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Mirrors > Home > MPE Home > Th. List > Mathboxes > lduallvec | Structured version Visualization version GIF version |
Description: The dual of a left vector space is also a left vector space. Note that scalar multiplication is reversed by df-oppr 19356; otherwise, the dual would be a right vector space as is sometimes the case in the literature. (Contributed by NM, 22-Oct-2014.) |
Ref | Expression |
---|---|
lduallvec.d | ⊢ 𝐷 = (LDual‘𝑊) |
lduallvec.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
Ref | Expression |
---|---|
lduallvec | ⊢ (𝜑 → 𝐷 ∈ LVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lduallvec.d | . . 3 ⊢ 𝐷 = (LDual‘𝑊) | |
2 | lduallvec.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
3 | lveclmod 19861 | . . . 4 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) |
5 | 1, 4 | lduallmod 36321 | . 2 ⊢ (𝜑 → 𝐷 ∈ LMod) |
6 | eqid 2821 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
7 | eqid 2821 | . . . 4 ⊢ (oppr‘(Scalar‘𝑊)) = (oppr‘(Scalar‘𝑊)) | |
8 | eqid 2821 | . . . 4 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
9 | 6, 7, 1, 8, 2 | ldualsca 36300 | . . 3 ⊢ (𝜑 → (Scalar‘𝐷) = (oppr‘(Scalar‘𝑊))) |
10 | 6 | lvecdrng 19860 | . . . . 5 ⊢ (𝑊 ∈ LVec → (Scalar‘𝑊) ∈ DivRing) |
11 | 2, 10 | syl 17 | . . . 4 ⊢ (𝜑 → (Scalar‘𝑊) ∈ DivRing) |
12 | 7 | opprdrng 19509 | . . . 4 ⊢ ((Scalar‘𝑊) ∈ DivRing ↔ (oppr‘(Scalar‘𝑊)) ∈ DivRing) |
13 | 11, 12 | sylib 220 | . . 3 ⊢ (𝜑 → (oppr‘(Scalar‘𝑊)) ∈ DivRing) |
14 | 9, 13 | eqeltrd 2913 | . 2 ⊢ (𝜑 → (Scalar‘𝐷) ∈ DivRing) |
15 | 8 | islvec 19859 | . 2 ⊢ (𝐷 ∈ LVec ↔ (𝐷 ∈ LMod ∧ (Scalar‘𝐷) ∈ DivRing)) |
16 | 5, 14, 15 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐷 ∈ LVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6341 Scalarcsca 16551 opprcoppr 19355 DivRingcdr 19485 LModclmod 19617 LVecclvec 19857 LDualcld 36291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-of 7395 df-om 7567 df-1st 7675 df-2nd 7676 df-tpos 7878 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-map 8394 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-n0 11885 df-z 11969 df-uz 12231 df-fz 12883 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-plusg 16561 df-mulr 16562 df-sca 16564 df-vsca 16565 df-0g 16698 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-grp 18089 df-minusg 18090 df-sbg 18091 df-cmn 18891 df-abl 18892 df-mgp 19223 df-ur 19235 df-ring 19282 df-oppr 19356 df-dvdsr 19374 df-unit 19375 df-drng 19487 df-lmod 19619 df-lvec 19858 df-lfl 36226 df-ldual 36292 |
This theorem is referenced by: lkreqN 36338 lkrlspeqN 36339 lcdlvec 38759 |
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