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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmaprnlem3N | Structured version Visualization version GIF version |
Description: Part of proof of part 12 in [Baer] p. 49 line 15, T β P. Our (β‘πβ(πΏβ{((πβπ’) β π )})) is Baer's P, where P* = G(u'+s). (Contributed by NM, 27-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hdmaprnlem1.h | β’ π» = (LHypβπΎ) |
hdmaprnlem1.u | β’ π = ((DVecHβπΎ)βπ) |
hdmaprnlem1.v | β’ π = (Baseβπ) |
hdmaprnlem1.n | β’ π = (LSpanβπ) |
hdmaprnlem1.c | β’ πΆ = ((LCDualβπΎ)βπ) |
hdmaprnlem1.l | β’ πΏ = (LSpanβπΆ) |
hdmaprnlem1.m | β’ π = ((mapdβπΎ)βπ) |
hdmaprnlem1.s | β’ π = ((HDMapβπΎ)βπ) |
hdmaprnlem1.k | β’ (π β (πΎ β HL β§ π β π»)) |
hdmaprnlem1.se | β’ (π β π β (π· β {π})) |
hdmaprnlem1.ve | β’ (π β π£ β π) |
hdmaprnlem1.e | β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) |
hdmaprnlem1.ue | β’ (π β π’ β π) |
hdmaprnlem1.un | β’ (π β Β¬ π’ β (πβ{π£})) |
hdmaprnlem1.d | β’ π· = (BaseβπΆ) |
hdmaprnlem1.q | β’ π = (0gβπΆ) |
hdmaprnlem1.o | β’ 0 = (0gβπ) |
hdmaprnlem1.a | β’ β = (+gβπΆ) |
Ref | Expression |
---|---|
hdmaprnlem3N | β’ (π β (πβ{π£}) β (β‘πβ(πΏβ{((πβπ’) β π )}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmaprnlem1.d | . . . . 5 β’ π· = (BaseβπΆ) | |
2 | hdmaprnlem1.l | . . . . 5 β’ πΏ = (LSpanβπΆ) | |
3 | hdmaprnlem1.h | . . . . . 6 β’ π» = (LHypβπΎ) | |
4 | hdmaprnlem1.c | . . . . . 6 β’ πΆ = ((LCDualβπΎ)βπ) | |
5 | hdmaprnlem1.k | . . . . . 6 β’ (π β (πΎ β HL β§ π β π»)) | |
6 | 3, 4, 5 | lcdlmod 41117 | . . . . 5 β’ (π β πΆ β LMod) |
7 | hdmaprnlem1.u | . . . . . . 7 β’ π = ((DVecHβπΎ)βπ) | |
8 | hdmaprnlem1.v | . . . . . . 7 β’ π = (Baseβπ) | |
9 | hdmaprnlem1.s | . . . . . . 7 β’ π = ((HDMapβπΎ)βπ) | |
10 | hdmaprnlem1.ue | . . . . . . 7 β’ (π β π’ β π) | |
11 | 3, 7, 8, 4, 1, 9, 5, 10 | hdmapcl 41355 | . . . . . 6 β’ (π β (πβπ’) β π·) |
12 | hdmaprnlem1.se | . . . . . . 7 β’ (π β π β (π· β {π})) | |
13 | 12 | eldifad 3953 | . . . . . 6 β’ (π β π β π·) |
14 | hdmaprnlem1.a | . . . . . . 7 β’ β = (+gβπΆ) | |
15 | 1, 14 | lmodvacl 20757 | . . . . . 6 β’ ((πΆ β LMod β§ (πβπ’) β π· β§ π β π·) β ((πβπ’) β π ) β π·) |
16 | 6, 11, 13, 15 | syl3anc 1368 | . . . . 5 β’ (π β ((πβπ’) β π ) β π·) |
17 | eqid 2725 | . . . . . 6 β’ (LSubSpβπΆ) = (LSubSpβπΆ) | |
18 | 1, 17, 2 | lspsncl 20860 | . . . . . . 7 β’ ((πΆ β LMod β§ π β π·) β (πΏβ{π }) β (LSubSpβπΆ)) |
19 | 6, 13, 18 | syl2anc 582 | . . . . . 6 β’ (π β (πΏβ{π }) β (LSubSpβπΆ)) |
20 | 1, 2 | lspsnid 20876 | . . . . . . 7 β’ ((πΆ β LMod β§ π β π·) β π β (πΏβ{π })) |
21 | 6, 13, 20 | syl2anc 582 | . . . . . 6 β’ (π β π β (πΏβ{π })) |
22 | hdmaprnlem1.q | . . . . . . 7 β’ π = (0gβπΆ) | |
23 | 3, 4, 5 | lcdlvec 41116 | . . . . . . 7 β’ (π β πΆ β LVec) |
24 | hdmaprnlem1.o | . . . . . . . 8 β’ 0 = (0gβπ) | |
25 | eqid 2725 | . . . . . . . . 9 β’ (LSubSpβπ) = (LSubSpβπ) | |
26 | 3, 7, 5 | dvhlmod 40635 | . . . . . . . . 9 β’ (π β π β LMod) |
27 | hdmaprnlem1.ve | . . . . . . . . . 10 β’ (π β π£ β π) | |
28 | hdmaprnlem1.n | . . . . . . . . . . 11 β’ π = (LSpanβπ) | |
29 | 8, 25, 28 | lspsncl 20860 | . . . . . . . . . 10 β’ ((π β LMod β§ π£ β π) β (πβ{π£}) β (LSubSpβπ)) |
