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| Mirrors > Home > MPE Home > Th. List > lssvnegcl | Structured version Visualization version GIF version | ||
| Description: Closure of negative vectors in a subspace. (Contributed by Stefan O'Rear, 11-Dec-2014.) |
| Ref | Expression |
|---|---|
| lssvnegcl.s | β’ π = (LSubSpβπ) |
| lssvnegcl.n | β’ π = (invgβπ) |
| Ref | Expression |
|---|---|
| lssvnegcl | β’ ((π β LMod β§ π β π β§ π β π) β (πβπ) β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2725 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
| 2 | lssvnegcl.s | . . . . 5 β’ π = (LSubSpβπ) | |
| 3 | 1, 2 | lssel 20825 | . . . 4 β’ ((π β π β§ π β π) β π β (Baseβπ)) |
| 4 | lssvnegcl.n | . . . . 5 β’ π = (invgβπ) | |
| 5 | eqid 2725 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
| 6 | eqid 2725 | . . . . 5 β’ ( Β·π βπ) = ( Β·π βπ) | |
| 7 | eqid 2725 | . . . . 5 β’ (1rβ(Scalarβπ)) = (1rβ(Scalarβπ)) | |
| 8 | eqid 2725 | . . . . 5 β’ (invgβ(Scalarβπ)) = (invgβ(Scalarβπ)) | |
| 9 | 1, 4, 5, 6, 7, 8 | lmodvneg1 20792 | . . . 4 β’ ((π β LMod β§ π β (Baseβπ)) β (((invgβ(Scalarβπ))β(1rβ(Scalarβπ)))( Β·π βπ)π) = (πβπ)) |
| 10 | 3, 9 | sylan2 591 | . . 3 β’ ((π β LMod β§ (π β π β§ π β π)) β (((invgβ(Scalarβπ))β(1rβ(Scalarβπ)))( Β·π βπ)π) = (πβπ)) |
| 11 | 10 | 3impb 1112 | . 2 β’ ((π β LMod β§ π β π β§ π β π) β (((invgβ(Scalarβπ))β(1rβ(Scalarβπ)))( Β·π βπ)π) = (πβπ)) |
| 12 | simp1 1133 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β π β LMod) | |
| 13 | simp2 1134 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β π β π) | |
| 14 | eqid 2725 | . . . 4 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
| 15 | 5 | lmodring 20755 | . . . . . 6 β’ (π β LMod β (Scalarβπ) β Ring) |
| 16 | 15 | 3ad2ant1 1130 | . . . . 5 β’ ((π β LMod β§ π β π β§ π β π) β (Scalarβπ) β Ring) |
| 17 | 16 | ringgrpd 20186 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β (Scalarβπ) β Grp) |
| 18 | 14, 7 | ringidcl 20206 | . . . . 5 β’ ((Scalarβπ) β Ring β (1rβ(Scalarβπ)) β (Baseβ(Scalarβπ))) |
| 19 | 16, 18 | syl 17 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β (1rβ(Scalarβπ)) β (Baseβ(Scalarβπ))) |
| 20 | 14, 8, 17, 19 | grpinvcld 18949 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β ((invgβ(Scalarβπ))β(1rβ(Scalarβπ))) β (Baseβ(Scalarβπ))) |
| 21 | simp3 1135 | . . 3 β’ ((π β LMod β§ π β π β§ π β π) β π β π) | |
| 22 | 5, 6, 14, 2 | lssvscl 20843 | . . 3 β’ (((π β LMod β§ π β π) β§ (((invgβ(Scalarβπ))β(1rβ(Scalarβπ))) β (Baseβ(Scalarβπ)) β§ π β π)) β (((invgβ(Scalarβπ))β(1rβ(Scalarβπ)))( Β·π βπ)π) β π) |
| 23 | 12, 13, 20, 21, 22 | syl22anc 837 | . 2 β’ ((π β LMod β§ π β π β§ π β π) β (((invgβ(Scalarβπ))β(1rβ(Scalarβπ)))( Β·π βπ)π) β π) |
| 24 | 11, 23 | eqeltrrd 2826 | 1 β’ ((π β LMod β§ π β π β§ π β π) β (πβπ) β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6543 (class class class)co 7416 Basecbs 17179 Scalarcsca 17235 Β·π cvsca 17236 invgcminusg 18895 1rcur 20125 Ringcrg 20177 LModclmod 20747 LSubSpclss 20819 |
| 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-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
| 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 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 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 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mgp 20079 df-ur 20126 df-ring 20179 df-lmod 20749 df-lss 20820 |
| This theorem is referenced by: lsssubg 20845 lidlnegcl 21122 mapdpglem14 41214 baerlem3lem1 41236 baerlem5amN 41245 baerlem5bmN 41246 baerlem5abmN 41247 |
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