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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 2737 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
2 | eqid 2737 | . . 3 β’ (invgβ(Scalarβπ)) = (invgβ(Scalarβπ)) | |
3 | eqid 2737 | . . 3 β’ (1rβ(Scalarβπ)) = (1rβ(Scalarβπ)) | |
4 | ldualvsubcl.f | . . 3 β’ πΉ = (LFnlβπ) | |
5 | ldualvsubcl.d | . . 3 β’ π· = (LDualβπ) | |
6 | eqid 2737 | . . 3 β’ (+gβπ·) = (+gβπ·) | |
7 | eqid 2737 | . . 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 37620 | . 2 β’ (π β (πΊ β π») = (πΊ(+gβπ·)(((invgβ(Scalarβπ))β(1rβ(Scalarβπ)))( Β·π βπ·)π»))) |
13 | eqid 2737 | . . . 4 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
14 | 1 | lmodring 20333 | . . . . . . 7 β’ (π β LMod β (Scalarβπ) β Ring) |
15 | 9, 14 | syl 17 | . . . . . 6 β’ (π β (Scalarβπ) β Ring) |
16 | ringgrp 19970 | . . . . . 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 19990 | . . . . . 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 18799 | . . . . 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 585 | . . . 4 β’ (π β ((invgβ(Scalarβπ))β(1rβ(Scalarβπ))) β (Baseβ(Scalarβπ))) |
22 | 4, 1, 13, 5, 7, 9, 21, 11 | ldualvscl 37604 | . . 3 β’ (π β (((invgβ(Scalarβπ))β(1rβ(Scalarβπ)))( Β·π βπ·)π») β πΉ) |
23 | 4, 5, 6, 9, 10, 22 | ldualvaddcl 37595 | . 2 β’ (π β (πΊ(+gβπ·)(((invgβ(Scalarβπ))β(1rβ(Scalarβπ)))( Β·π βπ·)π»)) β πΉ) |
24 | 12, 23 | eqeltrd 2838 | 1 β’ (π β (πΊ β π») β πΉ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 Basecbs 17084 +gcplusg 17134 Scalarcsca 17137 Β·π cvsca 17138 Grpcgrp 18749 invgcminusg 18750 -gcsg 18751 1rcur 19914 Ringcrg 19965 LModclmod 20325 LFnlclfn 37522 LDualcld 37588 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-5 12220 df-6 12221 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 df-struct 17020 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-mulr 17148 df-sca 17150 df-vsca 17151 df-0g 17324 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-grp 18752 df-minusg 18753 df-sbg 18754 df-cmn 19565 df-abl 19566 df-mgp 19898 df-ur 19915 df-ring 19967 df-oppr 20050 df-lmod 20327 df-lfl 37523 df-ldual 37589 |
This theorem is referenced by: lcfrlem3 40010 lcfrlem30 40038 |
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