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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualsca | Structured version Visualization version GIF version |
Description: The ring of scalars of the dual of a vector space. (Contributed by NM, 18-Oct-2014.) |
Ref | Expression |
---|---|
ldualsca.f | ⊢ 𝐹 = (Scalar‘𝑊) |
ldualsca.o | ⊢ 𝑂 = (oppr‘𝐹) |
ldualsca.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualsca.r | ⊢ 𝑅 = (Scalar‘𝐷) |
ldualsca.w | ⊢ (𝜑 → 𝑊 ∈ 𝑋) |
Ref | Expression |
---|---|
ldualsca | ⊢ (𝜑 → 𝑅 = 𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2799 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2799 | . . . 4 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
3 | eqid 2799 | . . . 4 ⊢ ( ∘𝑓 (+g‘𝐹) ↾ ((LFnl‘𝑊) × (LFnl‘𝑊))) = ( ∘𝑓 (+g‘𝐹) ↾ ((LFnl‘𝑊) × (LFnl‘𝑊))) | |
4 | eqid 2799 | . . . 4 ⊢ (LFnl‘𝑊) = (LFnl‘𝑊) | |
5 | ldualsca.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
6 | ldualsca.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
7 | eqid 2799 | . . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
8 | eqid 2799 | . . . 4 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
9 | ldualsca.o | . . . 4 ⊢ 𝑂 = (oppr‘𝐹) | |
10 | eqid 2799 | . . . 4 ⊢ (𝑘 ∈ (Base‘𝐹), 𝑓 ∈ (LFnl‘𝑊) ↦ (𝑓 ∘𝑓 (.r‘𝐹)((Base‘𝑊) × {𝑘}))) = (𝑘 ∈ (Base‘𝐹), 𝑓 ∈ (LFnl‘𝑊) ↦ (𝑓 ∘𝑓 (.r‘𝐹)((Base‘𝑊) × {𝑘}))) | |
11 | ldualsca.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | ldualset 35146 | . . 3 ⊢ (𝜑 → 𝐷 = ({〈(Base‘ndx), (LFnl‘𝑊)〉, 〈(+g‘ndx), ( ∘𝑓 (+g‘𝐹) ↾ ((LFnl‘𝑊) × (LFnl‘𝑊)))〉, 〈(Scalar‘ndx), 𝑂〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘𝐹), 𝑓 ∈ (LFnl‘𝑊) ↦ (𝑓 ∘𝑓 (.r‘𝐹)((Base‘𝑊) × {𝑘})))〉})) |
13 | 12 | fveq2d 6415 | . 2 ⊢ (𝜑 → (Scalar‘𝐷) = (Scalar‘({〈(Base‘ndx), (LFnl‘𝑊)〉, 〈(+g‘ndx), ( ∘𝑓 (+g‘𝐹) ↾ ((LFnl‘𝑊) × (LFnl‘𝑊)))〉, 〈(Scalar‘ndx), 𝑂〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘𝐹), 𝑓 ∈ (LFnl‘𝑊) ↦ (𝑓 ∘𝑓 (.r‘𝐹)((Base‘𝑊) × {𝑘})))〉}))) |
14 | ldualsca.r | . 2 ⊢ 𝑅 = (Scalar‘𝐷) | |
15 | 9 | fvexi 6425 | . . 3 ⊢ 𝑂 ∈ V |
16 | eqid 2799 | . . . 4 ⊢ ({〈(Base‘ndx), (LFnl‘𝑊)〉, 〈(+g‘ndx), ( ∘𝑓 (+g‘𝐹) ↾ ((LFnl‘𝑊) × (LFnl‘𝑊)))〉, 〈(Scalar‘ndx), 𝑂〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘𝐹), 𝑓 ∈ (LFnl‘𝑊) ↦ (𝑓 ∘𝑓 (.r‘𝐹)((Base‘𝑊) × {𝑘})))〉}) = ({〈(Base‘ndx), (LFnl‘𝑊)〉, 〈(+g‘ndx), ( ∘𝑓 (+g‘𝐹) ↾ ((LFnl‘𝑊) × (LFnl‘𝑊)))〉, 〈(Scalar‘ndx), 𝑂〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘𝐹), 𝑓 ∈ (LFnl‘𝑊) ↦ (𝑓 ∘𝑓 (.r‘𝐹)((Base‘𝑊) × {𝑘})))〉}) | |
17 | 16 | lmodsca 16341 | . . 3 ⊢ (𝑂 ∈ V → 𝑂 = (Scalar‘({〈(Base‘ndx), (LFnl‘𝑊)〉, 〈(+g‘ndx), ( ∘𝑓 (+g‘𝐹) ↾ ((LFnl‘𝑊) × (LFnl‘𝑊)))〉, 〈(Scalar‘ndx), 𝑂〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘𝐹), 𝑓 ∈ (LFnl‘𝑊) ↦ (𝑓 ∘𝑓 (.r‘𝐹)((Base‘𝑊) × {𝑘})))〉}))) |
18 | 15, 17 | ax-mp 5 | . 2 ⊢ 𝑂 = (Scalar‘({〈(Base‘ndx), (LFnl‘𝑊)〉, 〈(+g‘ndx), ( ∘𝑓 (+g‘𝐹) ↾ ((LFnl‘𝑊) × (LFnl‘𝑊)))〉, 〈(Scalar‘ndx), 𝑂〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘𝐹), 𝑓 ∈ (LFnl‘𝑊) ↦ (𝑓 ∘𝑓 (.r‘𝐹)((Base‘𝑊) × {𝑘})))〉})) |
19 | 13, 14, 18 | 3eqtr4g 2858 | 1 ⊢ (𝜑 → 𝑅 = 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 Vcvv 3385 ∪ cun 3767 {csn 4368 {ctp 4372 〈cop 4374 × cxp 5310 ↾ cres 5314 ‘cfv 6101 (class class class)co 6878 ↦ cmpt2 6880 ∘𝑓 cof 7129 ndxcnx 16181 Basecbs 16184 +gcplusg 16267 .rcmulr 16268 Scalarcsca 16270 ·𝑠 cvsca 16271 opprcoppr 18938 LFnlclfn 35078 LDualcld 35144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-plusg 16280 df-sca 16283 df-vsca 16284 df-ldual 35145 |
This theorem is referenced by: ldualsbase 35154 ldualsaddN 35155 ldualsmul 35156 ldual0 35168 ldual1 35169 ldualneg 35170 lduallmodlem 35173 lduallvec 35175 ldualvsub 35176 lcdsca 37620 |
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