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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lflvsdi1 | Structured version Visualization version GIF version |
Description: Distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.) |
Ref | Expression |
---|---|
lfldi.v | β’ π = (Baseβπ) |
lfldi.r | β’ π = (Scalarβπ) |
lfldi.k | β’ πΎ = (Baseβπ ) |
lfldi.p | β’ + = (+gβπ ) |
lfldi.t | β’ Β· = (.rβπ ) |
lfldi.f | β’ πΉ = (LFnlβπ) |
lfldi.w | β’ (π β π β LMod) |
lfldi.x | β’ (π β π β πΎ) |
lfldi1.g | β’ (π β πΊ β πΉ) |
lfldi1.h | β’ (π β π» β πΉ) |
Ref | Expression |
---|---|
lflvsdi1 | β’ (π β ((πΊ βf + π») βf Β· (π Γ {π})) = ((πΊ βf Β· (π Γ {π})) βf + (π» βf Β· (π Γ {π})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lfldi.v | . . . 4 β’ π = (Baseβπ) | |
2 | 1 | fvexi 6911 | . . 3 β’ π β V |
3 | 2 | a1i 11 | . 2 β’ (π β π β V) |
4 | lfldi.x | . . 3 β’ (π β π β πΎ) | |
5 | fconst6g 6786 | . . 3 β’ (π β πΎ β (π Γ {π}):πβΆπΎ) | |
6 | 4, 5 | syl 17 | . 2 β’ (π β (π Γ {π}):πβΆπΎ) |
7 | lfldi.w | . . 3 β’ (π β π β LMod) | |
8 | lfldi1.g | . . 3 β’ (π β πΊ β πΉ) | |
9 | lfldi.r | . . . 4 β’ π = (Scalarβπ) | |
10 | lfldi.k | . . . 4 β’ πΎ = (Baseβπ ) | |
11 | lfldi.f | . . . 4 β’ πΉ = (LFnlβπ) | |
12 | 9, 10, 1, 11 | lflf 38535 | . . 3 β’ ((π β LMod β§ πΊ β πΉ) β πΊ:πβΆπΎ) |
13 | 7, 8, 12 | syl2anc 583 | . 2 β’ (π β πΊ:πβΆπΎ) |
14 | lfldi1.h | . . 3 β’ (π β π» β πΉ) | |
15 | 9, 10, 1, 11 | lflf 38535 | . . 3 β’ ((π β LMod β§ π» β πΉ) β π»:πβΆπΎ) |
16 | 7, 14, 15 | syl2anc 583 | . 2 β’ (π β π»:πβΆπΎ) |
17 | 9 | lmodring 20750 | . . . 4 β’ (π β LMod β π β Ring) |
18 | 7, 17 | syl 17 | . . 3 β’ (π β π β Ring) |
19 | lfldi.p | . . . 4 β’ + = (+gβπ ) | |
20 | lfldi.t | . . . 4 β’ Β· = (.rβπ ) | |
21 | 10, 19, 20 | ringdir 20200 | . . 3 β’ ((π β Ring β§ (π₯ β πΎ β§ π¦ β πΎ β§ π§ β πΎ)) β ((π₯ + π¦) Β· π§) = ((π₯ Β· π§) + (π¦ Β· π§))) |
22 | 18, 21 | sylan 579 | . 2 β’ ((π β§ (π₯ β πΎ β§ π¦ β πΎ β§ π§ β πΎ)) β ((π₯ + π¦) Β· π§) = ((π₯ Β· π§) + (π¦ Β· π§))) |
23 | 3, 6, 13, 16, 22 | caofdir 7725 | 1 β’ (π β ((πΊ βf + π») βf Β· (π Γ {π})) = ((πΊ βf Β· (π Γ {π})) βf + (π» βf Β· (π Γ {π})))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 Vcvv 3471 {csn 4629 Γ cxp 5676 βΆwf 6544 βcfv 6548 (class class class)co 7420 βf cof 7683 Basecbs 17179 +gcplusg 17232 .rcmulr 17233 Scalarcsca 17235 Ringcrg 20172 LModclmod 20742 LFnlclfn 38529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-map 8846 df-ring 20174 df-lmod 20744 df-lfl 38530 |
This theorem is referenced by: ldualvsdi1 38615 |
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