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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvh3dim | Structured version Visualization version GIF version |
Description: There is a vector that is outside the span of 2 others. (Contributed by NM, 24-Apr-2015.) |
Ref | Expression |
---|---|
dvh3dim.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dvh3dim.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dvh3dim.v | ⊢ 𝑉 = (Base‘𝑈) |
dvh3dim.n | ⊢ 𝑁 = (LSpan‘𝑈) |
dvh3dim.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dvh3dim.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
dvh3dim.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
Ref | Expression |
---|---|
dvh3dim | ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvh3dim.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dvh3dim.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | dvh3dim.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
4 | dvh3dim.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
5 | dvh3dim.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | dvh3dim.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
7 | 1, 2, 3, 4, 5, 6 | dvh2dim 38461 | . . . 4 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑌})) |
8 | 7 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑌})) |
9 | prcom 4660 | . . . . . . . . 9 ⊢ {𝑋, 𝑌} = {𝑌, 𝑋} | |
10 | preq2 4662 | . . . . . . . . 9 ⊢ (𝑋 = (0g‘𝑈) → {𝑌, 𝑋} = {𝑌, (0g‘𝑈)}) | |
11 | 9, 10 | syl5eq 2865 | . . . . . . . 8 ⊢ (𝑋 = (0g‘𝑈) → {𝑋, 𝑌} = {𝑌, (0g‘𝑈)}) |
12 | 11 | fveq2d 6667 | . . . . . . 7 ⊢ (𝑋 = (0g‘𝑈) → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌, (0g‘𝑈)})) |
13 | eqid 2818 | . . . . . . . 8 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
14 | 1, 2, 5 | dvhlmod 38126 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
15 | 3, 13, 4, 14, 6 | lsppr0 19793 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑌, (0g‘𝑈)}) = (𝑁‘{𝑌})) |
16 | 12, 15 | sylan9eqr 2875 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑌})) |
17 | 16 | eleq2d 2895 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → (𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ 𝑧 ∈ (𝑁‘{𝑌}))) |
18 | 17 | notbid 319 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ ¬ 𝑧 ∈ (𝑁‘{𝑌}))) |
19 | 18 | rexbidv 3294 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → (∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑌}))) |
20 | 8, 19 | mpbird 258 | . 2 ⊢ ((𝜑 ∧ 𝑋 = (0g‘𝑈)) → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) |
21 | dvh3dim.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
22 | 1, 2, 3, 4, 5, 21 | dvh2dim 38461 | . . . 4 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋})) |
23 | 22 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑈)) → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋})) |
24 | preq2 4662 | . . . . . . . 8 ⊢ (𝑌 = (0g‘𝑈) → {𝑋, 𝑌} = {𝑋, (0g‘𝑈)}) | |
25 | 24 | fveq2d 6667 | . . . . . . 7 ⊢ (𝑌 = (0g‘𝑈) → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑋, (0g‘𝑈)})) |
26 | 3, 13, 4, 14, 21 | lsppr0 19793 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋, (0g‘𝑈)}) = (𝑁‘{𝑋})) |
27 | 25, 26 | sylan9eqr 2875 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑈)) → (𝑁‘{𝑋, 𝑌}) = (𝑁‘{𝑋})) |
28 | 27 | eleq2d 2895 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑈)) → (𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ 𝑧 ∈ (𝑁‘{𝑋}))) |
29 | 28 | notbid 319 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑈)) → (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ ¬ 𝑧 ∈ (𝑁‘{𝑋}))) |
30 | 29 | rexbidv 3294 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑈)) → (∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ↔ ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋}))) |
31 | 23, 30 | mpbird 258 | . 2 ⊢ ((𝜑 ∧ 𝑌 = (0g‘𝑈)) → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) |
32 | 5 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
33 | 21 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → 𝑋 ∈ 𝑉) |
34 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → 𝑌 ∈ 𝑉) |
35 | simprl 767 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → 𝑋 ≠ (0g‘𝑈)) | |
36 | simprr 769 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → 𝑌 ≠ (0g‘𝑈)) | |
37 | 1, 2, 3, 4, 32, 33, 34, 13, 35, 36 | dvhdimlem 38460 | . 2 ⊢ ((𝜑 ∧ (𝑋 ≠ (0g‘𝑈) ∧ 𝑌 ≠ (0g‘𝑈))) → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) |
38 | 20, 31, 37 | pm2.61da2ne 3102 | 1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∃wrex 3136 {csn 4557 {cpr 4559 ‘cfv 6348 Basecbs 16471 0gc0g 16701 LSpanclspn 19672 HLchlt 36366 LHypclh 37000 DVecHcdvh 38094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-riotaBAD 35969 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-undef 7928 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-0g 16703 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 df-lsatoms 35992 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 df-lvols 36516 df-lines 36517 df-psubsp 36519 df-pmap 36520 df-padd 36812 df-lhyp 37004 df-laut 37005 df-ldil 37120 df-ltrn 37121 df-trl 37175 df-tgrp 37759 df-tendo 37771 df-edring 37773 df-dveca 38019 df-disoa 38045 df-dvech 38095 df-dib 38155 df-dic 38189 df-dih 38245 df-doch 38364 df-djh 38411 |
This theorem is referenced by: dvh4dimN 38463 dvh3dim2 38464 mapdh6iN 38760 mapdh8e 38800 mapdh9a 38805 mapdh9aOLDN 38806 hdmap1l6i 38834 hdmapval0 38849 hdmapval3N 38854 hdmap10lem 38855 hdmap11lem2 38858 hdmap14lem11 38894 |
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