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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvsub | Structured version Visualization version GIF version |
Description: The value of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.) |
Ref | Expression |
---|---|
ldualvsub.r | β’ π = (Scalarβπ) |
ldualvsub.n | β’ π = (invgβπ ) |
ldualvsub.u | β’ 1 = (1rβπ ) |
ldualvsub.f | β’ πΉ = (LFnlβπ) |
ldualvsub.d | β’ π· = (LDualβπ) |
ldualvsub.p | β’ + = (+gβπ·) |
ldualvsub.t | β’ Β· = ( Β·π βπ·) |
ldualvsub.m | β’ β = (-gβπ·) |
ldualvsub.w | β’ (π β π β LMod) |
ldualvsub.g | β’ (π β πΊ β πΉ) |
ldualvsub.h | β’ (π β π» β πΉ) |
Ref | Expression |
---|---|
ldualvsub | β’ (π β (πΊ β π») = (πΊ + ((πβ 1 ) Β· π»))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualvsub.d | . . . 4 β’ π· = (LDualβπ) | |
2 | ldualvsub.w | . . . 4 β’ (π β π β LMod) | |
3 | 1, 2 | lduallmod 38681 | . . 3 β’ (π β π· β LMod) |
4 | ldualvsub.f | . . . 4 β’ πΉ = (LFnlβπ) | |
5 | eqid 2725 | . . . 4 β’ (Baseβπ·) = (Baseβπ·) | |
6 | ldualvsub.g | . . . 4 β’ (π β πΊ β πΉ) | |
7 | 4, 1, 5, 2, 6 | ldualelvbase 38655 | . . 3 β’ (π β πΊ β (Baseβπ·)) |
8 | ldualvsub.h | . . . 4 β’ (π β π» β πΉ) | |
9 | 4, 1, 5, 2, 8 | ldualelvbase 38655 | . . 3 β’ (π β π» β (Baseβπ·)) |
10 | ldualvsub.p | . . . 4 β’ + = (+gβπ·) | |
11 | ldualvsub.m | . . . 4 β’ β = (-gβπ·) | |
12 | eqid 2725 | . . . 4 β’ (Scalarβπ·) = (Scalarβπ·) | |
13 | ldualvsub.t | . . . 4 β’ Β· = ( Β·π βπ·) | |
14 | eqid 2725 | . . . 4 β’ (invgβ(Scalarβπ·)) = (invgβ(Scalarβπ·)) | |
15 | eqid 2725 | . . . 4 β’ (1rβ(Scalarβπ·)) = (1rβ(Scalarβπ·)) | |
16 | 5, 10, 11, 12, 13, 14, 15 | lmodvsubval2 20804 | . . 3 β’ ((π· β LMod β§ πΊ β (Baseβπ·) β§ π» β (Baseβπ·)) β (πΊ β π») = (πΊ + (((invgβ(Scalarβπ·))β(1rβ(Scalarβπ·))) Β· π»))) |
17 | 3, 7, 9, 16 | syl3anc 1368 | . 2 β’ (π β (πΊ β π») = (πΊ + (((invgβ(Scalarβπ·))β(1rβ(Scalarβπ·))) Β· π»))) |
18 | eqid 2725 | . . . . . . 7 β’ (opprβπ ) = (opprβπ ) | |
19 | ldualvsub.n | . . . . . . 7 β’ π = (invgβπ ) | |
20 | 18, 19 | opprneg 20294 | . . . . . 6 β’ π = (invgβ(opprβπ )) |
21 | ldualvsub.r | . . . . . . . 8 β’ π = (Scalarβπ) | |
22 | 21, 18, 1, 12, 2 | ldualsca 38660 | . . . . . . 7 β’ (π β (Scalarβπ·) = (opprβπ )) |
23 | 22 | fveq2d 6896 | . . . . . 6 β’ (π β (invgβ(Scalarβπ·)) = (invgβ(opprβπ ))) |
24 | 20, 23 | eqtr4id 2784 | . . . . 5 β’ (π β π = (invgβ(Scalarβπ·))) |
25 | ldualvsub.u | . . . . . . 7 β’ 1 = (1rβπ ) | |
26 | 18, 25 | oppr1 20293 | . . . . . 6 β’ 1 = (1rβ(opprβπ )) |
27 | 22 | fveq2d 6896 | . . . . . 6 β’ (π β (1rβ(Scalarβπ·)) = (1rβ(opprβπ ))) |
28 | 26, 27 | eqtr4id 2784 | . . . . 5 β’ (π β 1 = (1rβ(Scalarβπ·))) |
29 | 24, 28 | fveq12d 6899 | . . . 4 β’ (π β (πβ 1 ) = ((invgβ(Scalarβπ·))β(1rβ(Scalarβπ·)))) |
30 | 29 | oveq1d 7431 | . . 3 β’ (π β ((πβ 1 ) Β· π») = (((invgβ(Scalarβπ·))β(1rβ(Scalarβπ·))) Β· π»)) |
31 | 30 | oveq2d 7432 | . 2 β’ (π β (πΊ + ((πβ 1 ) Β· π»)) = (πΊ + (((invgβ(Scalarβπ·))β(1rβ(Scalarβπ·))) Β· π»))) |
32 | 17, 31 | eqtr4d 2768 | 1 β’ (π β (πΊ β π») = (πΊ + ((πβ 1 ) Β· π»))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6543 (class class class)co 7416 Basecbs 17179 +gcplusg 17232 Scalarcsca 17235 Β·π cvsca 17236 invgcminusg 18895 -gcsg 18896 1rcur 20125 opprcoppr 20276 LModclmod 20747 LFnlclfn 38585 LDualcld 38651 |
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-rep 5280 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-tp 4629 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-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 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-3 12306 df-4 12307 df-5 12308 df-6 12309 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-sbg 18899 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-oppr 20277 df-lmod 20749 df-lfl 38586 df-ldual 38652 |
This theorem is referenced by: ldualvsubcl 38684 lcfrlem2 41072 |
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