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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapval3lemN | Structured version Visualization version GIF version |
Description: Value of map from vectors to functionals at arguments not colinear with the reference vector 𝐸. (Contributed by NM, 17-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hdmapval3.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmapval3.e | ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 |
hdmapval3.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmapval3.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmapval3.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmapval3.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmapval3.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmapval3.j | ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) |
hdmapval3.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
hdmapval3.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmapval3.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmapval3.te | ⊢ (𝜑 → (𝑁‘{𝑇}) ≠ (𝑁‘{𝐸})) |
hdmapval3lem.t | ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
hdmapval3lem.x | ⊢ (𝜑 → 𝑥 ∈ 𝑉) |
hdmapval3lem.xn | ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘{𝐸, 𝑇})) |
Ref | Expression |
---|---|
hdmapval3lemN | ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑇〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapval3.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmapval3.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmapval3.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | eqid 2825 | . . 3 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
5 | hdmapval3.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
6 | hdmapval3.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
7 | hdmapval3.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
8 | eqid 2825 | . . 3 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶) | |
9 | eqid 2825 | . . 3 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊) | |
10 | hdmapval3.i | . . 3 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
11 | hdmapval3.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
12 | eqid 2825 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
13 | hdmapval3.j | . . . . . 6 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) | |
14 | eqid 2825 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
15 | eqid 2825 | . . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
16 | hdmapval3.e | . . . . . . 7 ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
17 | 1, 14, 15, 2, 3, 4, 16, 11 | dvheveccl 37182 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
18 | 1, 2, 3, 4, 6, 7, 12, 13, 11, 17 | hvmapcl2 37836 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0g‘𝐶)})) |
19 | 18 | eldifad 3810 | . . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷) |
20 | 1, 2, 3, 4, 5, 6, 8, 9, 13, 11, 17 | mapdhvmap 37839 | . . . 4 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝐸})) = ((LSpan‘𝐶)‘{(𝐽‘𝐸)})) |
21 | 1, 2, 11 | dvhlvec 37179 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec) |
22 | hdmapval3lem.x | . . . . . . 7 ⊢ (𝜑 → 𝑥 ∈ 𝑉) | |
23 | 17 | eldifad 3810 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
24 | hdmapval3lem.t | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ {(0g‘𝑈)})) | |
25 | 24 | eldifad 3810 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
26 | hdmapval3lem.xn | . . . . . . 7 ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘{𝐸, 𝑇})) | |
27 | 3, 5, 21, 22, 23, 25, 26 | lspindpi 19499 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑥}) ≠ (𝑁‘{𝐸}) ∧ (𝑁‘{𝑥}) ≠ (𝑁‘{𝑇}))) |
28 | 27 | simpld 490 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑥}) ≠ (𝑁‘{𝐸})) |
29 | 28 | necomd 3054 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑥})) |
30 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 20, 29, 17, 22 | hdmap1cl 37874 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉) ∈ 𝐷) |
31 | eqidd 2826 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉)) | |
32 | eqid 2825 | . . . . . 6 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
33 | eqid 2825 | . . . . . 6 ⊢ (-g‘𝐶) = (-g‘𝐶) | |
34 | eqid 2825 | . . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
35 | 1, 2, 11 | dvhlmod 37180 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
36 | 3, 34, 5, 35, 23, 25 | lspprcl 19344 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝐸, 𝑇}) ∈ (LSubSp‘𝑈)) |
37 | 4, 34, 35, 36, 22, 26 | lssneln0 19316 | . . . . . 6 ⊢ (𝜑 → 𝑥 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
