Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilocv | Structured version Visualization version GIF version | ||
| Description: The orthocomplement for the final constructed Hilbert space. (Contributed by NM, 23-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) |
| Ref | Expression |
|---|---|
| hlhil0.h | β’ π» = (LHypβπΎ) |
| hlhil0.l | β’ πΏ = ((DVecHβπΎ)βπ) |
| hlhil0.u | β’ π = ((HLHilβπΎ)βπ) |
| hlhil0.k | β’ (π β (πΎ β HL β§ π β π»)) |
| hlhilocv.v | β’ π = (BaseβπΏ) |
| hlhilocv.n | β’ π = ((ocHβπΎ)βπ) |
| hlhilocv.o | β’ π = (ocvβπ) |
| hlhilocv.x | β’ (π β π β π) |
| Ref | Expression |
|---|---|
| hlhilocv | β’ (π β (πβπ) = (πβπ)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hlhil0.h | . . . . 5 β’ π» = (LHypβπΎ) | |
| 2 | hlhil0.u | . . . . 5 β’ π = ((HLHilβπΎ)βπ) | |
| 3 | hlhil0.k | . . . . 5 β’ (π β (πΎ β HL β§ π β π»)) | |
| 4 | hlhil0.l | . . . . 5 β’ πΏ = ((DVecHβπΎ)βπ) | |
| 5 | hlhilocv.v | . . . . 5 β’ π = (BaseβπΏ) | |
| 6 | 1, 2, 3, 4, 5 | hlhilbase 40795 | . . . 4 β’ (π β π = (Baseβπ)) |
| 7 | rabeq 3446 | . . . 4 β’ (π = (Baseβπ) β {π¦ β π β£ βπ§ β π (π¦(Β·πβπ)π§) = (0gβ(Scalarβπ))} = {π¦ β (Baseβπ) β£ βπ§ β π (π¦(Β·πβπ)π§) = (0gβ(Scalarβπ))}) | |
| 8 | 6, 7 | syl 17 | . . 3 β’ (π β {π¦ β π β£ βπ§ β π (π¦(Β·πβπ)π§) = (0gβ(Scalarβπ))} = {π¦ β (Baseβπ) β£ βπ§ β π (π¦(Β·πβπ)π§) = (0gβ(Scalarβπ))}) |
| 9 | eqid 2732 | . . . . . . 7 β’ ((HDMapβπΎ)βπ) = ((HDMapβπΎ)βπ) | |
| 10 | 3 | ad2antrr 724 | . . . . . . 7 β’ (((π β§ π¦ β π) β§ π§ β π) β (πΎ β HL β§ π β π»)) |
| 11 | eqid 2732 | . . . . . . 7 β’ (Β·πβπ) = (Β·πβπ) | |
| 12 | simplr 767 | . . . . . . 7 β’ (((π β§ π¦ β π) β§ π§ β π) β π¦ β π) | |
| 13 | hlhilocv.x | . . . . . . . . 9 β’ (π β π β π) | |
| 14 | 13 | adantr 481 | . . . . . . . 8 β’ ((π β§ π¦ β π) β π β π) |
| 15 | 14 | sselda 3981 | . . . . . . 7 β’ (((π β§ π¦ β π) β§ π§ β π) β π§ β π) |
| 16 | 1, 4, 5, 9, 2, 10, 11, 12, 15 | hlhilipval 40812 | . . . . . 6 β’ (((π β§ π¦ β π) β§ π§ β π) β (π¦(Β·πβπ)π§) = ((((HDMapβπΎ)βπ)βπ§)βπ¦)) |
| 17 | eqid 2732 | . . . . . . . . 9 β’ (ScalarβπΏ) = (ScalarβπΏ) | |
| 18 | eqid 2732 | . . . . . . . . 9 β’ (Scalarβπ) = (Scalarβπ) | |
| 19 | eqid 2732 | . . . . . . . . 9 β’ (0gβ(ScalarβπΏ)) = (0gβ(ScalarβπΏ)) | |
| 20 | 1, 4, 17, 2, 18, 3, 19 | hlhils0 40808 | . . . . . . . 8 β’ (π β (0gβ(ScalarβπΏ)) = (0gβ(Scalarβπ))) |
| 21 | 20 | eqcomd 2738 | . . . . . . 7 β’ (π β (0gβ(Scalarβπ)) = (0gβ(ScalarβπΏ))) |
| 22 | 21 | ad2antrr 724 | . . . . . 6 β’ (((π β§ π¦ β π) β§ π§ β π) β (0gβ(Scalarβπ)) = (0gβ(ScalarβπΏ))) |
| 23 | 16, 22 | eqeq12d 2748 | . . . . 5 β’ (((π β§ π¦ β π) β§ π§ β π) β ((π¦(Β·πβπ)π§) = (0gβ(Scalarβπ)) β ((((HDMapβπΎ)βπ)βπ§)βπ¦) = (0gβ(ScalarβπΏ)))) |
