![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvaddcl | Structured version Visualization version GIF version |
Description: The value of vector addition in the dual of a vector space is a functional. (Contributed by NM, 21-Oct-2014.) |
Ref | Expression |
---|---|
ldualvaddcl.f | ⊢ 𝐹 = (LFnl‘𝑊) |
ldualvaddcl.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualvaddcl.p | ⊢ + = (+g‘𝐷) |
ldualvaddcl.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
ldualvaddcl.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
ldualvaddcl.h | ⊢ (𝜑 → 𝐻 ∈ 𝐹) |
Ref | Expression |
---|---|
ldualvaddcl | ⊢ (𝜑 → (𝐺 + 𝐻) ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualvaddcl.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑊) | |
2 | eqid 2824 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | eqid 2824 | . . 3 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊)) | |
4 | ldualvaddcl.d | . . 3 ⊢ 𝐷 = (LDual‘𝑊) | |
5 | ldualvaddcl.p | . . 3 ⊢ + = (+g‘𝐷) | |
6 | ldualvaddcl.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
7 | ldualvaddcl.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
8 | ldualvaddcl.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | ldualvadd 35203 | . 2 ⊢ (𝜑 → (𝐺 + 𝐻) = (𝐺 ∘𝑓 (+g‘(Scalar‘𝑊))𝐻)) |
10 | 2, 3, 1, 6, 7, 8 | lfladdcl 35145 | . 2 ⊢ (𝜑 → (𝐺 ∘𝑓 (+g‘(Scalar‘𝑊))𝐻) ∈ 𝐹) |
11 | 9, 10 | eqeltrd 2905 | 1 ⊢ (𝜑 → (𝐺 + 𝐻) ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ‘cfv 6122 (class class class)co 6904 ∘𝑓 cof 7154 +gcplusg 16304 Scalarcsca 16307 LModclmod 19218 LFnlclfn 35131 LDualcld 35197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-of 7156 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-er 8008 df-map 8123 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-5 11416 df-6 11417 df-n0 11618 df-z 11704 df-uz 11968 df-fz 12619 df-struct 16223 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-plusg 16317 df-sca 16320 df-vsca 16321 df-0g 16454 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-grp 17778 df-minusg 17779 df-sbg 17780 df-cmn 18547 df-abl 18548 df-mgp 18843 df-ur 18855 df-ring 18902 df-lmod 19220 df-lfl 35132 df-ldual 35198 |
This theorem is referenced by: ldualvsdi1 35217 ldualgrplem 35219 ldualvsubcl 35230 lkrin 35238 lclkrlem2e 37585 lclkrlem2f 37586 lclkrlem2h 37588 lclkrlem2m 37593 lclkrlem2n 37594 lclkrlem2o 37595 lclkrlem2p 37596 lclkrlem2s 37599 lclkrlem2v 37602 lclkrlem2 37606 lclkrslem2 37612 |
Copyright terms: Public domain | W3C validator |