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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualgrp | Structured version Visualization version GIF version |
Description: The dual of a vector space is a group. (Contributed by NM, 21-Oct-2014.) |
Ref | Expression |
---|---|
ldualgrp.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualgrp.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
Ref | Expression |
---|---|
ldualgrp | ⊢ (𝜑 → 𝐷 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualgrp.d | . 2 ⊢ 𝐷 = (LDual‘𝑊) | |
2 | ldualgrp.w | . 2 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | eqid 2771 | . 2 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | eqid 2771 | . 2 ⊢ ∘𝑓 (+g‘𝑊) = ∘𝑓 (+g‘𝑊) | |
5 | eqid 2771 | . 2 ⊢ (LFnl‘𝑊) = (LFnl‘𝑊) | |
6 | eqid 2771 | . 2 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
7 | eqid 2771 | . 2 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
8 | eqid 2771 | . 2 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊)) | |
9 | eqid 2771 | . 2 ⊢ (oppr‘(Scalar‘𝑊)) = (oppr‘(Scalar‘𝑊)) | |
10 | eqid 2771 | . 2 ⊢ ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘𝐷) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | ldualgrplem 34954 | 1 ⊢ (𝜑 → 𝐷 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ‘cfv 6031 ∘𝑓 cof 7042 Basecbs 16064 +gcplusg 16149 .rcmulr 16150 Scalarcsca 16152 ·𝑠 cvsca 16153 Grpcgrp 17630 opprcoppr 18830 LModclmod 19073 LFnlclfn 34866 LDualcld 34932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-n0 11495 df-z 11580 df-uz 11889 df-fz 12534 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-plusg 16162 df-sca 16165 df-vsca 16166 df-0g 16310 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-grp 17633 df-minusg 17634 df-sbg 17635 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-ring 18757 df-lmod 19075 df-lfl 34867 df-ldual 34933 |
This theorem is referenced by: ldual0v 34959 lduallmodlem 34961 lcfrlem33 37385 |
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