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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualkrsc | Structured version Visualization version GIF version |
Description: The kernel of a nonzero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 28-Dec-2014.) |
Ref | Expression |
---|---|
ldualkrsc.r | β’ π = (Scalarβπ) |
ldualkrsc.k | β’ πΎ = (Baseβπ ) |
ldualkrsc.o | β’ 0 = (0gβπ ) |
ldualkrsc.f | β’ πΉ = (LFnlβπ) |
ldualkrsc.l | β’ πΏ = (LKerβπ) |
ldualkrsc.d | β’ π· = (LDualβπ) |
ldualkrsc.s | β’ Β· = ( Β·π βπ·) |
ldualkrsc.w | β’ (π β π β LVec) |
ldualkrsc.g | β’ (π β πΊ β πΉ) |
ldualkrsc.x | β’ (π β π β πΎ) |
ldualkrsc.e | β’ (π β π β 0 ) |
Ref | Expression |
---|---|
ldualkrsc | β’ (π β (πΏβ(π Β· πΊ)) = (πΏβπΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualkrsc.f | . . . 4 β’ πΉ = (LFnlβπ) | |
2 | eqid 2725 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
3 | ldualkrsc.r | . . . 4 β’ π = (Scalarβπ) | |
4 | ldualkrsc.k | . . . 4 β’ πΎ = (Baseβπ ) | |
5 | eqid 2725 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
6 | ldualkrsc.d | . . . 4 β’ π· = (LDualβπ) | |
7 | ldualkrsc.s | . . . 4 β’ Β· = ( Β·π βπ·) | |
8 | ldualkrsc.w | . . . 4 β’ (π β π β LVec) | |
9 | ldualkrsc.x | . . . 4 β’ (π β π β πΎ) | |
10 | ldualkrsc.g | . . . 4 β’ (π β πΊ β πΉ) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | ldualvs 38665 | . . 3 β’ (π β (π Β· πΊ) = (πΊ βf (.rβπ )((Baseβπ) Γ {π}))) |
12 | 11 | fveq2d 6896 | . 2 β’ (π β (πΏβ(π Β· πΊ)) = (πΏβ(πΊ βf (.rβπ )((Baseβπ) Γ {π})))) |
13 | ldualkrsc.l | . . 3 β’ πΏ = (LKerβπ) | |
14 | ldualkrsc.o | . . 3 β’ 0 = (0gβπ ) | |
15 | ldualkrsc.e | . . 3 β’ (π β π β 0 ) | |
16 | 2, 3, 4, 5, 1, 13, 8, 10, 9, 14, 15 | lkrsc 38625 | . 2 β’ (π β (πΏβ(πΊ βf (.rβπ )((Baseβπ) Γ {π}))) = (πΏβπΊ)) |
17 | 12, 16 | eqtrd 2765 | 1 β’ (π β (πΏβ(π Β· πΊ)) = (πΏβπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wne 2930 {csn 4624 Γ cxp 5670 βcfv 6543 (class class class)co 7416 βf cof 7680 Basecbs 17179 .rcmulr 17233 Scalarcsca 17235 Β·π cvsca 17236 0gc0g 17420 LVecclvec 20991 LFnlclfn 38585 LKerclk 38613 LDualcld 38651 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-drng 20630 df-lmod 20749 df-lvec 20992 df-lfl 38586 df-lkr 38614 df-ldual 38652 |
This theorem is referenced by: lclkrlem1 41035 lcfrlem31 41102 |
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