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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmapevec2 | Structured version Visualization version GIF version |
Description: The inner product of the reference vector 𝐸 with itself is nonzero. This shows the inner product condition in the proof of Theorem 3.6 of [Holland95] p. 14 line 32, [ e , e ] ≠ 0 is satisfied. TODO: remove redundant hypothesis hdmapevec.j. (Contributed by NM, 1-Jun-2015.) |
Ref | Expression |
---|---|
hdmapevec.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmapevec.e | ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 |
hdmapevec.j | ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) |
hdmapevec.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmapevec.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmapevec2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmapevec2.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmapevec2.i | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
hdmapevec2 | ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmapevec.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmapevec.e | . . . . 5 ⊢ 𝐸 = 〈( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))〉 | |
3 | hdmapevec.j | . . . . 5 ⊢ 𝐽 = ((HVMap‘𝐾)‘𝑊) | |
4 | hdmapevec.s | . . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
5 | hdmapevec.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | 1, 2, 3, 4, 5 | hdmapevec 38998 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐸) = (𝐽‘𝐸)) |
7 | hdmapevec2.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
8 | eqid 2820 | . . . . 5 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊) | |
9 | eqid 2820 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
10 | eqid 2820 | . . . . 5 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
11 | eqid 2820 | . . . . 5 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
12 | eqid 2820 | . . . . 5 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
13 | hdmapevec2.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑈) | |
14 | eqid 2820 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
15 | eqid 2820 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
16 | eqid 2820 | . . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
17 | 1, 15, 16, 7, 9, 12, 2, 5 | dvheveccl 38275 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)})) |
18 | 1, 7, 8, 9, 10, 11, 12, 13, 14, 3, 5, 17 | hvmapval 38923 | . . . 4 ⊢ (𝜑 → (𝐽‘𝐸) = (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑘 ∈ (Base‘𝑅)∃𝑤 ∈ (((ocH‘𝐾)‘𝑊)‘{𝐸})𝑣 = (𝑤(+g‘𝑈)(𝑘( ·𝑠 ‘𝑈)𝐸))))) |
19 | 6, 18 | eqtrd 2855 | . . 3 ⊢ (𝜑 → (𝑆‘𝐸) = (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑘 ∈ (Base‘𝑅)∃𝑤 ∈ (((ocH‘𝐾)‘𝑊)‘{𝐸})𝑣 = (𝑤(+g‘𝑈)(𝑘( ·𝑠 ‘𝑈)𝐸))))) |
20 | 19 | fveq1d 6653 | . 2 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = ((𝑣 ∈ (Base‘𝑈) ↦ (℩𝑘 ∈ (Base‘𝑅)∃𝑤 ∈ (((ocH‘𝐾)‘𝑊)‘{𝐸})𝑣 = (𝑤(+g‘𝑈)(𝑘( ·𝑠 ‘𝑈)𝐸))))‘𝐸)) |
21 | hdmapevec2.i | . . 3 ⊢ 1 = (1r‘𝑅) | |
22 | eqid 2820 | . . 3 ⊢ (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑘 ∈ (Base‘𝑅)∃𝑤 ∈ (((ocH‘𝐾)‘𝑊)‘{𝐸})𝑣 = (𝑤(+g‘𝑈)(𝑘( ·𝑠 ‘𝑈)𝐸)))) = (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑘 ∈ (Base‘𝑅)∃𝑤 ∈ (((ocH‘𝐾)‘𝑊)‘{𝐸})𝑣 = (𝑤(+g‘𝑈)(𝑘( ·𝑠 ‘𝑈)𝐸)))) | |
23 | 1, 8, 7, 9, 10, 11, 12, 13, 14, 21, 5, 17, 22 | dochfl1 38639 | . 2 ⊢ (𝜑 → ((𝑣 ∈ (Base‘𝑈) ↦ (℩𝑘 ∈ (Base‘𝑅)∃𝑤 ∈ (((ocH‘𝐾)‘𝑊)‘{𝐸})𝑣 = (𝑤(+g‘𝑈)(𝑘( ·𝑠 ‘𝑈)𝐸))))‘𝐸) = 1 ) |
24 | 20, 23 | eqtrd 2855 | 1 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3134 {csn 4548 〈cop 4554 ↦ cmpt 5127 I cid 5440 ↾ cres 5538 ‘cfv 6336 ℩crio 7094 (class class class)co 7137 Basecbs 16461 +gcplusg 16543 Scalarcsca 16546 ·𝑠 cvsca 16547 0gc0g 16691 1rcur 19229 HLchlt 36513 LHypclh 37147 LTrncltrn 37264 DVecHcdvh 38241 ocHcoch 38510 HVMapchvm 38919 HDMapchdma 38955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 ax-riotaBAD 36116 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-ot 4557 df-uni 4820 df-int 4858 df-iun 4902 df-iin 4903 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-of 7390 df-om 7562 df-1st 7670 df-2nd 7671 df-tpos 7873 df-undef 7920 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-1o 8083 df-oadd 8087 df-er 8270 df-map 8389 df-en 8491 df-dom 8492 df-sdom 8493 df-fin 8494 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-nn 11620 df-2 11682 df-3 11683 df-4 11684 df-5 11685 df-6 11686 df-n0 11880 df-z 11964 df-uz 12226 df-fz 12878 df-struct 16463 df-ndx 16464 df-slot 16465 df-base 16467 df-sets 16468 df-ress 16469 df-plusg 16556 df-mulr 16557 df-sca 16559 df-vsca 16560 df-0g 16693 df-mre 16835 df-mrc 16836 df-acs 16838 df-proset 17516 df-poset 17534 df-plt 17546 df-lub 17562 df-glb 17563 df-join 17564 df-meet 17565 df-p0 17627 df-p1 17628 df-lat 17634 df-clat 17696 df-mgm 17830 df-sgrp 17879 df-mnd 17890 df-submnd 17935 df-grp 18084 df-minusg 18085 df-sbg 18086 df-subg 18254 df-cntz 18425 df-oppg 18452 df-lsm 18739 df-cmn 18886 df-abl 18887 df-mgp 19218 df-ur 19230 df-ring 19277 df-oppr 19351 df-dvdsr 19369 df-unit 19370 df-invr 19400 df-dvr 19411 df-drng 19482 df-lmod 19614 df-lss 19682 df-lsp 19722 df-lvec 19853 df-lsatoms 36139 df-lshyp 36140 df-lcv 36182 df-lfl 36221 df-lkr 36249 df-ldual 36287 df-oposet 36339 df-ol 36341 df-oml 36342 df-covers 36429 df-ats 36430 df-atl 36461 df-cvlat 36485 df-hlat 36514 df-llines 36661 df-lplanes 36662 df-lvols 36663 df-lines 36664 df-psubsp 36666 df-pmap 36667 df-padd 36959 df-lhyp 37151 df-laut 37152 df-ldil 37267 df-ltrn 37268 df-trl 37322 df-tgrp 37906 df-tendo 37918 df-edring 37920 df-dveca 38166 df-disoa 38192 df-dvech 38242 df-dib 38302 df-dic 38336 df-dih 38392 df-doch 38511 df-djh 38558 df-lcdual 38750 df-mapd 38788 df-hvmap 38920 df-hdmap1 38956 df-hdmap 38957 |
This theorem is referenced by: hdmapinvlem3 39083 hdmapinvlem4 39084 hdmapglem7b 39091 |
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