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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 | โข ๐ป = (LHypโ๐พ) |
hlhillvec.u | โข ๐ = ((HLHilโ๐พ)โ๐) |
hlhillvec.k | โข (๐ โ (๐พ โ HL โง ๐ โ ๐ป)) |
Ref | Expression |
---|---|
hlhillvec | โข (๐ โ ๐ โ LVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhillvec.h | . . 3 โข ๐ป = (LHypโ๐พ) | |
2 | eqid 2726 | . . 3 โข ((DVecHโ๐พ)โ๐) = ((DVecHโ๐พ)โ๐) | |
3 | hlhillvec.k | . . 3 โข (๐ โ (๐พ โ HL โง ๐ โ ๐ป)) | |
4 | 1, 2, 3 | dvhlvec 40493 | . 2 โข (๐ โ ((DVecHโ๐พ)โ๐) โ LVec) |
5 | eqidd 2727 | . . 3 โข (๐ โ (Baseโ((DVecHโ๐พ)โ๐)) = (Baseโ((DVecHโ๐พ)โ๐))) | |
6 | hlhillvec.u | . . . 4 โข ๐ = ((HLHilโ๐พ)โ๐) | |
7 | eqid 2726 | . . . 4 โข (Baseโ((DVecHโ๐พ)โ๐)) = (Baseโ((DVecHโ๐พ)โ๐)) | |
8 | 1, 6, 3, 2, 7 | hlhilbase 41320 | . . 3 โข (๐ โ (Baseโ((DVecHโ๐พ)โ๐)) = (Baseโ๐)) |
9 | eqid 2726 | . . 3 โข (Scalarโ((DVecHโ๐พ)โ๐)) = (Scalarโ((DVecHโ๐พ)โ๐)) | |
10 | eqid 2726 | . . 3 โข (Scalarโ๐) = (Scalarโ๐) | |
11 | eqidd 2727 | . . 3 โข (๐ โ (Baseโ(Scalarโ((DVecHโ๐พ)โ๐))) = (Baseโ(Scalarโ((DVecHโ๐พ)โ๐)))) | |
12 | eqid 2726 | . . . 4 โข (Baseโ(Scalarโ((DVecHโ๐พ)โ๐))) = (Baseโ(Scalarโ((DVecHโ๐พ)โ๐))) | |
13 | 1, 2, 9, 6, 10, 3, 12 | hlhilsbase2 41330 | . . 3 โข (๐ โ (Baseโ(Scalarโ((DVecHโ๐พ)โ๐))) = (Baseโ(Scalarโ๐))) |
14 | eqid 2726 | . . . . 5 โข (+gโ((DVecHโ๐พ)โ๐)) = (+gโ((DVecHโ๐พ)โ๐)) | |
15 | 1, 6, 3, 2, 14 | hlhilplus 41321 | . . . 4 โข (๐ โ (+gโ((DVecHโ๐พ)โ๐)) = (+gโ๐)) |
16 | 15 | oveqdr 7433 | . . 3 โข ((๐ โง (๐ฅ โ (Baseโ((DVecHโ๐พ)โ๐)) โง ๐ฆ โ (Baseโ((DVecHโ๐พ)โ๐)))) โ (๐ฅ(+gโ((DVecHโ๐พ)โ๐))๐ฆ) = (๐ฅ(+gโ๐)๐ฆ)) |
17 | eqid 2726 | . . . . 5 โข (+gโ(Scalarโ((DVecHโ๐พ)โ๐))) = (+gโ(Scalarโ((DVecHโ๐พ)โ๐))) | |
18 | 1, 2, 9, 6, 10, 3, 17 | hlhilsplus2 41331 | . . . 4 โข (๐ โ (+gโ(Scalarโ((DVecHโ๐พ)โ๐))) = (+gโ(Scalarโ๐))) |
19 | 18 | oveqdr 7433 | . . 3 โข ((๐ โง (๐ฅ โ (Baseโ(Scalarโ((DVecHโ๐พ)โ๐))) โง ๐ฆ โ (Baseโ(Scalarโ((DVecHโ๐พ)โ๐))))) โ (๐ฅ(+gโ(Scalarโ((DVecHโ๐พ)โ๐)))๐ฆ) = (๐ฅ(+gโ(Scalarโ๐))๐ฆ)) |
20 | eqid 2726 | . . . . 5 โข (.rโ(Scalarโ((DVecHโ๐พ)โ๐))) = (.rโ(Scalarโ((DVecHโ๐พ)โ๐))) | |
21 | 1, 2, 9, 6, 10, 3, 20 | hlhilsmul2 41332 | . . . 4 โข (๐ โ (.rโ(Scalarโ((DVecHโ๐พ)โ๐))) = (.rโ(Scalarโ๐))) |
22 | 21 | oveqdr 7433 | . . 3 โข ((๐ โง (๐ฅ โ (Baseโ(Scalarโ((DVecHโ๐พ)โ๐))) โง ๐ฆ โ (Baseโ(Scalarโ((DVecHโ๐พ)โ๐))))) โ (๐ฅ(.rโ(Scalarโ((DVecHโ๐พ)โ๐)))๐ฆ) = (๐ฅ(.rโ(Scalarโ๐))๐ฆ)) |
23 | eqid 2726 | . . . . 5 โข ( ยท๐ โ((DVecHโ๐พ)โ๐)) = ( ยท๐ โ((DVecHโ๐พ)โ๐)) | |
24 | 1, 2, 23, 6, 3 | hlhilvsca 41335 | . . . 4 โข (๐ โ ( ยท๐ โ((DVecHโ๐พ)โ๐)) = ( ยท๐ โ๐)) |
25 | 24 | oveqdr 7433 | . . 3 โข ((๐ โง (๐ฅ โ (Baseโ(Scalarโ((DVecHโ๐พ)โ๐))) โง ๐ฆ โ (Baseโ((DVecHโ๐พ)โ๐)))) โ (๐ฅ( ยท๐ โ((DVecHโ๐พ)โ๐))๐ฆ) = (๐ฅ( ยท๐ โ๐)๐ฆ)) |
26 | 5, 8, 9, 10, 11, 13, 16, 19, 22, 25 | lvecprop2d 21017 | . 2 โข (๐ โ (((DVecHโ๐พ)โ๐) โ LVec โ ๐ โ LVec)) |
27 | 4, 26 | mpbid 231 | 1 โข (๐ โ ๐ โ LVec) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 395 = wceq 1533 โ wcel 2098 โcfv 6537 Basecbs 17153 +gcplusg 17206 .rcmulr 17207 Scalarcsca 17209 ยท๐ cvsca 17210 LVecclvec 20950 HLchlt 38733 LHypclh 39368 DVecHcdvh 40462 HLHilchlh 41316 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 38336 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-undef 8259 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-0g 17396 df-proset 18260 df-poset 18278 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-drng 20589 df-lmod 20708 df-lvec 20951 df-oposet 38559 df-ol 38561 df-oml 38562 df-covers 38649 df-ats 38650 df-atl 38681 df-cvlat 38705 df-hlat 38734 df-llines 38882 df-lplanes 38883 df-lvols 38884 df-lines 38885 df-psubsp 38887 df-pmap 38888 df-padd 39180 df-lhyp 39372 df-laut 39373 df-ldil 39488 df-ltrn 39489 df-trl 39543 df-tendo 40139 df-edring 40141 df-dvech 40463 df-hlhil 41317 |
This theorem is referenced by: hlhilphllem 41347 |
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