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 2610 | . . 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 2610 | . . 3 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶) | |
9 | eqid 2610 | . . 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 2610 | . . . . . 6 ⊢ (0g‘𝐶) = (0g‘𝐶) | |
13 | hdmapval3.j | . . . . . 6 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) | |
14 | eqid 2610 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
15 | eqid 2610 | . . . . . . 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 35419 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0g‘𝑈)})) |
18 | 1, 2, 3, 4, 6, 7, 12, 13, 11, 17 | hvmapcl2 36073 | . . . . 5 ⊢ (𝜑 → (𝐽‘𝐸) ∈ (𝐷 ∖ {(0g‘𝐶)})) |
19 | 18 | eldifad 3552 | . . . 4 ⊢ (𝜑 → (𝐽‘𝐸) ∈ 𝐷) |
20 | 1, 2, 3, 4, 5, 6, 8, 9, 13, 11, 17 | mapdhvmap 36076 | . . . 4 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘(𝑁‘{𝐸})) = ((LSpan‘𝐶)‘{(𝐽‘𝐸)})) |
21 | 1, 2, 11 | dvhlvec 35416 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LVec) |
22 | hdmapval3lem.x | . . . . . . 7 ⊢ (𝜑 → 𝑥 ∈ 𝑉) | |
23 | 17 | eldifad 3552 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑉) |
24 | hdmapval3lem.t | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ {(0g‘𝑈)})) | |
25 | 24 | eldifad 3552 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
26 | hdmapval3lem.xn | . . . . . . 7 ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘{𝐸, 𝑇})) | |
27 | 3, 5, 21, 22, 23, 25, 26 | lspindpi 18953 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑥}) ≠ (𝑁‘{𝐸}) ∧ (𝑁‘{𝑥}) ≠ (𝑁‘{𝑇}))) |
28 | 27 | simpld 474 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑥}) ≠ (𝑁‘{𝐸})) |
29 | 28 | necomd 2837 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑥})) |
30 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 20, 29, 17, 22 | hdmap1cl 36112 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉) ∈ 𝐷) |
31 | eqidd 2611 | . . . . 5 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉)) | |
32 | eqid 2610 | . . . . . 6 ⊢ (-g‘𝑈) = (-g‘𝑈) | |
33 | eqid 2610 | . . . . . 6 ⊢ (-g‘𝐶) = (-g‘𝐶) | |
34 | eqid 2610 | . . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
35 | 1, 2, 11 | dvhlmod 35417 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
36 | 3, 34, 5, 35, 23, 25 | lspprcl 18799 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝐸, 𝑇}) ∈ (LSubSp‘𝑈)) |
37 | 3, 4, 34, 35, 36, 22, 26 | lssneln0 18773 | . . . . . 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 36109 | . . . . 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 221 | . . . 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 474 | . . 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 2837 | . . 3 ⊢ (𝜑 → (𝑁‘{𝐸}) ≠ (𝑁‘{𝑇})) |
43 | hdmapval3.s | . . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
44 | 3, 5, 35, 23, 25 | lspprid1 18818 | . . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (𝑁‘{𝐸, 𝑇})) |
45 | 34, 5, 35, 36, 44 | lspsnel5a 18817 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝐸}) ⊆ (𝑁‘{𝐸, 𝑇})) |
46 | 45, 45 | unssd 3751 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸})) ⊆ (𝑁‘{𝐸, 𝑇})) |
47 | 46, 26 | ssneldd 3571 | . . . . 5 ⊢ (𝜑 → ¬ 𝑥 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝐸}))) |
48 | 1, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 23, 22, 47 | hdmapval2 36142 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝐸〉)) |
49 | 1, 16, 13, 43, 11 | hdmapevec 36145 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐽‘𝐸)) |
50 | 48, 49 | eqtr3d 2646 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝐸〉) = (𝐽‘𝐸)) |
51 | 3, 5, 35, 23, 25 | lspprid2 18819 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ (𝑁‘{𝐸, 𝑇})) |
52 | 34, 5, 35, 36, 51 | lspsnel5a 18817 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ (𝑁‘{𝐸, 𝑇})) |
53 | 45, 52 | unssd 3751 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇})) ⊆ (𝑁‘{𝐸, 𝑇})) |
54 | 53, 26 | ssneldd 3571 | . . . . 5 ⊢ (𝜑 → ¬ 𝑥 ∈ ((𝑁‘{𝐸}) ∪ (𝑁‘{𝑇}))) |
55 | 1, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 25, 22, 54 | hdmapval2 36142 | . . . 4 ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝑥, (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑥〉), 𝑇〉)) |
56 | 55 | eqcomd 2616 | . . 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 36114 | . 2 ⊢ (𝜑 → (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑇〉) = (𝑆‘𝑇)) |
58 | 57 | eqcomd 2616 | 1 ⊢ (𝜑 → (𝑆‘𝑇) = (𝐼‘〈𝐸, (𝐽‘𝐸), 𝑇〉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 ∪ cun 3538 {csn 4125 {cpr 4127 〈cop 4131 〈cotp 4133 I cid 4948 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 0gc0g 15923 -gcsg 17247 LSubSpclss 18753 LSpanclspn 18792 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 DVecHcdvh 35385 LCDualclcd 35893 mapdcmpd 35931 HVMapchvm 36063 HDMap1chdma1 36099 HDMapchdma 36100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-undef 7286 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-0g 15925 df-mre 16069 df-mrc 16070 df-acs 16072 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-oppg 17599 df-lsm 17874 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 df-lsatoms 33281 df-lshyp 33282 df-lcv 33324 df-lfl 33363 df-lkr 33391 df-ldual 33429 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tgrp 35049 df-tendo 35061 df-edring 35063 df-dveca 35309 df-disoa 35336 df-dvech 35386 df-dib 35446 df-dic 35480 df-dih 35536 df-doch 35655 df-djh 35702 df-lcdual 35894 df-mapd 35932 df-hvmap 36064 df-hdmap1 36101 df-hdmap 36102 |
This theorem is referenced by: hdmapval3N 36148 |
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