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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 2821 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2821 | . . . 4 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
3 | eqid 2821 | . . . 4 ⊢ ( ∘f (+g‘𝐹) ↾ ((LFnl‘𝑊) × (LFnl‘𝑊))) = ( ∘f (+g‘𝐹) ↾ ((LFnl‘𝑊) × (LFnl‘𝑊))) | |
4 | eqid 2821 | . . . 4 ⊢ (LFnl‘𝑊) = (LFnl‘𝑊) | |
5 | ldualsca.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
6 | ldualsca.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
7 | eqid 2821 | . . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
8 | eqid 2821 | . . . 4 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
9 | ldualsca.o | . . . 4 ⊢ 𝑂 = (oppr‘𝐹) | |
10 | eqid 2821 | . . . 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 36255 | . . 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 6668 | . 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 6678 | . . 3 ⊢ 𝑂 ∈ V |
16 | eqid 2821 | . . . 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 16633 | . . 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 2881 | 1 ⊢ (𝜑 → 𝑅 = 𝑂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∪ cun 3933 {csn 4560 {ctp 4564 〈cop 4566 × cxp 5547 ↾ cres 5551 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 ∘f cof 7401 ndxcnx 16474 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 Scalarcsca 16562 ·𝑠 cvsca 16563 opprcoppr 19366 LFnlclfn 36187 LDualcld 36253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-sca 16575 df-vsca 16576 df-ldual 36254 |
This theorem is referenced by: ldualsbase 36263 ldualsaddN 36264 ldualsmul 36265 ldual0 36277 ldual1 36278 ldualneg 36279 lduallmodlem 36282 lduallvec 36284 ldualvsub 36285 lcdsca 38729 |
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