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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilhillem | Structured version Visualization version GIF version |
Description: Lemma for hlhil 24823. (Contributed by NM, 23-Jun-2015.) |
Ref | Expression |
---|---|
hlhilphl.h | β’ π» = (LHypβπΎ) |
hlhilphllem.u | β’ π = ((HLHilβπΎ)βπ) |
hlhilphl.k | β’ (π β (πΎ β HL β§ π β π»)) |
hlhilphllem.f | β’ πΉ = (Scalarβπ) |
hlhilphllem.l | β’ πΏ = ((DVecHβπΎ)βπ) |
hlhilphllem.v | β’ π = (BaseβπΏ) |
hlhilphllem.a | β’ + = (+gβπΏ) |
hlhilphllem.s | β’ Β· = ( Β·π βπΏ) |
hlhilphllem.r | β’ π = (ScalarβπΏ) |
hlhilphllem.b | β’ π΅ = (Baseβπ ) |
hlhilphllem.p | ⒠⨣ = (+gβπ ) |
hlhilphllem.t | β’ Γ = (.rβπ ) |
hlhilphllem.q | β’ π = (0gβπ ) |
hlhilphllem.z | β’ 0 = (0gβπΏ) |
hlhilphllem.i | β’ , = (Β·πβπ) |
hlhilphllem.j | β’ π½ = ((HDMapβπΎ)βπ) |
hlhilphllem.g | β’ πΊ = ((HGMapβπΎ)βπ) |
hlhilphllem.e | β’ πΈ = (π₯ β π, π¦ β π β¦ ((π½βπ¦)βπ₯)) |
hlhilphllem.o | β’ π = (ocvβπ) |
hlhilphllem.c | β’ πΆ = (ClSubSpβπ) |
Ref | Expression |
---|---|
hlhilhillem | β’ (π β π β Hil) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhilphl.h | . . 3 β’ π» = (LHypβπΎ) | |
2 | hlhilphllem.u | . . 3 β’ π = ((HLHilβπΎ)βπ) | |
3 | hlhilphl.k | . . 3 β’ (π β (πΎ β HL β§ π β π»)) | |
4 | hlhilphllem.f | . . 3 β’ πΉ = (Scalarβπ) | |
5 | hlhilphllem.l | . . 3 β’ πΏ = ((DVecHβπΎ)βπ) | |
6 | hlhilphllem.v | . . 3 β’ π = (BaseβπΏ) | |
7 | hlhilphllem.a | . . 3 β’ + = (+gβπΏ) | |
8 | hlhilphllem.s | . . 3 β’ Β· = ( Β·π βπΏ) | |
9 | hlhilphllem.r | . . 3 β’ π = (ScalarβπΏ) | |
10 | hlhilphllem.b | . . 3 β’ π΅ = (Baseβπ ) | |
11 | hlhilphllem.p | . . 3 ⒠⨣ = (+gβπ ) | |
12 | hlhilphllem.t | . . 3 β’ Γ = (.rβπ ) | |
13 | hlhilphllem.q | . . 3 β’ π = (0gβπ ) | |
14 | hlhilphllem.z | . . 3 β’ 0 = (0gβπΏ) | |
15 | hlhilphllem.i | . . 3 β’ , = (Β·πβπ) | |
16 | hlhilphllem.j | . . 3 β’ π½ = ((HDMapβπΎ)βπ) | |
17 | hlhilphllem.g | . . 3 β’ πΊ = ((HGMapβπΎ)βπ) | |
18 | hlhilphllem.e | . . 3 β’ πΈ = (π₯ β π, π¦ β π β¦ ((π½βπ¦)βπ₯)) | |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 | hlhilphllem 40455 | . 2 β’ (π β π β PreHil) |
20 | 3 | adantr 482 | . . . . . . 7 β’ ((π β§ π₯ β πΆ) β (πΎ β HL β§ π β π»)) |
21 | eqid 2737 | . . . . . . 7 β’ ((ocHβπΎ)βπ) = ((ocHβπΎ)βπ) | |
