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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdspex | Structured version Visualization version GIF version |
Description: The map of a span equals the dual span of some vector (functional). (Contributed by NM, 15-Mar-2015.) |
Ref | Expression |
---|---|
mapdspex.h | β’ π» = (LHypβπΎ) |
mapdspex.m | β’ π = ((mapdβπΎ)βπ) |
mapdspex.u | β’ π = ((DVecHβπΎ)βπ) |
mapdspex.v | β’ π = (Baseβπ) |
mapdspex.n | β’ π = (LSpanβπ) |
mapdspex.c | β’ πΆ = ((LCDualβπΎ)βπ) |
mapdspex.b | β’ π΅ = (BaseβπΆ) |
mapdspex.j | β’ π½ = (LSpanβπΆ) |
mapdspex.k | β’ (π β (πΎ β HL β§ π β π»)) |
mapdspex.x | β’ (π β π β π) |
Ref | Expression |
---|---|
mapdspex | β’ (π β βπ β π΅ (πβ(πβ{π})) = (π½β{π})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdspex.h | . . . . 5 β’ π» = (LHypβπΎ) | |
2 | mapdspex.c | . . . . 5 β’ πΆ = ((LCDualβπΎ)βπ) | |
3 | mapdspex.k | . . . . 5 β’ (π β (πΎ β HL β§ π β π»)) | |
4 | 1, 2, 3 | lcdlmod 40966 | . . . 4 β’ (π β πΆ β LMod) |
5 | 4 | adantr 480 | . . 3 β’ ((π β§ (πβ{π}) β (LSAtomsβπ)) β πΆ β LMod) |
6 | mapdspex.m | . . . 4 β’ π = ((mapdβπΎ)βπ) | |
7 | mapdspex.u | . . . 4 β’ π = ((DVecHβπΎ)βπ) | |
8 | eqid 2724 | . . . 4 β’ (LSAtomsβπ) = (LSAtomsβπ) | |
9 | eqid 2724 | . . . 4 β’ (LSAtomsβπΆ) = (LSAtomsβπΆ) | |
10 | 3 | adantr 480 | . . . 4 β’ ((π β§ (πβ{π}) β (LSAtomsβπ)) β (πΎ β HL β§ π β π»)) |
11 | simpr 484 | . . . 4 β’ ((π β§ (πβ{π}) β (LSAtomsβπ)) β (πβ{π}) β (LSAtomsβπ)) | |
12 | 1, 6, 7, 8, 2, 9, 10, 11 | mapdat 41041 | . . 3 β’ ((π β§ (πβ{π}) β (LSAtomsβπ)) β (πβ(πβ{π})) β (LSAtomsβπΆ)) |
13 | mapdspex.b | . . . 4 β’ π΅ = (BaseβπΆ) | |
14 | mapdspex.j | . . . 4 β’ π½ = (LSpanβπΆ) | |
15 | 13, 14, 9 | islsati 38367 | . . 3 β’ ((πΆ β LMod β§ (πβ(πβ{π})) β (LSAtomsβπΆ)) β βπ β π΅ (πβ(πβ{π})) = (π½β{π})) |
16 | 5, 12, 15 | syl2anc 583 | . 2 β’ ((π β§ (πβ{π}) β (LSAtomsβπ)) β βπ β π΅ (πβ(πβ{π})) = (π½β{π})) |
17 | eqid 2724 | . . . . 5 β’ (0gβπΆ) = (0gβπΆ) | |
18 | 1, 2, 13, 17, 3 | lcd0vcl 40988 | . . . 4 β’ (π β (0gβπΆ) β π΅) |
19 | 18 | adantr 480 | . . 3 β’ ((π β§ (πβ{π}) = {(0gβπ)}) β (0gβπΆ) β π΅) |
20 | fveq2 6882 | . . . 4 β’ ((πβ{π}) = {(0gβπ)} β (πβ(πβ{π})) = (πβ{(0gβπ)})) | |
21 | eqid 2724 | . . . . . 6 β’ (0gβπ) = (0gβπ) | |
22 | 1, 6, 7, 21, 2, 17, 3 | mapd0 41039 | . . . . 5 β’ (π β (πβ{(0gβπ)}) = {(0gβπΆ)}) |
23 | 17, 14 | lspsn0 20851 | . . . . . 6 β’ (πΆ β LMod β (π½β{(0gβπΆ)}) = {(0gβπΆ)}) |
24 | 4, 23 | syl 17 | . . . . 5 β’ (π β (π½β{(0gβπΆ)}) = {(0gβπΆ)}) |
25 | 22, 24 | eqtr4d 2767 | . . . 4 β’ (π β (πβ{(0gβπ)}) = (π½β{(0gβπΆ)})) |
