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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia2dimlem10 | Structured version Visualization version GIF version |
Description: Lemma for dia2dim 40602. Convert membership in closed subspace (πΌβ(π β¨ π)) to a lattice ordering. (Contributed by NM, 8-Sep-2014.) |
Ref | Expression |
---|---|
dia2dimlem10.l | β’ β€ = (leβπΎ) |
dia2dimlem10.j | β’ β¨ = (joinβπΎ) |
dia2dimlem10.a | β’ π΄ = (AtomsβπΎ) |
dia2dimlem10.h | β’ π» = (LHypβπΎ) |
dia2dimlem10.t | β’ π = ((LTrnβπΎ)βπ) |
dia2dimlem10.r | β’ π = ((trLβπΎ)βπ) |
dia2dimlem10.y | β’ π = ((DVecAβπΎ)βπ) |
dia2dimlem10.s | β’ π = (LSubSpβπ) |
dia2dimlem10.n | β’ π = (LSpanβπ) |
dia2dimlem10.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
dia2dimlem10.k | β’ (π β (πΎ β HL β§ π β π»)) |
dia2dimlem10.u | β’ (π β (π β π΄ β§ π β€ π)) |
dia2dimlem10.v | β’ (π β (π β π΄ β§ π β€ π)) |
dia2dimlem10.f | β’ (π β πΉ β π) |
dia2dimlem10.fe | β’ (π β πΉ β (πΌβ(π β¨ π))) |
Ref | Expression |
---|---|
dia2dimlem10 | β’ (π β (π βπΉ) β€ (π β¨ π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dia2dimlem10.k | . . . 4 β’ (π β (πΎ β HL β§ π β π»)) | |
2 | dia2dimlem10.f | . . . 4 β’ (π β πΉ β π) | |
3 | dia2dimlem10.h | . . . . 5 β’ π» = (LHypβπΎ) | |
4 | dia2dimlem10.t | . . . . 5 β’ π = ((LTrnβπΎ)βπ) | |
5 | dia2dimlem10.r | . . . . 5 β’ π = ((trLβπΎ)βπ) | |
6 | dia2dimlem10.y | . . . . 5 β’ π = ((DVecAβπΎ)βπ) | |
7 | dia2dimlem10.i | . . . . 5 β’ πΌ = ((DIsoAβπΎ)βπ) | |
8 | dia2dimlem10.n | . . . . 5 β’ π = (LSpanβπ) | |
9 | 3, 4, 5, 6, 7, 8 | dia1dim2 40587 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (πΌβ(π βπΉ)) = (πβ{πΉ})) |
10 | 1, 2, 9 | syl2anc 582 | . . 3 β’ (π β (πΌβ(π βπΉ)) = (πβ{πΉ})) |
11 | dia2dimlem10.s | . . . 4 β’ π = (LSubSpβπ) | |
12 | 3, 6 | dvalvec 40551 | . . . . 5 β’ ((πΎ β HL β§ π β π») β π β LVec) |
13 | lveclmod 20990 | . . . . 5 β’ (π β LVec β π β LMod) | |
14 | 1, 12, 13 | 3syl 18 | . . . 4 β’ (π β π β LMod) |
15 | 1 | simpld 493 | . . . . . 6 β’ (π β πΎ β HL) |
16 | dia2dimlem10.u | . . . . . . 7 β’ (π β (π β π΄ β§ π β€ π)) | |
17 | 16 | simpld 493 | . . . . . 6 β’ (π β π β π΄) |
18 | dia2dimlem10.v | . . . . . . 7 β’ (π β (π β π΄ β§ π β€ π)) | |
19 | 18 | simpld 493 | . . . . . 6 β’ (π β π β π΄) |
20 | eqid 2725 | . . . . . . 7 β’ (BaseβπΎ) = (BaseβπΎ) | |
21 | dia2dimlem10.j | . . . . . . 7 β’ β¨ = (joinβπΎ) | |
22 | dia2dimlem10.a | . . . . . . 7 β’ π΄ = (AtomsβπΎ) | |
23 | 20, 21, 22 | hlatjcl 38891 | . . . . . 6 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π) β (BaseβπΎ)) |
24 | 15, 17, 19, 23 | syl3anc 1368 | . . . . 5 β’ (π β (π β¨ π) β (BaseβπΎ)) |
25 | 16 | simprd 494 | . . . . . 6 β’ (π β π β€ π) |
26 | 18 | simprd 494 | . . . . . 6 β’ (π β π β€ π) |
27 | 15 | hllatd 38888 | . . . . . . 7 β’ (π β πΎ β Lat) |
28 | 20, 22 | atbase 38813 | . . . . . . . 8 β’ (π β π΄ β π β (BaseβπΎ)) |
29 | 17, 28 | syl 17 | . . . . . . 7 β’ (π β π β (BaseβπΎ)) |
30 | 20, 22 | atbase 38813 | . . . . . . . 8 β’ (π β π΄ β π β (BaseβπΎ)) |
31 | 19, 30 | syl 17 | . . . . . . 7 β’ (π β π β (BaseβπΎ)) |