22 | hlhilphllem.o | . . . . . . 7 β’ π = (ocvβπ) | |
23 | eqid 2737 | . . . . . . . . . . 11 β’ ((DIsoHβπΎ)βπ) = ((DIsoHβπΎ)βπ) | |
24 | hlhilphllem.c | . . . . . . . . . . 11 β’ πΆ = (ClSubSpβπ) | |
25 | 1, 23, 2, 24, 3 | hlhillcs 40454 | . . . . . . . . . 10 β’ (π β πΆ = ran ((DIsoHβπΎ)βπ)) |
26 | 25 | eleq2d 2824 | . . . . . . . . 9 β’ (π β (π₯ β πΆ β π₯ β ran ((DIsoHβπΎ)βπ))) |
27 | 26 | biimpa 478 | . . . . . . . 8 β’ ((π β§ π₯ β πΆ) β π₯ β ran ((DIsoHβπΎ)βπ)) |
28 | 1, 5, 23, 6 | dihrnss 39770 | . . . . . . . . 9 β’ (((πΎ β HL β§ π β π») β§ π₯ β ran ((DIsoHβπΎ)βπ)) β π₯ β π) |
29 | 3, 28 | sylan 581 | . . . . . . . 8 β’ ((π β§ π₯ β ran ((DIsoHβπΎ)βπ)) β π₯ β π) |
30 | 27, 29 | syldan 592 | . . . . . . 7 β’ ((π β§ π₯ β πΆ) β π₯ β π) |
31 | 1, 5, 2, 20, 6, 21, 22, 30 | hlhilocv 40453 | . . . . . 6 β’ ((π β§ π₯ β πΆ) β (πβπ₯) = (((ocHβπΎ)βπ)βπ₯)) |
32 | 31 | oveq2d 7378 | . . . . 5 β’ ((π β§ π₯ β πΆ) β (π₯(LSSumβπΏ)(πβπ₯)) = (π₯(LSSumβπΏ)(((ocHβπΎ)βπ)βπ₯))) |
33 | eqid 2737 | . . . . . . . 8 β’ (LSSumβπΏ) = (LSSumβπΏ) | |
34 | 1, 5, 2, 3, 33 | hlhillsm 40452 | . . . . . . 7 β’ (π β (LSSumβπΏ) = (LSSumβπ)) |
35 | 34 | adantr 482 | . . . . . 6 β’ ((π β§ π₯ β πΆ) β (LSSumβπΏ) = (LSSumβπ)) |
36 | 35 | oveqd 7379 | . . . . 5 β’ ((π β§ π₯ β πΆ) β (π₯(LSSumβπΏ)(πβπ₯)) = (π₯(LSSumβπ)(πβπ₯))) |
37 | eqid 2737 | . . . . . . 7 β’ (LSubSpβπΏ) = (LSubSpβπΏ) | |
38 | 3 | adantr 482 | . . . . . . 7 β’ ((π β§ π₯ β ran ((DIsoHβπΎ)βπ)) β (πΎ β HL β§ π β π»)) |
39 | 1, 5, 23, 37 | dihrnlss 39769 | . . . . . . . 8 β’ (((πΎ β HL β§ π β π») β§ π₯ β ran ((DIsoHβπΎ)βπ)) β π₯ β (LSubSpβπΏ)) |
40 | 3, 39 | sylan 581 | . . . . . . 7 β’ ((π β§ π₯ β ran ((DIsoHβπΎ)βπ)) β π₯ β (LSubSpβπΏ)) |
41 | 1, 23, 5, 6, 21, 38, 29 | dochoccl 39861 | . . . . . . . . . . 11 β’ ((π β§ π₯ β ran ((DIsoHβπΎ)βπ)) β (π₯ β ran ((DIsoHβπΎ)βπ) β (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)βπ₯)) = π₯)) |
42 | 41 | biimpd 228 | . . . . . . . . . 10 β’ ((π β§ π₯ β ran ((DIsoHβπΎ)βπ)) β (π₯ β ran ((DIsoHβπΎ)βπ) β (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)βπ₯)) = π₯)) |
43 | 42 | ex 414 | . . . . . . . . 9 β’ (π β (π₯ β ran ((DIsoHβπΎ)βπ) β (π₯ β ran ((DIsoHβπΎ)βπ) β (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)βπ₯)) = π₯))) |
