![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 6860 | . . 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 6735 | . . 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 37575 | . . 3 β’ ((π β LMod β§ πΊ β πΉ) β πΊ:πβΆπΎ) |
13 | 7, 8, 12 | syl2anc 585 | . 2 β’ (π β πΊ:πβΆπΎ) |
14 | lfldi1.h | . . 3 β’ (π β π» β πΉ) | |
15 | 9, 10, 1, 11 | lflf 37575 | . . 3 β’ ((π β LMod β§ π» β πΉ) β π»:πβΆπΎ) |
16 | 7, 14, 15 | syl2anc 585 | . 2 β’ (π β π»:πβΆπΎ) |
17 | 9 | lmodring 20373 | . . . 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 19996 | . . 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 581 | . 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 7661 | 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 1088 = wceq 1542 β wcel 2107 Vcvv 3447 {csn 4590 Γ cxp 5635 βΆwf 6496 βcfv 6500 (class class class)co 7361 βf cof 7619 Basecbs 17091 +gcplusg 17141 .rcmulr 17142 Scalarcsca 17144 Ringcrg 19972 LModclmod 20365 LFnlclfn 37569 |
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 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-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-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 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-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-map 8773 df-ring 19974 df-lmod 20367 df-lfl 37570 |
This theorem is referenced by: ldualvsdi1 37655 |
Copyright terms: Public domain | W3C validator |