dochsncom - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dochsncom		Structured version Visualization version GIF version

Theorem dochsncom 41381

Description: Swap vectors in an orthocomplement of a singleton. (Contributed by NM, 17-Jun-2015.)

**Hypotheses**
Ref	Expression
dochsncom.h	⊢ 𝐻 = (LHyp‘𝐾)
dochsncom.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
dochsncom.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochsncom.v	⊢ 𝑉 = (Base‘𝑈)
dochsncom.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dochsncom.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
dochsncom.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)

**Assertion**
Ref	Expression
dochsncom	⊢ (𝜑 → (𝑋 ∈ ( ⊥ ‘{𝑌}) ↔ 𝑌 ∈ ( ⊥ ‘{𝑋})))

**Proof of Theorem dochsncom**
Step	Hyp	Ref	Expression
1		dochsncom.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		eqid 2729	. . . 4 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
3		dochsncom.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
4		dochsncom.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
5		dochsncom.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
6		dochsncom.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		dochsncom.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
8		eqid 2729	. . . . . 6 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
9	1, 6, 7, 8, 2	dihlsprn 41330	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
10	4, 5, 9	syl2anc 584	. . . 4 ⊢ (𝜑 → ((LSpan‘𝑈)‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
11		dochsncom.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
12	1, 6, 7, 8, 2	dihlsprn 41330	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
13	4, 11, 12	syl2anc 584	. . . 4 ⊢ (𝜑 → ((LSpan‘𝑈)‘{𝑌}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
14	1, 2, 3, 4, 10, 13	dochord3 41371	. . 3 ⊢ (𝜑 → (((LSpan‘𝑈)‘{𝑋}) ⊆ ( ⊥ ‘((LSpan‘𝑈)‘{𝑌})) ↔ ((LSpan‘𝑈)‘{𝑌}) ⊆ ( ⊥ ‘((LSpan‘𝑈)‘{𝑋}))))
15	11	snssd 4763	. . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
16	1, 6, 3, 7, 8, 4, 15	dochocsp 41378	. . . 4 ⊢ (𝜑 → ( ⊥ ‘((LSpan‘𝑈)‘{𝑌})) = ( ⊥ ‘{𝑌}))
17	16	sseq2d 3970	. . 3 ⊢ (𝜑 → (((LSpan‘𝑈)‘{𝑋}) ⊆ ( ⊥ ‘((LSpan‘𝑈)‘{𝑌})) ↔ ((LSpan‘𝑈)‘{𝑋}) ⊆ ( ⊥ ‘{𝑌})))
18	5	snssd 4763	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
19	1, 6, 3, 7, 8, 4, 18	dochocsp 41378	. . . 4 ⊢ (𝜑 → ( ⊥ ‘((LSpan‘𝑈)‘{𝑋})) = ( ⊥ ‘{𝑋}))
20	19	sseq2d 3970	. . 3 ⊢ (𝜑 → (((LSpan‘𝑈)‘{𝑌}) ⊆ ( ⊥ ‘((LSpan‘𝑈)‘{𝑋})) ↔ ((LSpan‘𝑈)‘{𝑌}) ⊆ ( ⊥ ‘{𝑋})))
21	14, 17, 20	3bitr3d 309	. 2 ⊢ (𝜑 → (((LSpan‘𝑈)‘{𝑋}) ⊆ ( ⊥ ‘{𝑌}) ↔ ((LSpan‘𝑈)‘{𝑌}) ⊆ ( ⊥ ‘{𝑋})))
22		eqid 2729	. . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
23	1, 6, 4	dvhlmod 41109	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
24	1, 6, 7, 22, 3	dochlss 41353	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑌} ⊆ 𝑉) → ( ⊥ ‘{𝑌}) ∈ (LSubSp‘𝑈))
25	4, 15, 24	syl2anc 584	. . 3 ⊢ (𝜑 → ( ⊥ ‘{𝑌}) ∈ (LSubSp‘𝑈))
26	7, 22, 8, 23, 25, 5	ellspsn5b 20917	. 2 ⊢ (𝜑 → (𝑋 ∈ ( ⊥ ‘{𝑌}) ↔ ((LSpan‘𝑈)‘{𝑋}) ⊆ ( ⊥ ‘{𝑌})))
27	1, 6, 7, 22, 3	dochlss 41353	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ (LSubSp‘𝑈))
28	4, 18, 27	syl2anc 584	. . 3 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ (LSubSp‘𝑈))
29	7, 22, 8, 23, 28, 11	ellspsn5b 20917	. 2 ⊢ (𝜑 → (𝑌 ∈ ( ⊥ ‘{𝑋}) ↔ ((LSpan‘𝑈)‘{𝑌}) ⊆ ( ⊥ ‘{𝑋})))
30	21, 26, 29	3bitr4d 311	1 ⊢ (𝜑 → (𝑋 ∈ ( ⊥ ‘{𝑌}) ↔ 𝑌 ∈ ( ⊥ ‘{𝑋})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ⊆ wss 3905 {csn 4579 ran crn 5624 ‘cfv 6486 Basecbs 17139 LSubSpclss 20853 LSpanclspn 20893 HLchlt 39348 LHypclh 39983 DVecHcdvh 41077 DIsoHcdih 41227 ocHcoch 41346

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 38951

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-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-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-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 11368 df-neg 11369 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-n0 12404 df-z 12491 df-uz 12755 df-fz 13430 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17140 df-ress 17161 df-plusg 17193 df-mulr 17194 df-sca 17196 df-vsca 17197 df-0g 17364 df-proset 18219 df-poset 18238 df-plt 18253 df-lub 18269 df-glb 18270 df-join 18271 df-meet 18272 df-p0 18348 df-p1 18349 df-lat 18357 df-clat 18424 df-mgm 18533 df-sgrp 18612 df-mnd 18628 df-submnd 18677 df-grp 18834 df-minusg 18835 df-sbg 18836 df-subg 19021 df-cntz 19215 df-lsm 19534 df-cmn 19680 df-abl 19681 df-mgp 20045 df-rng 20057 df-ur 20086 df-ring 20139 df-oppr 20241 df-dvdsr 20261 df-unit 20262 df-invr 20292 df-dvr 20305 df-drng 20635 df-lmod 20784 df-lss 20854 df-lsp 20894 df-lvec 21026 df-lsatoms 38974 df-oposet 39174 df-ol 39176 df-oml 39177 df-covers 39264 df-ats 39265 df-atl 39296 df-cvlat 39320 df-hlat 39349 df-llines 39497 df-lplanes 39498 df-lvols 39499 df-lines 39500 df-psubsp 39502 df-pmap 39503 df-padd 39795 df-lhyp 39987 df-laut 39988 df-ldil 40103 df-ltrn 40104 df-trl 40158 df-tendo 40754 df-edring 40756 df-disoa 41028 df-dvech 41078 df-dib 41138 df-dic 41172 df-dih 41228 df-doch 41347

This theorem is referenced by: hdmapip0com 41916 hdmapoc 41930

W3C validator