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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvsdi1 | Structured version Visualization version GIF version |
Description: Distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.) |
Ref | Expression |
---|---|
ldualvsdi1.f | β’ πΉ = (LFnlβπ) |
ldualvsdi1.r | β’ π = (Scalarβπ) |
ldualvsdi1.k | β’ πΎ = (Baseβπ ) |
ldualvsdi1.d | β’ π· = (LDualβπ) |
ldualvsdi1.p | β’ + = (+gβπ·) |
ldualvsdi1.s | β’ Β· = ( Β·π βπ·) |
ldualvsdi1.w | β’ (π β π β LMod) |
ldualvsdi1.x | β’ (π β π β πΎ) |
ldualvsdi1.g | β’ (π β πΊ β πΉ) |
ldualvsdi1.h | β’ (π β π» β πΉ) |
Ref | Expression |
---|---|
ldualvsdi1 | β’ (π β (π Β· (πΊ + π»)) = ((π Β· πΊ) + (π Β· π»))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualvsdi1.f | . . . 4 β’ πΉ = (LFnlβπ) | |
2 | eqid 2731 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
3 | ldualvsdi1.r | . . . 4 β’ π = (Scalarβπ) | |
4 | ldualvsdi1.k | . . . 4 β’ πΎ = (Baseβπ ) | |
5 | eqid 2731 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
6 | ldualvsdi1.d | . . . 4 β’ π· = (LDualβπ) | |
7 | ldualvsdi1.s | . . . 4 β’ Β· = ( Β·π βπ·) | |
8 | ldualvsdi1.w | . . . 4 β’ (π β π β LMod) | |
9 | ldualvsdi1.x | . . . 4 β’ (π β π β πΎ) | |
10 | ldualvsdi1.g | . . . 4 β’ (π β πΊ β πΉ) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | ldualvs 38311 | . . 3 β’ (π β (π Β· πΊ) = (πΊ βf (.rβπ )((Baseβπ) Γ {π}))) |
12 | ldualvsdi1.h | . . . 4 β’ (π β π» β πΉ) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 12 | ldualvs 38311 | . . 3 β’ (π β (π Β· π») = (π» βf (.rβπ )((Baseβπ) Γ {π}))) |
14 | 11, 13 | oveq12d 7430 | . 2 β’ (π β ((π Β· πΊ) βf (+gβπ )(π Β· π»)) = ((πΊ βf (.rβπ )((Baseβπ) Γ {π})) βf (+gβπ )(π» βf (.rβπ )((Baseβπ) Γ {π})))) |
15 | eqid 2731 | . . 3 β’ (+gβπ ) = (+gβπ ) | |
16 | ldualvsdi1.p | . . 3 β’ + = (+gβπ·) | |
17 | 1, 3, 4, 6, 7, 8, 9, 10 | ldualvscl 38313 | . . 3 β’ (π β (π Β· πΊ) β πΉ) |
18 | 1, 3, 4, 6, 7, 8, 9, 12 | ldualvscl 38313 | . . 3 β’ (π β (π Β· π») β πΉ) |
19 | 1, 3, 15, 6, 16, 8, 17, 18 | ldualvadd 38303 | . 2 β’ (π β ((π Β· πΊ) + (π Β· π»)) = ((π Β· πΊ) βf (+gβπ )(π Β· π»))) |
20 | 1, 6, 16, 8, 10, 12 | ldualvaddcl 38304 | . . . 4 β’ (π β (πΊ + π») β πΉ) |
21 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 20 | ldualvs 38311 | . . 3 β’ (π β (π Β· (πΊ + π»)) = ((πΊ + π») βf (.rβπ )((Baseβπ) Γ {π}))) |
22 | 1, 3, 15, 6, 16, 8, 10, 12 | ldualvadd 38303 | . . . 4 β’ (π β (πΊ + π») = (πΊ βf (+gβπ )π»)) |
23 | 22 | oveq1d 7427 | . . 3 β’ (π β ((πΊ + π») βf (.rβπ )((Baseβπ) Γ {π})) = ((πΊ βf (+gβπ )π») βf (.rβπ )((Baseβπ) Γ {π}))) |
24 | 2, 3, 4, 15, 5, 1, 8, 9, 10, 12 | lflvsdi1 38252 | . . 3 β’ (π β ((πΊ βf (+gβπ )π») βf (.rβπ )((Baseβπ) Γ {π})) = ((πΊ βf (.rβπ )((Baseβπ) Γ {π})) βf (+gβπ )(π» βf (.rβπ )((Baseβπ) Γ {π})))) |
25 | 21, 23, 24 | 3eqtrd 2775 | . 2 β’ (π β (π Β· (πΊ + π»)) = ((πΊ βf (.rβπ )((Baseβπ) Γ {π})) βf (+gβπ )(π» βf (.rβπ )((Baseβπ) Γ {π})))) |
26 | 14, 19, 25 | 3eqtr4rd 2782 | 1 β’ (π β (π Β· (πΊ + π»)) = ((π Β· πΊ) + (π Β· π»))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 {csn 4629 Γ cxp 5675 βcfv 6544 (class class class)co 7412 βf cof 7671 Basecbs 17149 +gcplusg 17202 .rcmulr 17203 Scalarcsca 17205 Β·π cvsca 17206 LModclmod 20615 LFnlclfn 38231 LDualcld 38297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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-tp 4634 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-sca 17218 df-vsca 17219 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-minusg 18860 df-sbg 18861 df-cmn 19692 df-abl 19693 df-mgp 20030 df-ur 20077 df-ring 20130 df-lmod 20617 df-lfl 38232 df-ldual 38298 |
This theorem is referenced by: lduallmodlem 38326 |
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