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Theorem ldualset 28004
 Description: Define the (left) dual of a left vector space (or module) in which the vectors are functionals. In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in [Holland95] p. 218. This allows us to reuse our existing collection of left vector space theorems. Note the operation reversal in the scalar product to allow us to use the original scalar ring instead of the oppr ring, for convenience. (Contributed by NM, 18-Oct-2014.)
Hypotheses
Ref Expression
ldualset.v
ldualset.a
ldualset.p
ldualset.f LFnl
ldualset.d LDual
ldualset.r Scalar
ldualset.k
ldualset.t
ldualset.o oppr
ldualset.s
ldualset.w
Assertion
Ref Expression
ldualset Scalar
Distinct variable group:   ,,
Allowed substitution hints:   (,)   (,)   (,)   (,)   (,)   (,)   (,)   (,)   (,)   (,)   (,)   (,)

Proof of Theorem ldualset
StepHypRef Expression
1 ldualset.w . 2
2 elex 2735 . 2
3 ldualset.d . . 3 LDual
4 fveq2 5377 . . . . . . . 8 LFnl LFnl
5 ldualset.f . . . . . . . 8 LFnl
64, 5syl6eqr 2303 . . . . . . 7 LFnl
76opeq2d 3703 . . . . . 6 LFnl
8 fveq2 5377 . . . . . . . . . . . . 13 Scalar Scalar
9 ldualset.r . . . . . . . . . . . . 13 Scalar
108, 9syl6eqr 2303 . . . . . . . . . . . 12 Scalar
1110fveq2d 5381 . . . . . . . . . . 11 Scalar
12 ldualset.a . . . . . . . . . . 11
1311, 12syl6eqr 2303 . . . . . . . . . 10 Scalar
14 ofeq 5932 . . . . . . . . . 10 Scalar Scalar
1513, 14syl 17 . . . . . . . . 9 Scalar
166, 6xpeq12d 4621 . . . . . . . . 9 LFnl LFnl
1715, 16reseq12d 4863 . . . . . . . 8 Scalar LFnl LFnl
18 ldualset.p . . . . . . . 8
1917, 18syl6eqr 2303 . . . . . . 7 Scalar LFnl LFnl
2019opeq2d 3703 . . . . . 6 Scalar LFnl LFnl
2110fveq2d 5381 . . . . . . . 8 opprScalar oppr
22 ldualset.o . . . . . . . 8 oppr
2321, 22syl6eqr 2303 . . . . . . 7 opprScalar
2423opeq2d 3703 . . . . . 6 Scalar opprScalar Scalar
257, 20, 24tpeq123d 3625 . . . . 5 LFnl Scalar LFnl LFnl Scalar opprScalar Scalar
2610fveq2d 5381 . . . . . . . . . 10 Scalar
27 ldualset.k . . . . . . . . . 10
2826, 27syl6eqr 2303 . . . . . . . . 9 Scalar
2910fveq2d 5381 . . . . . . . . . . . 12 Scalar
30 ldualset.t . . . . . . . . . . . 12
3129, 30syl6eqr 2303 . . . . . . . . . . 11 Scalar
32 ofeq 5932 . . . . . . . . . . 11 Scalar Scalar
3331, 32syl 17 . . . . . . . . . 10 Scalar
34 eqidd 2254 . . . . . . . . . 10
35 fveq2 5377 . . . . . . . . . . . 12
36 ldualset.v . . . . . . . . . . . 12
3735, 36syl6eqr 2303 . . . . . . . . . . 11
3837xpeq1d 4619 . . . . . . . . . 10
3933, 34, 38oveq123d 5731 . . . . . . . . 9 Scalar
4028, 6, 39mpt2eq123dv 5762 . . . . . . . 8 Scalar LFnl Scalar
41 ldualset.s . . . . . . . 8
4240, 41syl6eqr 2303 . . . . . . 7 Scalar LFnl Scalar
4342opeq2d 3703 . . . . . 6 Scalar LFnl Scalar
4443sneqd 3557 . . . . 5 Scalar LFnl Scalar
4525, 44uneq12d 3240 . . . 4 LFnl Scalar LFnl LFnl Scalar opprScalar Scalar LFnl Scalar Scalar
46 df-ldual 28003 . . . 4 LDual LFnl Scalar LFnl LFnl Scalar opprScalar Scalar LFnl Scalar
47 tpex 4410 . . . . 5 Scalar
48 snex 4110 . . . . 5
4947, 48unex 4409 . . . 4 Scalar
5045, 46, 49fvmpt 5454 . . 3 LDual Scalar
513, 50syl5eq 2297 . 2 Scalar
521, 2, 513syl 20 1 Scalar
 Colors of variables: wff set class Syntax hints:   wi 6   wceq 1619   wcel 1621  cvv 2727   cun 3076  csn 3544  ctp 3546  cop 3547   cxp 4578   cres 4582  cfv 4592  (class class class)co 5710   cmpt2 5712   cof 5928  cnx 13019  cbs 13022   cplusg 13082  cmulr 13083  Scalarcsca 13085  cvsca 13086  opprcoppr 15239  LFnlclfn 27936  LDualcld 28002 This theorem is referenced by:  ldualvbase  28005  ldualfvadd  28007  ldualsca  28011  ldualfvs  28015 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-ldual 28003
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