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Mathbox for Norm Megill |
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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 40428 | . . . 4 β’ (π β π = (Baseβπ)) |
7 | rabeq 3424 | . . . 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 2737 | . . . . . . 7 β’ ((HDMapβπΎ)βπ) = ((HDMapβπΎ)βπ) | |
10 | 3 | ad2antrr 725 | . . . . . . 7 β’ (((π β§ π¦ β π) β§ π§ β π) β (πΎ β HL β§ π β π»)) |
11 | eqid 2737 | . . . . . . 7 β’ (Β·πβπ) = (Β·πβπ) | |
12 | simplr 768 | . . . . . . 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 482 | . . . . . . . 8 β’ ((π β§ π¦ β π) β π β π) |
15 | 14 | sselda 3949 | . . . . . . 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 40445 | . . . . . 6 β’ (((π β§ π¦ β π) β§ π§ β π) β (π¦(Β·πβπ)π§) = ((((HDMapβπΎ)βπ)βπ§)βπ¦)) |
17 | eqid 2737 | . . . . . . . . 9 β’ (ScalarβπΏ) = (ScalarβπΏ) | |
18 | eqid 2737 | . . . . . . . . 9 β’ (Scalarβπ) = (Scalarβπ) | |
19 | eqid 2737 | . . . . . . . . 9 β’ (0gβ(ScalarβπΏ)) = (0gβ(ScalarβπΏ)) | |
20 | 1, 4, 17, 2, 18, 3, 19 | hlhils0 40441 | . . . . . . . 8 β’ (π β (0gβ(ScalarβπΏ)) = (0gβ(Scalarβπ))) |
21 | 20 | eqcomd 2743 | . . . . . . 7 β’ (π β (0gβ(Scalarβπ)) = (0gβ(ScalarβπΏ))) |
22 | 21 | ad2antrr 725 | . . . . . 6 β’ (((π β§ π¦ β π) β§ π§ β π) β (0gβ(Scalarβπ)) = (0gβ(ScalarβπΏ))) |
23 | 16, 22 | eqeq12d 2753 | . . . . 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 3173 | . . . 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 3417 | . . 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 2779 | . 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 3989 | . . 3 β’ (π β π β (Baseβπ)) |
28 | eqid 2737 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
29 | eqid 2737 | . . . 4 β’ (0gβ(Scalarβπ)) = (0gβ(Scalarβπ)) | |
30 | hlhilocv.o | . . . 4 β’ π = (ocvβπ) | |
31 | 28, 11, 18, 29, 30 | ocvval 21087 | . . 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 40423 | . 2 β’ (π β (πβπ) = {π¦ β π β£ βπ§ β π ((((HDMapβπΎ)βπ)βπ§)βπ¦) = (0gβ(ScalarβπΏ))}) |
35 | 26, 32, 34 | 3eqtr4d 2787 | 1 β’ (π β (πβπ) = (πβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 {crab 3410 β wss 3915 βcfv 6501 (class class class)co 7362 Basecbs 17090 Scalarcsca 17143 Β·πcip 17145 0gc0g 17328 ocvcocv 21080 HLchlt 37841 LHypclh 38476 DVecHcdvh 39570 ocHcoch 39839 HDMapchdma 40284 HLHilchlh 40424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-riotaBAD 37444 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-tpos 8162 df-undef 8209 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-ip 17158 df-0g 17330 df-mre 17473 df-mrc 17474 df-acs 17476 df-proset 18191 df-poset 18209 df-plt 18226 df-lub 18242 df-glb 18243 df-join 18244 df-meet 18245 df-p0 18321 df-p1 18322 df-lat 18328 df-clat 18395 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-grp 18758 df-minusg 18759 df-sbg 18760 df-subg 18932 df-cntz 19104 df-oppg 19131 df-lsm 19425 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-oppr 20056 df-dvdsr 20077 df-unit 20078 df-invr 20108 df-dvr 20119 df-drng 20201 df-lmod 20340 df-lss 20409 df-lsp 20449 df-lvec 20580 df-ocv 21083 df-lsatoms 37467 df-lshyp 37468 df-lcv 37510 df-lfl 37549 df-lkr 37577 df-ldual 37615 df-oposet 37667 df-ol 37669 df-oml 37670 df-covers 37757 df-ats 37758 df-atl 37789 df-cvlat 37813 df-hlat 37842 df-llines 37990 df-lplanes 37991 df-lvols 37992 df-lines 37993 df-psubsp 37995 df-pmap 37996 df-padd 38288 df-lhyp 38480 df-laut 38481 df-ldil 38596 df-ltrn 38597 df-trl 38651 df-tgrp 39235 df-tendo 39247 df-edring 39249 df-dveca 39495 df-disoa 39521 df-dvech 39571 df-dib 39631 df-dic 39665 df-dih 39721 df-doch 39840 df-djh 39887 df-lcdual 40079 df-mapd 40117 df-hvmap 40249 df-hdmap1 40285 df-hdmap 40286 df-hlhil 40425 |
This theorem is referenced by: hlhillcs 40454 hlhilhillem 40456 |
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