26 | 20, 25 | sylan9eqr 2786 | . . 3 β’ ((π β§ (πβ{π}) = {(0gβπ)}) β (πβ(πβ{π})) = (π½β{(0gβπΆ)})) |
27 | sneq 4631 | . . . . 5 β’ (π = (0gβπΆ) β {π} = {(0gβπΆ)}) | |
28 | 27 | fveq2d 6886 | . . . 4 β’ (π = (0gβπΆ) β (π½β{π}) = (π½β{(0gβπΆ)})) |
29 | 28 | rspceeqv 3626 | . . 3 β’ (((0gβπΆ) β π΅ β§ (πβ(πβ{π})) = (π½β{(0gβπΆ)})) β βπ β π΅ (πβ(πβ{π})) = (π½β{π})) |
30 | 19, 26, 29 | syl2anc 583 | . 2 β’ ((π β§ (πβ{π}) = {(0gβπ)}) β βπ β π΅ (πβ(πβ{π})) = (π½β{π})) |
31 | mapdspex.v | . . 3 β’ π = (Baseβπ) | |
32 | mapdspex.n | . . 3 β’ π = (LSpanβπ) | |
33 | 1, 7, 3 | dvhlmod 40484 | . . 3 β’ (π β π β LMod) |
34 | mapdspex.x | . . 3 β’ (π β π β π) | |
35 | 31, 32, 21, 8, 33, 34 | lsator0sp 38374 | . 2 β’ (π β ((πβ{π}) β (LSAtomsβπ) β¨ (πβ{π}) = {(0gβπ)})) |
36 | 16, 30, 35 | mpjaodan 955 | 1 β’ (π β βπ β π΅ (πβ(πβ{π})) = (π½β{π})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βwrex 3062 {csn 4621 βcfv 6534 Basecbs 17149 0gc0g 17390 LModclmod 20702 LSpanclspn 20814 LSAtomsclsa 38347 HLchlt 38723 LHypclh 39358 DVecHcdvh 40452 LCDualclcd 40960 mapdcmpd 40998 |
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 ax-riotaBAD 38326 |
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-rmo 3368 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-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 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-1st 7969 df-2nd 7970 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 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-3 12275 df-4 12276 df-5 12277 df-6 12278 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13486 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-0g 17392 df-mre 17535 df-mrc 17536 df-acs 17538 df-proset 18256 df-poset 18274 df-plt 18291 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-p0 18386 df-p1 18387 df-lat 18393 df-clat 18460 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-subg 19046 df-cntz 19229 df-oppg 19258 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lvec 20947 df-lsatoms 38349 df-lshyp 38350 df-lcv 38392 df-lfl 38431 df-lkr 38459 df-ldual 38497 df-oposet 38549 df-ol 38551 df-oml 38552 df-covers 38639 df-ats 38640 df-atl 38671 df-cvlat 38695 df-hlat 38724 df-llines 38872 df-lplanes 38873 df-lvols 38874 df-lines 38875 df-psubsp 38877 df-pmap 38878 df-padd 39170 df-lhyp 39362 df-laut 39363 df-ldil 39478 df-ltrn 39479 df-trl 39533 df-tgrp 40117 df-tendo 40129 df-edring 40131 df-dveca 40377 df-disoa 40403 df-dvech 40453 df-dib 40513 df-dic 40547 df-dih 40603 df-doch 40722 df-djh 40769 df-lcdual 40961 df-mapd 40999 |
This theorem is referenced by: mapdpglem2 41047 |
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