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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lfl1sc | Structured version Visualization version GIF version |
Description: The (right vector space) scalar product of a functional with one is the functional. (Contributed by NM, 21-Oct-2014.) |
Ref | Expression |
---|---|
lfl1sc.v | β’ π = (Baseβπ) |
lfl1sc.d | β’ π· = (Scalarβπ) |
lfl1sc.f | β’ πΉ = (LFnlβπ) |
lfl1sc.k | β’ πΎ = (Baseβπ·) |
lfl1sc.t | β’ Β· = (.rβπ·) |
lfl1sc.i | β’ 1 = (1rβπ·) |
lfl1sc.w | β’ (π β π β LMod) |
lfl1sc.g | β’ (π β πΊ β πΉ) |
Ref | Expression |
---|---|
lfl1sc | β’ (π β (πΊ βf Β· (π Γ { 1 })) = πΊ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lfl1sc.v | . . . 4 β’ π = (Baseβπ) | |
2 | 1 | fvexi 6905 | . . 3 β’ π β V |
3 | 2 | a1i 11 | . 2 β’ (π β π β V) |
4 | lfl1sc.w | . . 3 β’ (π β π β LMod) | |
5 | lfl1sc.g | . . 3 β’ (π β πΊ β πΉ) | |
6 | lfl1sc.d | . . . 4 β’ π· = (Scalarβπ) | |
7 | lfl1sc.k | . . . 4 β’ πΎ = (Baseβπ·) | |
8 | lfl1sc.f | . . . 4 β’ πΉ = (LFnlβπ) | |
9 | 6, 7, 1, 8 | lflf 37928 | . . 3 β’ ((π β LMod β§ πΊ β πΉ) β πΊ:πβΆπΎ) |
10 | 4, 5, 9 | syl2anc 584 | . 2 β’ (π β πΊ:πβΆπΎ) |
11 | lfl1sc.i | . . . 4 β’ 1 = (1rβπ·) | |
12 | 11 | fvexi 6905 | . . 3 β’ 1 β V |
13 | 12 | a1i 11 | . 2 β’ (π β 1 β V) |
14 | 6 | lmodring 20478 | . . . 4 β’ (π β LMod β π· β Ring) |
15 | 4, 14 | syl 17 | . . 3 β’ (π β π· β Ring) |
16 | lfl1sc.t | . . . 4 β’ Β· = (.rβπ·) | |
17 | 7, 16, 11 | ringridm 20086 | . . 3 β’ ((π· β Ring β§ π β πΎ) β (π Β· 1 ) = π) |
18 | 15, 17 | sylan 580 | . 2 β’ ((π β§ π β πΎ) β (π Β· 1 ) = π) |
19 | 3, 10, 13, 18 | caofid0r 7701 | 1 β’ (π β (πΊ βf Β· (π Γ { 1 })) = πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 Vcvv 3474 {csn 4628 Γ cxp 5674 βΆwf 6539 βcfv 6543 (class class class)co 7408 βf cof 7667 Basecbs 17143 .rcmulr 17197 Scalarcsca 17199 1rcur 20003 Ringcrg 20055 LModclmod 20470 LFnlclfn 37922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 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 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mgp 19987 df-ur 20004 df-ring 20057 df-lmod 20472 df-lfl 37923 |
This theorem is referenced by: lduallmodlem 38017 |
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