Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochnel | Structured version Visualization version GIF version |
Description: A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014.) |
Ref | Expression |
---|---|
dochnel.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochnel.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochnel.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochnel.v | ⊢ 𝑉 = (Base‘𝑈) |
dochnel.z | ⊢ 0 = (0g‘𝑈) |
dochnel.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochnel.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
Ref | Expression |
---|---|
dochnel | ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{𝑋})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochnel.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dochnel.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | eqid 2819 | . . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
4 | dochnel.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
5 | dochnel.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
6 | dochnel.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | 1, 2, 6 | dvhlmod 38238 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
8 | dochnel.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
9 | 8 | eldifad 3946 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
10 | dochnel.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
11 | eqid 2819 | . . . . 5 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈) | |
12 | 10, 3, 11 | lspsncl 19741 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑋}) ∈ (LSubSp‘𝑈)) |
13 | 7, 9, 12 | syl2anc 586 | . . 3 ⊢ (𝜑 → ((LSpan‘𝑈)‘{𝑋}) ∈ (LSubSp‘𝑈)) |
14 | 10, 11 | lspsnid 19757 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ ((LSpan‘𝑈)‘{𝑋})) |
15 | 7, 9, 14 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ((LSpan‘𝑈)‘{𝑋})) |
16 | eldifsni 4714 | . . . . 5 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 ) | |
17 | 8, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
18 | eldifsn 4711 | . . . 4 ⊢ (𝑋 ∈ (((LSpan‘𝑈)‘{𝑋}) ∖ { 0 }) ↔ (𝑋 ∈ ((LSpan‘𝑈)‘{𝑋}) ∧ 𝑋 ≠ 0 )) | |
19 | 15, 17, 18 | sylanbrc 585 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (((LSpan‘𝑈)‘{𝑋}) ∖ { 0 })) |
20 | 1, 2, 3, 4, 5, 6, 13, 19 | dochnel2 38520 | . 2 ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘((LSpan‘𝑈)‘{𝑋}))) |
21 | 9 | snssd 4734 | . . 3 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
22 | 1, 2, 5, 10, 11, 6, 21 | dochocsp 38507 | . 2 ⊢ (𝜑 → ( ⊥ ‘((LSpan‘𝑈)‘{𝑋})) = ( ⊥ ‘{𝑋})) |
23 | 20, 22 | neleqtrd 2932 | 1 ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{𝑋})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ∖ cdif 3931 {csn 4559 ‘cfv 6348 Basecbs 16475 0gc0g 16705 LModclmod 19626 LSubSpclss 19695 LSpanclspn 19735 HLchlt 36478 LHypclh 37112 DVecHcdvh 38206 ocHcoch 38475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-riotaBAD 36081 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-tpos 7884 df-undef 7931 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19232 df-ur 19244 df-ring 19291 df-oppr 19365 df-dvdsr 19383 df-unit 19384 df-invr 19414 df-dvr 19425 df-drng 19496 df-lmod 19628 df-lss 19696 df-lsp 19736 df-lvec 19867 df-lsatoms 36104 df-oposet 36304 df-ol 36306 df-oml 36307 df-covers 36394 df-ats 36395 df-atl 36426 df-cvlat 36450 df-hlat 36479 df-llines 36626 df-lplanes 36627 df-lvols 36628 df-lines 36629 df-psubsp 36631 df-pmap 36632 df-padd 36924 df-lhyp 37116 df-laut 37117 df-ldil 37232 df-ltrn 37233 df-trl 37287 df-tendo 37883 df-edring 37885 df-disoa 38157 df-dvech 38207 df-dib 38267 df-dic 38301 df-dih 38357 df-doch 38476 |
This theorem is referenced by: dochexmidat 38587 dochfln0 38605 lclkrlem2o 38649 lcfrlem19 38689 hdmapip0 39043 |
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