| 24 | 23 | ralbidva 3175 | . . . 4 β’ ((π β§ π¦ β π) β (βπ§ β π (π¦(Β·πβπ)π§) = (0gβ(Scalarβπ)) β βπ§ β π ((((HDMapβπΎ)βπ)βπ§)βπ¦) = (0gβ(ScalarβπΏ)))) |
| 25 | 24 | rabbidva 3439 | . . 3 β’ (π β {π¦ β π β£ βπ§ β π (π¦(Β·πβπ)π§) = (0gβ(Scalarβπ))} = {π¦ β π β£ βπ§ β π ((((HDMapβπΎ)βπ)βπ§)βπ¦) = (0gβ(ScalarβπΏ))}) |
| 26 | 8, 25 | eqtr3d 2774 | . 2 β’ (π β {π¦ β (Baseβπ) β£ βπ§ β π (π¦(Β·πβπ)π§) = (0gβ(Scalarβπ))} = {π¦ β π β£ βπ§ β π ((((HDMapβπΎ)βπ)βπ§)βπ¦) = (0gβ(ScalarβπΏ))}) |
| 27 | 13, 6 | sseqtrd 4021 | . . 3 β’ (π β π β (Baseβπ)) |
| 28 | eqid 2732 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
| 29 | eqid 2732 | . . . 4 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
| 30 | hlhilocv.o | . . . 4 β’ π = (ocvβπ) | |
| 31 | 28, 11, 18, 29, 30 | ocvval 21211 | . . 3 β’ (π β (Baseβπ) β (πβπ) = {π¦ β (Baseβπ) β£ βπ§ β π (π¦(Β·πβπ)π§) = (0gβ(Scalarβπ))}) |
| 32 | 27, 31 | syl 17 | . 2 β’ (π β (πβπ) = {π¦ β (Baseβπ) β£ βπ§ β π (π¦(Β·πβπ)π§) = (0gβ(Scalarβπ))}) |
| 33 | hlhilocv.n | . . 3 β’ π = ((ocHβπΎ)βπ) | |
| 34 | 1, 4, 5, 17, 19, 33, 9, 3, 13 | hdmapoc 40790 | . 2 β’ (π β (πβπ) = {π¦ β π β£ βπ§ β π ((((HDMapβπΎ)βπ)βπ§)βπ¦) = (0gβ(ScalarβπΏ))}) |
| 35 | 26, 32, 34 | 3eqtr4d 2782 | 1 β’ (π β (πβπ) = (πβπ)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 {crab 3432 β wss 3947 βcfv 6540 (class class class)co 7405 Basecbs 17140 Scalarcsca 17196 Β·πcip 17198 0gc0g 17381 ocvcocv 21204 HLchlt 38208 LHypclh 38843 DVecHcdvh 39937 ocHcoch 40206 HDMapchdma 40651 HLHilchlh 40791 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-riotaBAD 37811 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-0g 17383 df-mre 17526 df-mrc 17527 df-acs 17529 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-cntz 19175 df-oppg 19204 df-lsm 19498 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-drng 20309 df-lmod 20465 df-lss 20535 df-lsp 20575 df-lvec 20706 df-ocv 21207 df-lsatoms 37834 df-lshyp 37835 df-lcv 37877 df-lfl 37916 df-lkr 37944 df-ldual 37982 df-oposet 38034 df-ol 38036 df-oml 38037 df-covers 38124 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 df-llines 38357 df-lplanes 38358 df-lvols 38359 df-lines 38360 df-psubsp 38362 df-pmap 38363 df-padd 38655 df-lhyp 38847 df-laut 38848 df-ldil 38963 df-ltrn 38964 df-trl 39018 df-tgrp 39602 df-tendo 39614 df-edring 39616 df-dveca 39862 df-disoa 39888 df-dvech 39938 df-dib 39998 df-dic 40032 df-dih 40088 df-doch 40207 df-djh 40254 df-lcdual 40446 df-mapd 40484 df-hvmap 40616 df-hdmap1 40652 df-hdmap 40653 df-hlhil 40792 |
| This theorem is referenced by: hlhillcs 40821 hlhilhillem 40823 |
| Copyright terms: Public domain | W3C validator |