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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvaddval | Structured version Visualization version GIF version |
Description: The value of the value of vector addition in the dual of a vector space. (Contributed by NM, 7-Jan-2015.) |
Ref | Expression |
---|---|
ldualvaddval.v | β’ π = (Baseβπ) |
ldualvaddval.r | β’ π = (Scalarβπ) |
ldualvaddval.a | β’ + = (+gβπ ) |
ldualvaddval.f | β’ πΉ = (LFnlβπ) |
ldualvaddval.d | β’ π· = (LDualβπ) |
ldualvaddval.p | β’ β = (+gβπ·) |
ldualvaddval.w | β’ (π β π β LMod) |
ldualvaddval.g | β’ (π β πΊ β πΉ) |
ldualvaddval.h | β’ (π β π» β πΉ) |
ldualvaddval.x | β’ (π β π β π) |
Ref | Expression |
---|---|
ldualvaddval | β’ (π β ((πΊ β π»)βπ) = ((πΊβπ) + (π»βπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualvaddval.f | . . . 4 β’ πΉ = (LFnlβπ) | |
2 | ldualvaddval.r | . . . 4 β’ π = (Scalarβπ) | |
3 | ldualvaddval.a | . . . 4 β’ + = (+gβπ ) | |
4 | ldualvaddval.d | . . . 4 β’ π· = (LDualβπ) | |
5 | ldualvaddval.p | . . . 4 β’ β = (+gβπ·) | |
6 | ldualvaddval.w | . . . 4 β’ (π β π β LMod) | |
7 | ldualvaddval.g | . . . 4 β’ (π β πΊ β πΉ) | |
8 | ldualvaddval.h | . . . 4 β’ (π β π» β πΉ) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | ldualvadd 37641 | . . 3 β’ (π β (πΊ β π») = (πΊ βf + π»)) |
10 | 9 | fveq1d 6848 | . 2 β’ (π β ((πΊ β π»)βπ) = ((πΊ βf + π»)βπ)) |
11 | ldualvaddval.x | . . 3 β’ (π β π β π) | |
12 | eqid 2733 | . . . . . . 7 β’ (Baseβπ ) = (Baseβπ ) | |
13 | ldualvaddval.v | . . . . . . 7 β’ π = (Baseβπ) | |
14 | 2, 12, 13, 1 | lflf 37575 | . . . . . 6 β’ ((π β LMod β§ πΊ β πΉ) β πΊ:πβΆ(Baseβπ )) |
15 | 14 | ffnd 6673 | . . . . 5 β’ ((π β LMod β§ πΊ β πΉ) β πΊ Fn π) |
16 | 6, 7, 15 | syl2anc 585 | . . . 4 β’ (π β πΊ Fn π) |
17 | 2, 12, 13, 1 | lflf 37575 | . . . . . 6 β’ ((π β LMod β§ π» β πΉ) β π»:πβΆ(Baseβπ )) |
18 | 17 | ffnd 6673 | . . . . 5 β’ ((π β LMod β§ π» β πΉ) β π» Fn π) |
19 | 6, 8, 18 | syl2anc 585 | . . . 4 β’ (π β π» Fn π) |
20 | 13 | fvexi 6860 | . . . . 5 β’ π β V |
21 | 20 | a1i 11 | . . . 4 β’ (π β π β V) |
22 | inidm 4182 | . . . 4 β’ (π β© π) = π | |
23 | eqidd 2734 | . . . 4 β’ ((π β§ π β π) β (πΊβπ) = (πΊβπ)) | |
24 | eqidd 2734 | . . . 4 β’ ((π β§ π β π) β (π»βπ) = (π»βπ)) | |
25 | 16, 19, 21, 21, 22, 23, 24 | ofval 7632 | . . 3 β’ ((π β§ π β π) β ((πΊ βf + π»)βπ) = ((πΊβπ) + (π»βπ))) |
26 | 11, 25 | mpdan 686 | . 2 β’ (π β ((πΊ βf + π»)βπ) = ((πΊβπ) + (π»βπ))) |
27 | 10, 26 | eqtrd 2773 | 1 β’ (π β ((πΊ β π»)βπ) = ((πΊβπ) + (π»βπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3447 Fn wfn 6495 βcfv 6500 (class class class)co 7361 βf cof 7619 Basecbs 17091 +gcplusg 17141 Scalarcsca 17144 LModclmod 20365 LFnlclfn 37569 LDualcld 37635 |
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-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-struct 17027 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-sca 17157 df-vsca 17158 df-lfl 37570 df-ldual 37636 |
This theorem is referenced by: ldualvsubval 37669 lkrin 37676 lcdvaddval 40111 |
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