32 | 1 | simprd 494 | . . . . . . . 8 β’ (π β π β π») |
33 | 20, 3 | lhpbase 39523 | . . . . . . . 8 β’ (π β π» β π β (BaseβπΎ)) |
34 | 32, 33 | syl 17 | . . . . . . 7 β’ (π β π β (BaseβπΎ)) |
35 | dia2dimlem10.l | . . . . . . . 8 β’ β€ = (leβπΎ) | |
36 | 20, 35, 21 | latjle12 18436 | . . . . . . 7 β’ ((πΎ β Lat β§ (π β (BaseβπΎ) β§ π β (BaseβπΎ) β§ π β (BaseβπΎ))) β ((π β€ π β§ π β€ π) β (π β¨ π) β€ π)) |
37 | 27, 29, 31, 34, 36 | syl13anc 1369 | . . . . . 6 β’ (π β ((π β€ π β§ π β€ π) β (π β¨ π) β€ π)) |
38 | 25, 26, 37 | mpbi2and 710 | . . . . 5 β’ (π β (π β¨ π) β€ π) |
39 | 20, 35, 3, 6, 7, 11 | dialss 40571 | . . . . 5 β’ (((πΎ β HL β§ π β π») β§ ((π β¨ π) β (BaseβπΎ) β§ (π β¨ π) β€ π)) β (πΌβ(π β¨ π)) β π) |
40 | 1, 24, 38, 39 | syl12anc 835 | . . . 4 β’ (π β (πΌβ(π β¨ π)) β π) |
41 | dia2dimlem10.fe | . . . 4 β’ (π β πΉ β (πΌβ(π β¨ π))) | |
42 | 11, 8, 14, 40, 41 | lspsnel5a 20879 | . . 3 β’ (π β (πβ{πΉ}) β (πΌβ(π β¨ π))) |
43 | 10, 42 | eqsstrd 4012 | . 2 β’ (π β (πΌβ(π βπΉ)) β (πΌβ(π β¨ π))) |
44 | 20, 3, 4, 5 | trlcl 39689 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π βπΉ) β (BaseβπΎ)) |
45 | 1, 2, 44 | syl2anc 582 | . . 3 β’ (π β (π βπΉ) β (BaseβπΎ)) |
46 | 35, 3, 4, 5 | trlle 39709 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ πΉ β π) β (π βπΉ) β€ π) |
47 | 1, 2, 46 | syl2anc 582 | . . 3 β’ (π β (π βπΉ) β€ π) |
48 | 20, 35, 3, 7 | diaord 40572 | . . 3 β’ (((πΎ β HL β§ π β π») β§ ((π βπΉ) β (BaseβπΎ) β§ (π βπΉ) β€ π) β§ ((π β¨ π) β (BaseβπΎ) β§ (π β¨ π) β€ π)) β ((πΌβ(π βπΉ)) β (πΌβ(π β¨ π)) β (π βπΉ) β€ (π β¨ π))) |
49 | 1, 45, 47, 24, 38, 48 | syl122anc 1376 | . 2 β’ (π β ((πΌβ(π βπΉ)) β (πΌβ(π β¨ π)) β (π βπΉ) β€ (π β¨ π))) |
50 | 43, 49 | mpbid 231 | 1 β’ (π β (π βπΉ) β€ (π β¨ π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β wss 3941 {csn 4625 class class class wbr 5144 βcfv 6543 (class class class)co 7413 Basecbs 17174 lecple 17234 joincjn 18297 Latclat 18417 LModclmod 20742 LSubSpclss 20814 LSpanclspn 20854 LVecclvec 20986 Atomscatm 38787 HLchlt 38874 LHypclh 39509 LTrncltrn 39626 trLctrl 39683 DVecAcdveca 40527 DIsoAcdia 40553 |
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 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-riotaBAD 38477 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-0g 17417 df-proset 18281 df-poset 18299 df-plt 18316 df-lub 18332 df-glb 18333 df-join 18334 df-meet 18335 df-p0 18411 df-p1 18412 df-lat 18418 df-clat 18485 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18892 df-minusg 18893 df-sbg 18894 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-drng 20625 df-lmod 20744 df-lss 20815 df-lsp 20855 df-lvec 20987 df-oposet 38700 df-ol 38702 df-oml 38703 df-covers 38790 df-ats 38791 df-atl 38822 df-cvlat 38846 df-hlat 38875 df-llines 39023 df-lplanes 39024 df-lvols 39025 df-lines 39026 df-psubsp 39028 df-pmap 39029 df-padd 39321 df-lhyp 39513 df-laut 39514 df-ldil 39629 df-ltrn 39630 df-trl 39684 df-tgrp 40268 df-tendo 40280 df-edring 40282 df-dveca 40528 df-disoa 40554 |
This theorem is referenced by: dia2dimlem11 40599 |
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