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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvsubcl | Structured version Visualization version GIF version |
Description: Closure of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.) |
Ref | Expression |
---|---|
ldualvsubcl.f | β’ πΉ = (LFnlβπ) |
ldualvsubcl.d | β’ π· = (LDualβπ) |
ldualvsubcl.m | β’ β = (-gβπ·) |
ldualvsubcl.w | β’ (π β π β LMod) |
ldualvsubcl.g | β’ (π β πΊ β πΉ) |
ldualvsubcl.h | β’ (π β π» β πΉ) |
Ref | Expression |
---|---|
ldualvsubcl | β’ (π β (πΊ β π») β πΉ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
2 | eqid 2725 | . . 3 β’ (invgβ(Scalarβπ)) = (invgβ(Scalarβπ)) | |
3 | eqid 2725 | . . 3 β’ (1rβ(Scalarβπ)) = (1rβ(Scalarβπ)) | |
4 | ldualvsubcl.f | . . 3 β’ πΉ = (LFnlβπ) | |
5 | ldualvsubcl.d | . . 3 β’ π· = (LDualβπ) | |
6 | eqid 2725 | . . 3 β’ (+gβπ·) = (+gβπ·) | |
7 | eqid 2725 | . . 3 β’ ( Β·π βπ·) = ( Β·π βπ·) | |
8 | ldualvsubcl.m | . . 3 β’ β = (-gβπ·) | |
9 | ldualvsubcl.w | . . 3 β’ (π β π β LMod) | |
10 | ldualvsubcl.g | . . 3 β’ (π β πΊ β πΉ) | |
11 | ldualvsubcl.h | . . 3 β’ (π β π» β πΉ) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | ldualvsub 38682 | . 2 β’ (π β (πΊ β π») = (πΊ(+gβπ·)(((invgβ(Scalarβπ))β(1rβ(Scalarβπ)))( Β·π βπ·)π»))) |
13 | eqid 2725 | . . . 4 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
14 | 1 | lmodring 20753 | . . . . . . 7 β’ (π β LMod β (Scalarβπ) β Ring) |
15 | 9, 14 | syl 17 | . . . . . 6 β’ (π β (Scalarβπ) β Ring) |
16 | ringgrp 20180 | . . . . . 6 β’ ((Scalarβπ) β Ring β (Scalarβπ) β Grp) | |
17 | 15, 16 | syl 17 | . . . . 5 β’ (π β (Scalarβπ) β Grp) |
18 | 13, 3 | ringidcl 20204 | . . . . . 6 β’ ((Scalarβπ) β Ring β (1rβ(Scalarβπ)) β (Baseβ(Scalarβπ))) |
19 | 15, 18 | syl 17 | . . . . 5 β’ (π β (1rβ(Scalarβπ)) β (Baseβ(Scalarβπ))) |
20 | 13, 2 | grpinvcl 18946 | . . . . 5 β’ (((Scalarβπ) β Grp β§ (1rβ(Scalarβπ)) β (Baseβ(Scalarβπ))) β ((invgβ(Scalarβπ))β(1rβ(Scalarβπ))) β (Baseβ(Scalarβπ))) |
21 | 17, 19, 20 | syl2anc 582 | . . . 4 β’ (π β ((invgβ(Scalarβπ))β(1rβ(Scalarβπ))) β (Baseβ(Scalarβπ))) |
22 | 4, 1, 13, 5, 7, 9, 21, 11 | ldualvscl 38666 | . . 3 β’ (π β (((invgβ(Scalarβπ))β(1rβ(Scalarβπ)))( Β·π βπ·)π») β πΉ) |
23 | 4, 5, 6, 9, 10, 22 | ldualvaddcl 38657 | . 2 β’ (π β (πΊ(+gβπ·)(((invgβ(Scalarβπ))β(1rβ(Scalarβπ)))( Β·π βπ·)π»)) β πΉ) |
24 | 12, 23 | eqeltrd 2825 | 1 β’ (π β (πΊ β π») β πΉ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6542 (class class class)co 7415 Basecbs 17177 +gcplusg 17230 Scalarcsca 17233 Β·π cvsca 17234 Grpcgrp 18892 invgcminusg 18893 -gcsg 18894 1rcur 20123 Ringcrg 20175 LModclmod 20745 LFnlclfn 38584 LDualcld 38650 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-0g 17420 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-minusg 18896 df-sbg 18897 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-oppr 20275 df-lmod 20747 df-lfl 38585 df-ldual 38651 |
This theorem is referenced by: lcfrlem3 41072 lcfrlem30 41100 |
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