44 | 43 | pm2.43d 53 | . . . . . . . 8 β’ (π β (π₯ β ran ((DIsoHβπΎ)βπ) β (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)βπ₯)) = π₯)) |
45 | 44 | imp 408 | . . . . . . 7 β’ ((π β§ π₯ β ran ((DIsoHβπΎ)βπ)) β (((ocHβπΎ)βπ)β(((ocHβπΎ)βπ)βπ₯)) = π₯) |
46 | 1, 21, 5, 6, 37, 33, 38, 40, 45 | dochexmid 39960 | . . . . . 6 β’ ((π β§ π₯ β ran ((DIsoHβπΎ)βπ)) β (π₯(LSSumβπΏ)(((ocHβπΎ)βπ)βπ₯)) = π) |
47 | 27, 46 | syldan 592 | . . . . 5 β’ ((π β§ π₯ β πΆ) β (π₯(LSSumβπΏ)(((ocHβπΎ)βπ)βπ₯)) = π) |
48 | 32, 36, 47 | 3eqtr3d 2785 | . . . 4 β’ ((π β§ π₯ β πΆ) β (π₯(LSSumβπ)(πβπ₯)) = π) |
49 | 1, 2, 3, 5, 6 | hlhilbase 40428 | . . . . 5 β’ (π β π = (Baseβπ)) |
50 | 49 | adantr 482 | . . . 4 β’ ((π β§ π₯ β πΆ) β π = (Baseβπ)) |
51 | 48, 50 | eqtrd 2777 | . . 3 β’ ((π β§ π₯ β πΆ) β (π₯(LSSumβπ)(πβπ₯)) = (Baseβπ)) |
52 | 51 | ralrimiva 3144 | . 2 β’ (π β βπ₯ β πΆ (π₯(LSSumβπ)(πβπ₯)) = (Baseβπ)) |
53 | eqid 2737 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
54 | eqid 2737 | . . 3 β’ (LSSumβπ) = (LSSumβπ) | |
55 | 53, 54, 22, 24 | ishil2 21141 | . 2 β’ (π β Hil β (π β PreHil β§ βπ₯ β πΆ (π₯(LSSumβπ)(πβπ₯)) = (Baseβπ))) |
56 | 19, 52, 55 | sylanbrc 584 | 1 β’ (π β π β Hil) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 β wss 3915 ran crn 5639 βcfv 6501 (class class class)co 7362 β cmpo 7364 Basecbs 17090 +gcplusg 17140 .rcmulr 17141 Scalarcsca 17143 Β·π cvsca 17144 Β·πcip 17145 0gc0g 17328 LSSumclsm 19423 LSubSpclss 20408 PreHilcphl 21044 ocvcocv 21080 ClSubSpccss 21081 Hilchil 21123 HLchlt 37841 LHypclh 38476 DVecHcdvh 39570 DIsoHcdih 39720 ocHcoch 39839 HDMapchdma 40284 HGMapchg 40375 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-mhm 18608 df-submnd 18609 df-grp 18758 df-minusg 18759 df-sbg 18760 df-subg 18932 df-ghm 19013 df-cntz 19104 df-oppg 19131 df-lsm 19425 df-pj1 19426 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-rnghom 20155 df-drng 20201 df-subrg 20236 df-staf 20320 df-srng 20321 df-lmod 20340 df-lss 20409 df-lsp 20449 df-lmhm 20499 df-lvec 20580 df-sra 20649 df-rgmod 20650 df-phl 21046 df-ocv 21083 df-css 21084 df-pj 21125 df-hil 21126 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-hgmap 40376 df-hlhil 40425 |
This theorem is referenced by: hlathil 40457 |
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