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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldual0v | Structured version Visualization version GIF version |
Description: The zero vector of the dual of a vector space. (Contributed by NM, 24-Oct-2014.) |
Ref | Expression |
---|---|
ldualv0.v | β’ π = (Baseβπ) |
ldualv0.r | β’ π = (Scalarβπ) |
ldualv0.z | β’ 0 = (0gβπ ) |
ldualv0.d | β’ π· = (LDualβπ) |
ldualv0.o | β’ π = (0gβπ·) |
ldualv0.w | β’ (π β π β LMod) |
Ref | Expression |
---|---|
ldual0v | β’ (π β π = (π Γ { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . . 4 β’ (LFnlβπ) = (LFnlβπ) | |
2 | ldualv0.r | . . . 4 β’ π = (Scalarβπ) | |
3 | eqid 2731 | . . . 4 β’ (+gβπ ) = (+gβπ ) | |
4 | ldualv0.d | . . . 4 β’ π· = (LDualβπ) | |
5 | eqid 2731 | . . . 4 β’ (+gβπ·) = (+gβπ·) | |
6 | ldualv0.w | . . . 4 β’ (π β π β LMod) | |
7 | ldualv0.z | . . . . . 6 β’ 0 = (0gβπ ) | |
8 | ldualv0.v | . . . . . 6 β’ π = (Baseβπ) | |
9 | 2, 7, 8, 1 | lfl0f 37637 | . . . . 5 β’ (π β LMod β (π Γ { 0 }) β (LFnlβπ)) |
10 | 6, 9 | syl 17 | . . . 4 β’ (π β (π Γ { 0 }) β (LFnlβπ)) |
11 | 1, 2, 3, 4, 5, 6, 10, 10 | ldualvadd 37697 | . . 3 β’ (π β ((π Γ { 0 })(+gβπ·)(π Γ { 0 })) = ((π Γ { 0 }) βf (+gβπ )(π Γ { 0 }))) |
12 | 8, 2, 3, 7, 1, 6, 10 | lfladd0l 37642 | . . 3 β’ (π β ((π Γ { 0 }) βf (+gβπ )(π Γ { 0 })) = (π Γ { 0 })) |
13 | 11, 12 | eqtrd 2771 | . 2 β’ (π β ((π Γ { 0 })(+gβπ·)(π Γ { 0 })) = (π Γ { 0 })) |
14 | 4, 6 | ldualgrp 37714 | . . 3 β’ (π β π· β Grp) |
15 | eqid 2731 | . . . 4 β’ (Baseβπ·) = (Baseβπ·) | |
16 | 1, 4, 15, 6, 10 | ldualelvbase 37695 | . . 3 β’ (π β (π Γ { 0 }) β (Baseβπ·)) |
17 | ldualv0.o | . . . 4 β’ π = (0gβπ·) | |
18 | 15, 5, 17 | grpid 18815 | . . 3 β’ ((π· β Grp β§ (π Γ { 0 }) β (Baseβπ·)) β (((π Γ { 0 })(+gβπ·)(π Γ { 0 })) = (π Γ { 0 }) β π = (π Γ { 0 }))) |
19 | 14, 16, 18 | syl2anc 584 | . 2 β’ (π β (((π Γ { 0 })(+gβπ·)(π Γ { 0 })) = (π Γ { 0 }) β π = (π Γ { 0 }))) |
20 | 13, 19 | mpbid 231 | 1 β’ (π β π = (π Γ { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1541 β wcel 2106 {csn 4606 Γ cxp 5651 βcfv 6516 (class class class)co 7377 βf cof 7635 Basecbs 17109 +gcplusg 17162 Scalarcsca 17165 0gc0g 17350 Grpcgrp 18777 LModclmod 20393 LFnlclfn 37625 LDualcld 37691 |
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 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-of 7637 df-om 7823 df-1st 7941 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8670 df-map 8789 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-nn 12178 df-2 12240 df-3 12241 df-4 12242 df-5 12243 df-6 12244 df-n0 12438 df-z 12524 df-uz 12788 df-fz 13450 df-struct 17045 df-sets 17062 df-slot 17080 df-ndx 17092 df-base 17110 df-plusg 17175 df-sca 17178 df-vsca 17179 df-0g 17352 df-mgm 18526 df-sgrp 18575 df-mnd 18586 df-grp 18780 df-minusg 18781 df-sbg 18782 df-cmn 19593 df-abl 19594 df-mgp 19926 df-ur 19943 df-ring 19995 df-lmod 20395 df-lfl 37626 df-ldual 37692 |
This theorem is referenced by: ldual0vcl 37719 lkr0f2 37729 lduallkr3 37730 lclkrlem1 40075 lclkrlem2j 40085 lcd0v 40180 lcd0v2 40181 |
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