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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 2736 | . . 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 2736 | . . 3 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶) | |
| 9 | eqid 2736 | . . 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 2736 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
| 13 | hdmapval3.j | . . . . . 6 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) | |
| 14 | eqid 2736 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 15 | eqid 2736 | . . . . . . 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 41115 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) | 
| 18 | 1, 2, 3, 4, 6, 7, 12, 13, 11, 17 | hvmapcl2 41769 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0g‘𝐶)})) | 
| 19 | 18 | eldifad 3962 | . . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷) | 
| 20 | 1, 2, 3, 4, 5, 6, 8, 9, 13, 11, 17 | mapdhvmap 41772 | . . . 4 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝐸})) = ((LSpan‘𝐶)‘{(𝐽‘𝐸)})) | 
| 21 | 1, 2, 11 | dvhlvec 41112 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec) | 
| 22 | hdmapval3lem.x | . . . . . . 7 ⊢ (𝜑 → 𝑥 ∈ 𝑉) | |
| 23 | 17 | eldifad 3962 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑉) | 
| 24 | hdmapval3lem.t | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ {(0g‘𝑈)})) | |
| 25 | 24 | eldifad 3962 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | 
| 26 | hdmapval3lem.xn | . . . . . . 7 ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘{𝐸, 𝑇})) | |
| 27 | 3, 5, 21, 22, 23, 25, 26 | lspindpi 21135 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑥}) ≠ (𝑁‘{𝐸}) ∧ (𝑁‘{𝑥}) ≠ (𝑁‘{𝑇}))) | 
| 28 | 27 | simpld 494 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑥}) ≠ (𝑁‘{𝐸})) | 
| 29 | 28 | necomd 2995 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑥})) | 
| 30 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 20, 29, 17, 22 | hdmap1cl 41807 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉) ∈ 𝐷) | 
| 31 | eqidd 2737 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉)) | |
| 32 | eqid 2736 | . . . . . 6 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
| 33 | eqid 2736 | . . . . . 6 ⊢ (-g‘𝐶) = (-g‘𝐶) | |
| 34 | eqid 2736 | . . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
| 35 | 1, 2, 11 | dvhlmod 41113 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) | 
| 36 | 3, 34, 5, 35, 23, 25 | lspprcl 20977 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝐸, 𝑇}) ∈ (LSubSp‘𝑈)) | 
| 37 | 4, 34, 35, 36, 22, 26 | lssneln0 20952 | . . . . . 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 41804 | . . . . 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 232 | . . . 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 494 | . . 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 2995 | . . 3 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑇})) | 
| 43 | hdmapval3.s | . . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
| 44 | 3, 5, 35, 23, 25 | lspprid1 20996 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (𝑁‘{𝐸, 𝑇})) | 
| 45 | 34, 5, 35, 36, 44 | ellspsn5 20995 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝐸}) ⊆ (𝑁‘{𝐸, 𝑇})) | 
| 46 | 45, 45 | unssd 4191 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸})) ⊆ (𝑁‘{𝐸, 𝑇})) | 
| 47 | 46, 26 | ssneldd 3985 | . . . . 5 ⊢ (𝜑 → ¬ 𝑥 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸}))) | 
| 48 | 1, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 23, 22, 47 | hdmapval2 41835 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝐸〉)) | 
| 49 | 1, 16, 13, 43, 11 | hdmapevec 41838 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐽‘𝐸)) | 
| 50 | 48, 49 | eqtr3d 2778 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝐸〉) = (𝐽‘𝐸)) | 
| 51 | 3, 5, 35, 23, 25 | lspprid2 20997 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑁‘{𝐸, 𝑇})) | 
| 52 | 34, 5, 35, 36, 51 | ellspsn5 20995 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝐸, 𝑇})) | 
| 53 | 45, 52 | unssd 4191 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) ⊆ (𝑁‘{𝐸, 𝑇})) | 
| 54 | 53, 26 | ssneldd 3985 | . . . . 5 ⊢ (𝜑 → ¬ 𝑥 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇}))) | 
| 55 | 1, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 25, 22, 54 | hdmapval2 41835 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝑇〉)) | 
| 56 | 55 | eqcomd 2742 | . . 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 41809 | . 2 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑇〉) = (𝑆‘𝑇)) | 
| 58 | 57 | eqcomd 2742 | 1 ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑇〉)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∖ cdif 3947 ∪ cun 3948 {csn 4625 {cpr 4627 〈cop 4631 〈cotp 4633 I cid 5576 ↾ cres 5686 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 0gc0g 17485 -gcsg 18954 LSubSpclss 20930 LSpanclspn 20970 HLchlt 39352 LHypclh 39987 LTrncltrn 40104 DVecHcdvh 41081 LCDualclcd 41589 mapdcmpd 41627 HVMapchvm 41759 HDMap1chdma1 41794 HDMapchdma 41795 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-riotaBAD 38955 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-tpos 8252 df-undef 8299 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17487 df-mre 17630 df-mrc 17631 df-acs 17633 df-proset 18341 df-poset 18360 df-plt 18376 df-lub 18392 df-glb 18393 df-join 18394 df-meet 18395 df-p0 18471 df-p1 18472 df-lat 18478 df-clat 18545 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-subg 19142 df-cntz 19336 df-oppg 19365 df-lsm 19655 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-dvr 20402 df-nzr 20514 df-rlreg 20695 df-domn 20696 df-drng 20732 df-lmod 20861 df-lss 20931 df-lsp 20971 df-lvec 21103 df-lsatoms 38978 df-lshyp 38979 df-lcv 39021 df-lfl 39060 df-lkr 39088 df-ldual 39126 df-oposet 39178 df-ol 39180 df-oml 39181 df-covers 39268 df-ats 39269 df-atl 39300 df-cvlat 39324 df-hlat 39353 df-llines 39501 df-lplanes 39502 df-lvols 39503 df-lines 39504 df-psubsp 39506 df-pmap 39507 df-padd 39799 df-lhyp 39991 df-laut 39992 df-ldil 40107 df-ltrn 40108 df-trl 40162 df-tgrp 40746 df-tendo 40758 df-edring 40760 df-dveca 41006 df-disoa 41032 df-dvech 41082 df-dib 41142 df-dic 41176 df-dih 41232 df-doch 41351 df-djh 41398 df-lcdual 41590 df-mapd 41628 df-hvmap 41760 df-hdmap1 41796 df-hdmap 41797 | 
| This theorem is referenced by: hdmapval3N 41841 | 
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