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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lflvsass | Structured version Visualization version GIF version |
Description: Associative law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.) |
Ref | Expression |
---|---|
lflass.v | β’ π = (Baseβπ) |
lflass.r | β’ π = (Scalarβπ) |
lflass.k | β’ πΎ = (Baseβπ ) |
lflass.t | β’ Β· = (.rβπ ) |
lflass.f | β’ πΉ = (LFnlβπ) |
lflass.w | β’ (π β π β LMod) |
lflass.x | β’ (π β π β πΎ) |
lflass.y | β’ (π β π β πΎ) |
lflass.g | β’ (π β πΊ β πΉ) |
Ref | Expression |
---|---|
lflvsass | β’ (π β (πΊ βf Β· (π Γ {(π Β· π)})) = ((πΊ βf Β· (π Γ {π})) βf Β· (π Γ {π}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lflass.v | . . . . 5 β’ π = (Baseβπ) | |
2 | 1 | fvexi 6896 | . . . 4 β’ π β V |
3 | 2 | a1i 11 | . . 3 β’ (π β π β V) |
4 | lflass.w | . . . 4 β’ (π β π β LMod) | |
5 | lflass.g | . . . 4 β’ (π β πΊ β πΉ) | |
6 | lflass.r | . . . . 5 β’ π = (Scalarβπ) | |
7 | lflass.k | . . . . 5 β’ πΎ = (Baseβπ ) | |
8 | lflass.f | . . . . 5 β’ πΉ = (LFnlβπ) | |
9 | 6, 7, 1, 8 | lflf 38436 | . . . 4 β’ ((π β LMod β§ πΊ β πΉ) β πΊ:πβΆπΎ) |
10 | 4, 5, 9 | syl2anc 583 | . . 3 β’ (π β πΊ:πβΆπΎ) |
11 | lflass.x | . . . 4 β’ (π β π β πΎ) | |
12 | fconst6g 6771 | . . . 4 β’ (π β πΎ β (π Γ {π}):πβΆπΎ) | |
13 | 11, 12 | syl 17 | . . 3 β’ (π β (π Γ {π}):πβΆπΎ) |
14 | lflass.y | . . . 4 β’ (π β π β πΎ) | |
15 | fconst6g 6771 | . . . 4 β’ (π β πΎ β (π Γ {π}):πβΆπΎ) | |
16 | 14, 15 | syl 17 | . . 3 β’ (π β (π Γ {π}):πβΆπΎ) |
17 | 6 | lmodring 20710 | . . . . 5 β’ (π β LMod β π β Ring) |
18 | 4, 17 | syl 17 | . . . 4 β’ (π β π β Ring) |
19 | lflass.t | . . . . 5 β’ Β· = (.rβπ ) | |
20 | 7, 19 | ringass 20154 | . . . 4 β’ ((π β Ring β§ (π₯ β πΎ β§ π¦ β πΎ β§ π§ β πΎ)) β ((π₯ Β· π¦) Β· π§) = (π₯ Β· (π¦ Β· π§))) |
21 | 18, 20 | sylan 579 | . . 3 β’ ((π β§ (π₯ β πΎ β§ π¦ β πΎ β§ π§ β πΎ)) β ((π₯ Β· π¦) Β· π§) = (π₯ Β· (π¦ Β· π§))) |
22 | 3, 10, 13, 16, 21 | caofass 7701 | . 2 β’ (π β ((πΊ βf Β· (π Γ {π})) βf Β· (π Γ {π})) = (πΊ βf Β· ((π Γ {π}) βf Β· (π Γ {π})))) |
23 | 3, 11, 14 | ofc12 7692 | . . 3 β’ (π β ((π Γ {π}) βf Β· (π Γ {π})) = (π Γ {(π Β· π)})) |
24 | 23 | oveq2d 7418 | . 2 β’ (π β (πΊ βf Β· ((π Γ {π}) βf Β· (π Γ {π}))) = (πΊ βf Β· (π Γ {(π Β· π)}))) |
25 | 22, 24 | eqtr2d 2765 | 1 β’ (π β (πΊ βf Β· (π Γ {(π Β· π)})) = ((πΊ βf Β· (π Γ {π})) 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 .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 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 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-pss 3960 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-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 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-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 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-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-sgrp 18648 df-mnd 18664 df-mgp 20036 df-ring 20136 df-lmod 20704 df-lfl 38431 |
This theorem is referenced by: ldualvsass 38514 |
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