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 2738 | . . 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 2738 | . . 3 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶) | |
9 | eqid 2738 | . . 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 2738 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
13 | hdmapval3.j | . . . . . 6 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) | |
14 | eqid 2738 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
15 | eqid 2738 | . . . . . . 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 38738 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
18 | 1, 2, 3, 4, 6, 7, 12, 13, 11, 17 | hvmapcl2 39392 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0g‘𝐶)})) |
19 | 18 | eldifad 3853 | . . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷) |
20 | 1, 2, 3, 4, 5, 6, 8, 9, 13, 11, 17 | mapdhvmap 39395 | . . . 4 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝐸})) = ((LSpan‘𝐶)‘{(𝐽‘𝐸)})) |
21 | 1, 2, 11 | dvhlvec 38735 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec) |
22 | hdmapval3lem.x | . . . . . . 7 ⊢ (𝜑 → 𝑥 ∈ 𝑉) | |
23 | 17 | eldifad 3853 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
24 | hdmapval3lem.t | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ {(0g‘𝑈)})) | |
25 | 24 | eldifad 3853 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
26 | hdmapval3lem.xn | . . . . . . 7 ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘{𝐸, 𝑇})) | |
27 | 3, 5, 21, 22, 23, 25, 26 | lspindpi 20016 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑥}) ≠ (𝑁‘{𝐸}) ∧ (𝑁‘{𝑥}) ≠ (𝑁‘{𝑇}))) |
28 | 27 | simpld 498 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑥}) ≠ (𝑁‘{𝐸})) |
29 | 28 | necomd 2989 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑥})) |
30 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 20, 29, 17, 22 | hdmap1cl 39430 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉) ∈ 𝐷) |
31 | eqidd 2739 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉)) | |
32 | eqid 2738 | . . . . . 6 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
33 | eqid 2738 | . . . . . 6 ⊢ (-g‘𝐶) = (-g‘𝐶) | |
34 | eqid 2738 | . . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
35 | 1, 2, 11 | dvhlmod 38736 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
36 | 3, 34, 5, 35, 23, 25 | lspprcl 19862 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝐸, 𝑇}) ∈ (LSubSp‘𝑈)) |
37 | 4, 34, 35, 36, 22, 26 | lssneln0 19836 | . . . . . 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 39427 | . . . . 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 235 | . . . 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 498 | . . 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 2989 | . . 3 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑇})) |
43 | hdmapval3.s | . . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
44 | 3, 5, 35, 23, 25 | lspprid1 19881 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (𝑁‘{𝐸, 𝑇})) |
45 | 34, 5, 35, 36, 44 | lspsnel5a 19880 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝐸}) ⊆ (𝑁‘{𝐸, 𝑇})) |
46 | 45, 45 | unssd 4074 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸})) ⊆ (𝑁‘{𝐸, 𝑇})) |
47 | 46, 26 | ssneldd 3878 | . . . . 5 ⊢ (𝜑 → ¬ 𝑥 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸}))) |
48 | 1, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 23, 22, 47 | hdmapval2 39458 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝐸〉)) |
49 | 1, 16, 13, 43, 11 | hdmapevec 39461 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐽‘𝐸)) |
50 | 48, 49 | eqtr3d 2775 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝐸〉) = (𝐽‘𝐸)) |
51 | 3, 5, 35, 23, 25 | lspprid2 19882 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑁‘{𝐸, 𝑇})) |
52 | 34, 5, 35, 36, 51 | lspsnel5a 19880 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝐸, 𝑇})) |
53 | 45, 52 | unssd 4074 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) ⊆ (𝑁‘{𝐸, 𝑇})) |
54 | 53, 26 | ssneldd 3878 | . . . . 5 ⊢ (𝜑 → ¬ 𝑥 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇}))) |
55 | 1, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 25, 22, 54 | hdmapval2 39458 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝑇〉)) |
56 | 55 | eqcomd 2744 | . . 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 39432 | . 2 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑇〉) = (𝑆‘𝑇)) |
58 | 57 | eqcomd 2744 | 1 ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑇〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ∖ cdif 3838 ∪ cun 3839 {csn 4513 {cpr 4515 〈cop 4519 〈cotp 4521 I cid 5424 ↾ cres 5521 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 0gc0g 16809 -gcsg 18214 LSubSpclss 19815 LSpanclspn 19855 HLchlt 36976 LHypclh 37610 LTrncltrn 37727 DVecHcdvh 38704 LCDualclcd 39212 mapdcmpd 39250 HVMapchvm 39382 HDMap1chdma1 39417 HDMapchdma 39418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-riotaBAD 36579 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-ot 4522 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-tpos 7914 df-undef 7961 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-sca 16677 df-vsca 16678 df-0g 16811 df-mre 16953 df-mrc 16954 df-acs 16956 df-proset 17647 df-poset 17665 df-plt 17677 df-lub 17693 df-glb 17694 df-join 17695 df-meet 17696 df-p0 17758 df-p1 17759 df-lat 17765 df-clat 17827 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-grp 18215 df-minusg 18216 df-sbg 18217 df-subg 18387 df-cntz 18558 df-oppg 18585 df-lsm 18872 df-cmn 19019 df-abl 19020 df-mgp 19352 df-ur 19364 df-ring 19411 df-oppr 19488 df-dvdsr 19506 df-unit 19507 df-invr 19537 df-dvr 19548 df-drng 19616 df-lmod 19748 df-lss 19816 df-lsp 19856 df-lvec 19987 df-lsatoms 36602 df-lshyp 36603 df-lcv 36645 df-lfl 36684 df-lkr 36712 df-ldual 36750 df-oposet 36802 df-ol 36804 df-oml 36805 df-covers 36892 df-ats 36893 df-atl 36924 df-cvlat 36948 df-hlat 36977 df-llines 37124 df-lplanes 37125 df-lvols 37126 df-lines 37127 df-psubsp 37129 df-pmap 37130 df-padd 37422 df-lhyp 37614 df-laut 37615 df-ldil 37730 df-ltrn 37731 df-trl 37785 df-tgrp 38369 df-tendo 38381 df-edring 38383 df-dveca 38629 df-disoa 38655 df-dvech 38705 df-dib 38765 df-dic 38799 df-dih 38855 df-doch 38974 df-djh 39021 df-lcdual 39213 df-mapd 39251 df-hvmap 39383 df-hdmap1 39419 df-hdmap 39420 |
This theorem is referenced by: hdmapval3N 39464 |
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