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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 2724 | . . 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 2724 | . . 3 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶) | |
9 | eqid 2724 | . . 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 2724 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
13 | hdmapval3.j | . . . . . 6 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) | |
14 | eqid 2724 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
15 | eqid 2724 | . . . . . . 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 40439 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
18 | 1, 2, 3, 4, 6, 7, 12, 13, 11, 17 | hvmapcl2 41093 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0g‘𝐶)})) |
19 | 18 | eldifad 3952 | . . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷) |
20 | 1, 2, 3, 4, 5, 6, 8, 9, 13, 11, 17 | mapdhvmap 41096 | . . . 4 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝐸})) = ((LSpan‘𝐶)‘{(𝐽‘𝐸)})) |
21 | 1, 2, 11 | dvhlvec 40436 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec) |
22 | hdmapval3lem.x | . . . . . . 7 ⊢ (𝜑 → 𝑥 ∈ 𝑉) | |
23 | 17 | eldifad 3952 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
24 | hdmapval3lem.t | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ {(0g‘𝑈)})) | |
25 | 24 | eldifad 3952 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
26 | hdmapval3lem.xn | . . . . . . 7 ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘{𝐸, 𝑇})) | |
27 | 3, 5, 21, 22, 23, 25, 26 | lspindpi 20972 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑥}) ≠ (𝑁‘{𝐸}) ∧ (𝑁‘{𝑥}) ≠ (𝑁‘{𝑇}))) |
28 | 27 | simpld 494 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑥}) ≠ (𝑁‘{𝐸})) |
29 | 28 | necomd 2988 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑥})) |
30 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 20, 29, 17, 22 | hdmap1cl 41131 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉) ∈ 𝐷) |
31 | eqidd 2725 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉)) | |
32 | eqid 2724 | . . . . . 6 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
33 | eqid 2724 | . . . . . 6 ⊢ (-g‘𝐶) = (-g‘𝐶) | |
34 | eqid 2724 | . . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
35 | 1, 2, 11 | dvhlmod 40437 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
36 | 3, 34, 5, 35, 23, 25 | lspprcl 20814 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝐸, 𝑇}) ∈ (LSubSp‘𝑈)) |
37 | 4, 34, 35, 36, 22, 26 | lssneln0 20789 | . . . . . 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 41128 | . . . . 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 231 | . . . 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 2988 | . . 3 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑇})) |
43 | hdmapval3.s | . . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
44 | 3, 5, 35, 23, 25 | lspprid1 20833 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (𝑁‘{𝐸, 𝑇})) |
45 | 34, 5, 35, 36, 44 | lspsnel5a 20832 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝐸}) ⊆ (𝑁‘{𝐸, 𝑇})) |
46 | 45, 45 | unssd 4178 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸})) ⊆ (𝑁‘{𝐸, 𝑇})) |
47 | 46, 26 | ssneldd 3977 | . . . . 5 ⊢ (𝜑 → ¬ 𝑥 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸}))) |
48 | 1, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 23, 22, 47 | hdmapval2 41159 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝐸〉)) |
49 | 1, 16, 13, 43, 11 | hdmapevec 41162 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐽‘𝐸)) |
50 | 48, 49 | eqtr3d 2766 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝐸〉) = (𝐽‘𝐸)) |
51 | 3, 5, 35, 23, 25 | lspprid2 20834 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑁‘{𝐸, 𝑇})) |
52 | 34, 5, 35, 36, 51 | lspsnel5a 20832 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝐸, 𝑇})) |
53 | 45, 52 | unssd 4178 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) ⊆ (𝑁‘{𝐸, 𝑇})) |
54 | 53, 26 | ssneldd 3977 | . . . . 5 ⊢ (𝜑 → ¬ 𝑥 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇}))) |
55 | 1, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 25, 22, 54 | hdmapval2 41159 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝑇〉)) |
56 | 55 | eqcomd 2730 | . . 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 41133 | . 2 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑇〉) = (𝑆‘𝑇)) |
58 | 57 | eqcomd 2730 | 1 ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑇〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∖ cdif 3937 ∪ cun 3938 {csn 4620 {cpr 4622 〈cop 4626 〈cotp 4628 I cid 5563 ↾ cres 5668 ‘cfv 6533 (class class class)co 7401 Basecbs 17142 0gc0g 17383 -gcsg 18854 LSubSpclss 20767 LSpanclspn 20807 HLchlt 38676 LHypclh 39311 LTrncltrn 39428 DVecHcdvh 40405 LCDualclcd 40913 mapdcmpd 40951 HVMapchvm 41083 HDMap1chdma1 41118 HDMapchdma 41119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-riotaBAD 38279 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-ot 4629 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-tpos 8206 df-undef 8253 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-sca 17211 df-vsca 17212 df-0g 17385 df-mre 17528 df-mrc 17529 df-acs 17531 df-proset 18249 df-poset 18267 df-plt 18284 df-lub 18300 df-glb 18301 df-join 18302 df-meet 18303 df-p0 18379 df-p1 18380 df-lat 18386 df-clat 18453 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-submnd 18703 df-grp 18855 df-minusg 18856 df-sbg 18857 df-subg 19039 df-cntz 19222 df-oppg 19251 df-lsm 19545 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-dvr 20292 df-drng 20578 df-lmod 20697 df-lss 20768 df-lsp 20808 df-lvec 20940 df-lsatoms 38302 df-lshyp 38303 df-lcv 38345 df-lfl 38384 df-lkr 38412 df-ldual 38450 df-oposet 38502 df-ol 38504 df-oml 38505 df-covers 38592 df-ats 38593 df-atl 38624 df-cvlat 38648 df-hlat 38677 df-llines 38825 df-lplanes 38826 df-lvols 38827 df-lines 38828 df-psubsp 38830 df-pmap 38831 df-padd 39123 df-lhyp 39315 df-laut 39316 df-ldil 39431 df-ltrn 39432 df-trl 39486 df-tgrp 40070 df-tendo 40082 df-edring 40084 df-dveca 40330 df-disoa 40356 df-dvech 40406 df-dib 40466 df-dic 40500 df-dih 40556 df-doch 40675 df-djh 40722 df-lcdual 40914 df-mapd 40952 df-hvmap 41084 df-hdmap1 41120 df-hdmap 41121 |
This theorem is referenced by: hdmapval3N 41165 |
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