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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 2726 | . . . . 5 β’ (Baseβπ) = (Baseβπ) | |
| 2 | lssvnegcl.s | . . . . 5 β’ π = (LSubSpβπ) | |
| 3 | 1, 2 | lssel 20784 | . . . 4 β’ ((π β π β§ π β π) β π β (Baseβπ)) |
| 4 | lssvnegcl.n | . . . . 5 β’ π = (invgβπ) | |
| 5 | eqid 2726 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
| 6 | eqid 2726 | . . . . 5 β’ ( Β·π βπ) = ( Β·π βπ) | |
| 7 | eqid 2726 | . . . . 5 β’ (1rβ(Scalarβπ)) = (1rβ(Scalarβπ)) | |
| 8 | eqid 2726 | . . . . 5 β’ (invgβ(Scalarβπ)) = (invgβ(Scalarβπ)) | |
| 9 | 1, 4, 5, 6, 7, 8 | lmodvneg1 20751 | . . . 4 β’ ((π β LMod β§ π β (Baseβπ)) β (((invgβ(Scalarβπ))β(1rβ(Scalarβπ)))( Β·π βπ)π) = (πβπ)) |
| 10 | 3, 9 | sylan2 592 | . . 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 2726 | . . . 4 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
| 15 | 5 | lmodring 20714 | . . . . . 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 20147 | . . . 4 β’ ((π β LMod β§ π β π β§ π β π) β (Scalarβπ) β Grp) |
| 18 | 14, 7 | ringidcl 20165 | . . . . 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 18918 | . . 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 20802 | . . 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 836 | . 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 2828 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6537 (class class class)co 7405 Basecbs 17153 Scalarcsca 17209 Β·π cvsca 17210 invgcminusg 18864 1rcur 20086 Ringcrg 20138 LModclmod 20706 LSubSpclss 20778 |
| 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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mgp 20040 df-ur 20087 df-ring 20140 df-lmod 20708 df-lss 20779 |
| This theorem is referenced by: lsssubg 20804 lidlnegcl 21081 mapdpglem14 41069 baerlem3lem1 41091 baerlem5amN 41100 baerlem5bmN 41101 baerlem5abmN 41102 |
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