp 39370 df-laut 39371 df-ldil 39486 df-ltrn 39487 df-trl 39541 df-tendo 40137 df-edring 40139 df-disoa 40411 df-dvech 40461 df-dib 40521 df-dic 40555 df-dih 40611 df-doch 40730

This theorem is referenced by: dochlss 40736 dochssv 40737 dochfN 40738 dochvalr3 40745 dochocss 40748 dochoccl 40751 dochord 40752 dochord2N 40753 dochord3 40754 dochn0nv 40757 dihoml4c 40758 dihoml4 40759 dochocsp 40761 dochdmj1 40772 dochnoncon 40773 djhcl 40782 dochsatshp 40833 dochsatshpb 40834 dochshpsat 40836 dochexmidlem6 40847 lcfl6 40882 lcfl8 40884 lcfl9a 40887 lclkrlem2c 40891 lclkrlem2d 40892 lclkrlem2e 40893 lclkrlem2j 40898 lclkrlem2s 40907 lclkrslem2 40920 lcfrlem23 40947 mapdordlem2 41019 hdmapoc 41313

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