30 | 26, 27, 29 | syl2anc 582 | . . . . . . . . 9 β’ (π β (πβ{π£}) β (LSubSpβπ)) |
31 | hdmaprnlem1.un | . . . . . . . . 9 β’ (π β Β¬ π’ β (πβ{π£})) | |
32 | 24, 25, 26, 30, 10, 31 | lssneln0 20836 | . . . . . . . 8 β’ (π β π’ β (π β { 0 })) |
33 | 3, 7, 8, 24, 4, 22, 1, 9, 5, 32 | hdmapnzcl 41370 | . . . . . . 7 β’ (π β (πβπ’) β (π· β {π})) |
34 | hdmaprnlem1.m | . . . . . . . 8 β’ π = ((mapdβπΎ)βπ) | |
35 | hdmaprnlem1.e | . . . . . . . 8 β’ (π β (πβ(πβ{π£})) = (πΏβ{π })) | |
36 | 3, 7, 8, 28, 4, 2, 34, 9, 5, 12, 27, 35, 10, 31 | hdmaprnlem1N 41374 | . . . . . . 7 β’ (π β (πΏβ{(πβπ’)}) β (πΏβ{π })) |
37 | 1, 22, 2, 23, 33, 13, 36 | lspsnne1 21004 | . . . . . 6 β’ (π β Β¬ (πβπ’) β (πΏβ{π })) |
38 | 1, 14, 17, 6, 19, 21, 11, 37 | lssvancl2 20829 | . . . . 5 β’ (π β Β¬ ((πβπ’) β π ) β (πΏβ{π })) |
39 | 1, 2, 6, 16, 13, 38 | lspsnne2 21005 | . . . 4 β’ (π β (πΏβ{((πβπ’) β π )}) β (πΏβ{π })) |
40 | 39 | necomd 2986 | . . 3 β’ (π β (πΏβ{π }) β (πΏβ{((πβπ’) β π )})) |
41 | 1, 17, 2 | lspsncl 20860 | . . . . . 6 β’ ((πΆ β LMod β§ ((πβπ’) β π ) β π·) β (πΏβ{((πβπ’) β π )}) β (LSubSpβπΆ)) |
42 | 6, 16, 41 | syl2anc 582 | . . . . 5 β’ (π β (πΏβ{((πβπ’) β π )}) β (LSubSpβπΆ)) |
43 | 3, 34, 4, 17, 5 | mapdrn2 41176 | . . . . 5 β’ (π β ran π = (LSubSpβπΆ)) |
44 | 42, 43 | eleqtrrd 2828 | . . . 4 β’ (π β (πΏβ{((πβπ’) β π )}) β ran π) |
45 | 3, 34, 5, 44 | mapdcnvid2 41182 | . . 3 β’ (π β (πβ(β‘πβ(πΏβ{((πβπ’) β π )}))) = (πΏβ{((πβπ’) β π )})) |
46 | 40, 35, 45 | 3netr4d 3008 | . 2 β’ (π β (πβ(πβ{π£})) β (πβ(β‘πβ(πΏβ{((πβπ’) β π )})))) |
47 | 3, 34, 7, 25, 5, 44 | mapdcnvcl 41177 | . . . 4 β’ (π β (β‘πβ(πΏβ{((πβπ’) β π )})) β (LSubSpβπ)) |
48 | 3, 7, 25, 34, 5, 30, 47 | mapd11 41164 | . . 3 β’ (π β ((πβ(πβ{π£})) = (πβ(β‘πβ(πΏβ{((πβπ’) β π )}))) β (πβ{π£}) = (β‘πβ(πΏβ{((πβπ’) β π )})))) |
49 | 48 | necon3bid 2975 | . 2 β’ (π β ((πβ(πβ{π£})) β (πβ(β‘πβ(πΏβ{((πβπ’) β π )}))) β (πβ{π£}) β (β‘πβ(πΏβ{((πβπ’) β π )})))) |
50 | 46, 49 | mpbid 231 | 1 β’ (π β (πβ{π£}) β (β‘πβ(πΏβ{((πβπ’) β π )}))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 β cdif 3938 {csn 4625 β‘ccnv 5672 ran crn 5674 βcfv 6543 (class class class)co 7413 Basecbs 17174 +gcplusg 17227 0gc0g 17415 LModclmod 20742 LSubSpclss 20814 LSpanclspn 20854 HLchlt 38874 LHypclh 39509 DVecHcdvh 40603 LCDualclcd 41111 mapdcmpd 41149 HDMapchdma 41317 |
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-ot 4634 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-of 7679 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-mre 17560 df-mrc 17561 df-acs 17563 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-submnd 18735 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-cntz 19267 df-oppg 19296 df-lsm 19590 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-lsatoms 38500 df-lshyp 38501 df-lcv 38543 df-lfl 38582 df-lkr 38610 df-ldual 38648 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 df-dvech 40604 df-dib 40664 df-dic 40698 df-dih 40754 df-doch 40873 df-djh 40920 df-lcdual 41112 df-mapd 41150 df-hvmap 41282 df-hdmap1 41318 df-hdmap 41319 |
This theorem is referenced by: hdmaprnlem9N 41382 hdmaprnlem3eN 41383 |
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