38 | 1, 2, 3, 32, 4, 5, 6, 7, 33, 8, 9, 10, 11, 17, 19, 37, 30, 29, 20 | hdmap1eq 37871 | . . . . 5 ⊢ (𝜑 → ((𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉) ↔ ((((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑥})) = ((LSpan‘𝐶)‘{(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉)}) ∧ (((mapd‘𝐾)‘𝑊)‘(𝑁‘{(𝐸(-g‘𝑈)𝑥)})) = ((LSpan‘𝐶)‘{((𝐽‘𝐸)(-g‘𝐶)(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉))})))) |
39 | 31, 38 | mpbid 224 | . . . 4 ⊢ (𝜑 → ((((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑥})) = ((LSpan‘𝐶)‘{(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉)}) ∧ (((mapd‘𝐾)‘𝑊)‘(𝑁‘{(𝐸(-g‘𝑈)𝑥)})) = ((LSpan‘𝐶)‘{((𝐽‘𝐸)(-g‘𝐶)(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉))}))) |
40 | 39 | simpld 490 | . . 3 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝑥})) = ((LSpan‘𝐶)‘{(𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉)})) |
41 | hdmapval3.te | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑇}) ≠ (𝑁‘{𝐸})) | |
42 | 41 | necomd 3054 | . . 3 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑇})) |
43 | hdmapval3.s | . . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
44 | 3, 5, 35, 23, 25 | lspprid1 19363 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (𝑁‘{𝐸, 𝑇})) |
45 | 34, 5, 35, 36, 44 | lspsnel5a 19362 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝐸}) ⊆ (𝑁‘{𝐸, 𝑇})) |
46 | 45, 45 | unssd 4018 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸})) ⊆ (𝑁‘{𝐸, 𝑇})) |
47 | 46, 26 | ssneldd 3830 | . . . . 5 ⊢ (𝜑 → ¬ 𝑥 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸}))) |
48 | 1, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 23, 22, 47 | hdmapval2 37902 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝐸〉)) |
49 | 1, 16, 13, 43, 11 | hdmapevec 37905 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐽‘𝐸)) |
50 | 48, 49 | eqtr3d 2863 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝐸〉) = (𝐽‘𝐸)) |
51 | 3, 5, 35, 23, 25 | lspprid2 19364 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑁‘{𝐸, 𝑇})) |
52 | 34, 5, 35, 36, 51 | lspsnel5a 19362 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝐸, 𝑇})) |
53 | 45, 52 | unssd 4018 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) ⊆ (𝑁‘{𝐸, 𝑇})) |
54 | 53, 26 | ssneldd 3830 | . . . . 5 ⊢ (𝜑 → ¬ 𝑥 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇}))) |
55 | 1, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 25, 22, 54 | hdmapval2 37902 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝑇〉)) |
56 | 55 | eqcomd 2831 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝑇〉) = (𝑆‘𝑇)) |
57 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 30, 40, 37, 17, 24, 42, 26, 50, 56 | hdmap1eq4N 37876 | . 2 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑇〉) = (𝑆‘𝑇)) |
58 | 57 | eqcomd 2831 | 1 ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑇〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ∖ cdif 3795 ∪ cun 3796 {csn 4399 {cpr 4401 〈cop 4405 〈cotp 4407 I cid 5251 ↾ cres 5348 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 0gc0g 16460 -gcsg 17785 LSubSpclss 19295 LSpanclspn 19337 HLchlt 35420 LHypclh 36054 LTrncltrn 36171 DVecHcdvh 37148 LCDualclcd 37656 mapdcmpd 37694 HVMapchvm 37826 HDMap1chdma1 37861 HDMapchdma 37862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-riotaBAD 35023 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-ot 4408 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-undef 7669 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-0g 16462 df-mre 16606 df-mrc 16607 df-acs 16609 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-p1 17400 df-lat 17406 df-clat 17468 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-cntz 18107 df-oppg 18133 df-lsm 18409 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-drng 19112 df-lmod 19228 df-lss 19296 df-lsp 19338 df-lvec 19469 df-lsatoms 35046 df-lshyp 35047 df-lcv 35089 df-lfl 35128 df-lkr 35156 df-ldual 35194 df-oposet 35246 df-ol 35248 df-oml 35249 df-covers 35336 df-ats 35337 df-atl 35368 df-cvlat 35392 df-hlat 35421 df-llines 35568 df-lplanes 35569 df-lvols 35570 df-lines 35571 df-psubsp 35573 df-pmap 35574 df-padd 35866 df-lhyp 36058 df-laut 36059 df-ldil 36174 df-ltrn 36175 df-trl 36229 df-tgrp 36813 df-tendo 36825 df-edring 36827 df-dveca 37073 df-disoa 37099 df-dvech 37149 df-dib 37209 df-dic 37243 df-dih 37299 df-doch 37418 df-djh 37465 df-lcdual 37657 df-mapd 37695 df-hvmap 37827 df-hdmap1 37863 df-hdmap 37864 |
This theorem is referenced by: hdmapval3N 37908 |
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