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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhillvec | Structured version Visualization version GIF version |
Description: The final constructed Hilbert space is a vector space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) |
Ref | Expression |
---|---|
hlhillvec.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hlhillvec.u | ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) |
hlhillvec.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
hlhillvec | ⊢ (𝜑 → 𝑈 ∈ LVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhillvec.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2825 | . . 3 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
3 | hlhillvec.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | dvhlvec 37179 | . 2 ⊢ (𝜑 → ((DVecH‘𝐾)‘𝑊) ∈ LVec) |
5 | eqidd 2826 | . . 3 ⊢ (𝜑 → (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊))) | |
6 | hlhillvec.u | . . . 4 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) | |
7 | eqid 2825 | . . . 4 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊)) | |
8 | 1, 6, 3, 2, 7 | hlhilbase 38006 | . . 3 ⊢ (𝜑 → (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘𝑈)) |
9 | eqid 2825 | . . 3 ⊢ (Scalar‘((DVecH‘𝐾)‘𝑊)) = (Scalar‘((DVecH‘𝐾)‘𝑊)) | |
10 | eqid 2825 | . . 3 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
11 | eqidd 2826 | . . 3 ⊢ (𝜑 → (Base‘(Scalar‘((DVecH‘𝐾)‘𝑊))) = (Base‘(Scalar‘((DVecH‘𝐾)‘𝑊)))) | |
12 | eqid 2825 | . . . 4 ⊢ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑊))) = (Base‘(Scalar‘((DVecH‘𝐾)‘𝑊))) | |
13 | 1, 2, 9, 6, 10, 3, 12 | hlhilsbase2 38012 | . . 3 ⊢ (𝜑 → (Base‘(Scalar‘((DVecH‘𝐾)‘𝑊))) = (Base‘(Scalar‘𝑈))) |
14 | eqid 2825 | . . . . 5 ⊢ (+g‘((DVecH‘𝐾)‘𝑊)) = (+g‘((DVecH‘𝐾)‘𝑊)) | |
15 | 1, 6, 3, 2, 14 | hlhilplus 38007 | . . . 4 ⊢ (𝜑 → (+g‘((DVecH‘𝐾)‘𝑊)) = (+g‘𝑈)) |
16 | 15 | oveqdr 6938 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ 𝑦 ∈ (Base‘((DVecH‘𝐾)‘𝑊)))) → (𝑥(+g‘((DVecH‘𝐾)‘𝑊))𝑦) = (𝑥(+g‘𝑈)𝑦)) |
17 | eqid 2825 | . . . . 5 ⊢ (+g‘(Scalar‘((DVecH‘𝐾)‘𝑊))) = (+g‘(Scalar‘((DVecH‘𝐾)‘𝑊))) | |
18 | 1, 2, 9, 6, 10, 3, 17 | hlhilsplus2 38013 | . . . 4 ⊢ (𝜑 → (+g‘(Scalar‘((DVecH‘𝐾)‘𝑊))) = (+g‘(Scalar‘𝑈))) |
19 | 18 | oveqdr 6938 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑊))) ∧ 𝑦 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑊))))) → (𝑥(+g‘(Scalar‘((DVecH‘𝐾)‘𝑊)))𝑦) = (𝑥(+g‘(Scalar‘𝑈))𝑦)) |
20 | eqid 2825 | . . . . 5 ⊢ (.r‘(Scalar‘((DVecH‘𝐾)‘𝑊))) = (.r‘(Scalar‘((DVecH‘𝐾)‘𝑊))) | |
21 | 1, 2, 9, 6, 10, 3, 20 | hlhilsmul2 38014 | . . . 4 ⊢ (𝜑 → (.r‘(Scalar‘((DVecH‘𝐾)‘𝑊))) = (.r‘(Scalar‘𝑈))) |
22 | 21 | oveqdr 6938 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑊))) ∧ 𝑦 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑊))))) → (𝑥(.r‘(Scalar‘((DVecH‘𝐾)‘𝑊)))𝑦) = (𝑥(.r‘(Scalar‘𝑈))𝑦)) |
23 | eqid 2825 | . . . . 5 ⊢ ( ·𝑠 ‘((DVecH‘𝐾)‘𝑊)) = ( ·𝑠 ‘((DVecH‘𝐾)‘𝑊)) | |
24 | 1, 2, 23, 6, 3 | hlhilvsca 38017 | . . . 4 ⊢ (𝜑 → ( ·𝑠 ‘((DVecH‘𝐾)‘𝑊)) = ( ·𝑠 ‘𝑈)) |
25 | 24 | oveqdr 6938 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑊))) ∧ 𝑦 ∈ (Base‘((DVecH‘𝐾)‘𝑊)))) → (𝑥( ·𝑠 ‘((DVecH‘𝐾)‘𝑊))𝑦) = (𝑥( ·𝑠 ‘𝑈)𝑦)) |
26 | 5, 8, 9, 10, 11, 13, 16, 19, 22, 25 | lvecprop2d 19534 | . 2 ⊢ (𝜑 → (((DVecH‘𝐾)‘𝑊) ∈ LVec ↔ 𝑈 ∈ LVec)) |
27 | 4, 26 | mpbid 224 | 1 ⊢ (𝜑 → 𝑈 ∈ LVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 Basecbs 16229 +gcplusg 16312 .rcmulr 16313 Scalarcsca 16315 ·𝑠 cvsca 16316 LVecclvec 19468 HLchlt 35420 LHypclh 36054 DVecHcdvh 37148 HLHilchlh 38002 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-riotaBAD 35023 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-undef 7669 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-ip 16330 df-0g 16462 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-p1 17400 df-lat 17406 df-clat 17468 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-minusg 17787 df-mgp 18851 df-ur 18863 df-ring 18910 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-drng 19112 df-lmod 19228 df-lvec 19469 df-oposet 35246 df-ol 35248 df-oml 35249 df-covers 35336 df-ats 35337 df-atl 35368 df-cvlat 35392 df-hlat 35421 df-llines 35568 df-lplanes 35569 df-lvols 35570 df-lines 35571 df-psubsp 35573 df-pmap 35574 df-padd 35866 df-lhyp 36058 df-laut 36059 df-ldil 36174 df-ltrn 36175 df-trl 36229 df-tendo 36825 df-edring 36827 df-dvech 37149 df-hlhil 38003 |
This theorem is referenced by: hlhilphllem 38029 |
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