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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 38536 | . . 3 β’ (π β π· β LMod) |
4 | ldualvsub.f | . . . 4 β’ πΉ = (LFnlβπ) | |
5 | eqid 2726 | . . . 4 β’ (Baseβπ·) = (Baseβπ·) | |
6 | ldualvsub.g | . . . 4 β’ (π β πΊ β πΉ) | |
7 | 4, 1, 5, 2, 6 | ldualelvbase 38510 | . . 3 β’ (π β πΊ β (Baseβπ·)) |
8 | ldualvsub.h | . . . 4 β’ (π β π» β πΉ) | |
9 | 4, 1, 5, 2, 8 | ldualelvbase 38510 | . . 3 β’ (π β π» β (Baseβπ·)) |
10 | ldualvsub.p | . . . 4 β’ + = (+gβπ·) | |
11 | ldualvsub.m | . . . 4 β’ β = (-gβπ·) | |
12 | eqid 2726 | . . . 4 β’ (Scalarβπ·) = (Scalarβπ·) | |
13 | ldualvsub.t | . . . 4 β’ Β· = ( Β·π βπ·) | |
14 | eqid 2726 | . . . 4 β’ (invgβ(Scalarβπ·)) = (invgβ(Scalarβπ·)) | |
15 | eqid 2726 | . . . 4 β’ (1rβ(Scalarβπ·)) = (1rβ(Scalarβπ·)) | |
16 | 5, 10, 11, 12, 13, 14, 15 | lmodvsubval2 20763 | . . 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 2726 | . . . . . . 7 β’ (opprβπ ) = (opprβπ ) | |
19 | ldualvsub.n | . . . . . . 7 β’ π = (invgβπ ) | |
20 | 18, 19 | opprneg 20253 | . . . . . 6 β’ π = (invgβ(opprβπ )) |
21 | ldualvsub.r | . . . . . . . 8 β’ π = (Scalarβπ) | |
22 | 21, 18, 1, 12, 2 | ldualsca 38515 | . . . . . . 7 β’ (π β (Scalarβπ·) = (opprβπ )) |
23 | 22 | fveq2d 6889 | . . . . . 6 β’ (π β (invgβ(Scalarβπ·)) = (invgβ(opprβπ ))) |
24 | 20, 23 | eqtr4id 2785 | . . . . 5 β’ (π β π = (invgβ(Scalarβπ·))) |
25 | ldualvsub.u | . . . . . . 7 β’ 1 = (1rβπ ) | |
26 | 18, 25 | oppr1 20252 | . . . . . 6 β’ 1 = (1rβ(opprβπ )) |
27 | 22 | fveq2d 6889 | . . . . . 6 β’ (π β (1rβ(Scalarβπ·)) = (1rβ(opprβπ ))) |
28 | 26, 27 | eqtr4id 2785 | . . . . 5 β’ (π β 1 = (1rβ(Scalarβπ·))) |
29 | 24, 28 | fveq12d 6892 | . . . 4 β’ (π β (πβ 1 ) = ((invgβ(Scalarβπ·))β(1rβ(Scalarβπ·)))) |
30 | 29 | oveq1d 7420 | . . 3 β’ (π β ((πβ 1 ) Β· π») = (((invgβ(Scalarβπ·))β(1rβ(Scalarβπ·))) Β· π»)) |
31 | 30 | oveq2d 7421 | . 2 β’ (π β (πΊ + ((πβ 1 ) Β· π»)) = (πΊ + (((invgβ(Scalarβπ·))β(1rβ(Scalarβπ·))) Β· π»))) |
32 | 17, 31 | eqtr4d 2769 | 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 6537 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 Scalarcsca 17209 Β·π cvsca 17210 invgcminusg 18864 -gcsg 18865 1rcur 20086 opprcoppr 20235 LModclmod 20706 LFnlclfn 38440 LDualcld 38506 |
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-rep 5278 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-tp 4628 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-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 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-3 12280 df-4 12281 df-5 12282 df-6 12283 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-sbg 18868 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-oppr 20236 df-lmod 20708 df-lfl 38441 df-ldual 38507 |
This theorem is referenced by: ldualvsubcl 38539 lcfrlem2 40927 |
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