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| Mirrors > Home > MPE Home > Th. List > lssvscl | Structured version Visualization version GIF version | ||
| Description: Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
| Ref | Expression |
|---|---|
| lssvscl.f | β’ πΉ = (Scalarβπ) |
| lssvscl.t | β’ Β· = ( Β·π βπ) |
| lssvscl.b | β’ π΅ = (BaseβπΉ) |
| lssvscl.s | β’ π = (LSubSpβπ) |
| Ref | Expression |
|---|---|
| lssvscl | β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β (π Β· π) β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpll 766 | . . 3 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β π β LMod) | |
| 2 | simprl 770 | . . . 4 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β π β π΅) | |
| 3 | eqid 2733 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
| 4 | lssvscl.s | . . . . . 6 β’ π = (LSubSpβπ) | |
| 5 | 3, 4 | lssel 20548 | . . . . 5 β’ ((π β π β§ π β π) β π β (Baseβπ)) |
| 6 | 5 | ad2ant2l 745 | . . . 4 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β π β (Baseβπ)) |
| 7 | lssvscl.f | . . . . 5 β’ πΉ = (Scalarβπ) | |
| 8 | lssvscl.t | . . . . 5 β’ Β· = ( Β·π βπ) | |
| 9 | lssvscl.b | . . . . 5 β’ π΅ = (BaseβπΉ) | |
| 10 | 3, 7, 8, 9 | lmodvscl 20489 | . . . 4 β’ ((π β LMod β§ π β π΅ β§ π β (Baseβπ)) β (π Β· π) β (Baseβπ)) |
| 11 | 1, 2, 6, 10 | syl3anc 1372 | . . 3 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β (π Β· π) β (Baseβπ)) |
| 12 | eqid 2733 | . . . 4 β’ (+gβπ) = (+gβπ) | |
| 13 | eqid 2733 | . . . 4 β’ (0gβπ) = (0gβπ) | |
| 14 | 3, 12, 13 | lmod0vrid 20503 | . . 3 β’ ((π β LMod β§ (π Β· π) β (Baseβπ)) β ((π Β· π)(+gβπ)(0gβπ)) = (π Β· π)) |
| 15 | 1, 11, 14 | syl2anc 585 | . 2 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β ((π Β· π)(+gβπ)(0gβπ)) = (π Β· π)) |
| 16 | simplr 768 | . . 3 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β π β π) | |
| 17 | simprr 772 | . . 3 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β π β π) | |
| 18 | 13, 4 | lss0cl 20557 | . . . 4 β’ ((π β LMod β§ π β π) β (0gβπ) β π) |
| 19 | 18 | adantr 482 | . . 3 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β (0gβπ) β π) |
| 20 | 7, 9, 12, 8, 4 | lsscl 20553 | . . 3 β’ ((π β π β§ (π β π΅ β§ π β π β§ (0gβπ) β π)) β ((π Β· π)(+gβπ)(0gβπ)) β π) |
| 21 | 16, 2, 17, 19, 20 | syl13anc 1373 | . 2 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β ((π Β· π)(+gβπ)(0gβπ)) β π) |
| 22 | 15, 21 | eqeltrrd 2835 | 1 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β (π Β· π) β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βcfv 6544 (class class class)co 7409 Basecbs 17144 +gcplusg 17197 Scalarcsca 17200 Β·π cvsca 17201 0gc0g 17385 LModclmod 20471 LSubSpclss 20542 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-plusg 17210 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mgp 19988 df-ur 20005 df-ring 20058 df-lmod 20473 df-lss 20543 |
| This theorem is referenced by: lssvnegcl 20567 islss3 20570 islss4 20573 lspsneli 20612 lspsn 20613 lmhmima 20658 lmhmpreima 20659 reslmhm 20663 lsmcl 20694 pj1lmhm 20711 lssvs0or 20723 lspfixed 20741 lspexch 20742 lspsolv 20756 lidlmcl 20840 frlmssuvc1 21349 frlmsslsp 21351 mplbas2 21597 lssnlm 24218 minveclem2 24943 pjthlem1 24954 eqgvscpbl 32465 lindsunlem 32709 lshpkrlem5 37984 ldualssvscl 38028 dochkr1 40349 dochkr1OLDN 40350 lclkrlem2o 40392 lcfrlem5 40417 lcdlssvscl 40477 hgmapvvlem3 40796 gsumlsscl 47059 |
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