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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lflvsdi2 | Structured version Visualization version GIF version |
Description: Reverse 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) |
lfldi2.y | β’ (π β π β πΎ) |
lfldi2.g | β’ (π β πΊ β πΉ) |
Ref | Expression |
---|---|
lflvsdi2 | β’ (π β (πΊ βf Β· ((π Γ {π}) βf + (π Γ {π}))) = ((πΊ βf Β· (π Γ {π})) βf + (πΊ βf Β· (π Γ {π})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lfldi.v | . . . 4 β’ π = (Baseβπ) | |
2 | 1 | fvexi 6896 | . . 3 β’ π β V |
3 | 2 | a1i 11 | . 2 β’ (π β π β V) |
4 | lfldi.w | . . 3 β’ (π β π β LMod) | |
5 | lfldi2.g | . . 3 β’ (π β πΊ β πΉ) | |
6 | lfldi.r | . . . 4 β’ π = (Scalarβπ) | |
7 | lfldi.k | . . . 4 β’ πΎ = (Baseβπ ) | |
8 | lfldi.f | . . . 4 β’ πΉ = (LFnlβπ) | |
9 | 6, 7, 1, 8 | lflf 38436 | . . 3 β’ ((π β LMod β§ πΊ β πΉ) β πΊ:πβΆπΎ) |
10 | 4, 5, 9 | syl2anc 583 | . 2 β’ (π β πΊ:πβΆπΎ) |
11 | lfldi.x | . . 3 β’ (π β π β πΎ) | |
12 | fconst6g 6771 | . . 3 β’ (π β πΎ β (π Γ {π}):πβΆπΎ) | |
13 | 11, 12 | syl 17 | . 2 β’ (π β (π Γ {π}):πβΆπΎ) |
14 | lfldi2.y | . . 3 β’ (π β π β πΎ) | |
15 | fconst6g 6771 | . . 3 β’ (π β πΎ β (π Γ {π}):πβΆπΎ) | |
16 | 14, 15 | syl 17 | . 2 β’ (π β (π Γ {π}):πβΆπΎ) |
17 | 6 | lmodring 20710 | . . . 4 β’ (π β LMod β π β Ring) |
18 | 4, 17 | syl 17 | . . 3 β’ (π β π β Ring) |
19 | lfldi.p | . . . 4 β’ + = (+gβπ ) | |
20 | lfldi.t | . . . 4 β’ Β· = (.rβπ ) | |
21 | 7, 19, 20 | ringdi 20159 | . . 3 β’ ((π β Ring β§ (π₯ β πΎ β§ π¦ β πΎ β§ π§ β πΎ)) β (π₯ Β· (π¦ + 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₯ Β· (π¦ + π§)) = ((π₯ Β· π¦) + (π₯ Β· π§))) |
23 | 3, 10, 13, 16, 22 | caofdi 7703 | 1 β’ (π β (πΊ βf Β· ((π Γ {π}) βf + (π Γ {π}))) = ((πΊ βf Β· (π Γ {π})) βf + (πΊ βf Β· (π Γ {π})))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3466 {csn 4621 Γ cxp 5665 βΆwf 6530 βcfv 6534 (class class class)co 7402 βf cof 7662 Basecbs 17149 +gcplusg 17202 .rcmulr 17203 Scalarcsca 17205 Ringcrg 20134 LModclmod 20702 LFnlclfn 38430 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-map 8819 df-ring 20136 df-lmod 20704 df-lfl 38431 |
This theorem is referenced by: lflvsdi2a 38453 |
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