Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ldualset Structured version   Unicode version

Theorem ldualset 29860
 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
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ldualset.w . 2
2 elex 2956 . 2
3 ldualset.d . . 3 LDual
4 fveq2 5720 . . . . . . . 8 LFnl LFnl
5 ldualset.f . . . . . . . 8 LFnl
64, 5syl6eqr 2485 . . . . . . 7 LFnl
76opeq2d 3983 . . . . . 6 LFnl
8 fveq2 5720 . . . . . . . . . . . . 13 Scalar Scalar
9 ldualset.r . . . . . . . . . . . . 13 Scalar
108, 9syl6eqr 2485 . . . . . . . . . . . 12 Scalar
1110fveq2d 5724 . . . . . . . . . . 11 Scalar
12 ldualset.a . . . . . . . . . . 11
1311, 12syl6eqr 2485 . . . . . . . . . 10 Scalar
14 ofeq 6299 . . . . . . . . . 10 Scalar Scalar
1513, 14syl 16 . . . . . . . . 9 Scalar
166, 6xpeq12d 4895 . . . . . . . . 9 LFnl LFnl
1715, 16reseq12d 5139 . . . . . . . 8 Scalar LFnl LFnl
18 ldualset.p . . . . . . . 8
1917, 18syl6eqr 2485 . . . . . . 7 Scalar LFnl LFnl
2019opeq2d 3983 . . . . . 6 Scalar LFnl LFnl
2110fveq2d 5724 . . . . . . . 8 opprScalar oppr
22 ldualset.o . . . . . . . 8 oppr
2321, 22syl6eqr 2485 . . . . . . 7 opprScalar
2423opeq2d 3983 . . . . . 6 Scalar opprScalar Scalar
257, 20, 24tpeq123d 3890 . . . . 5 LFnl Scalar LFnl LFnl Scalar opprScalar Scalar
2610fveq2d 5724 . . . . . . . . . 10 Scalar
27 ldualset.k . . . . . . . . . 10
2826, 27syl6eqr 2485 . . . . . . . . 9 Scalar
2910fveq2d 5724 . . . . . . . . . . . 12 Scalar
30 ldualset.t . . . . . . . . . . . 12
3129, 30syl6eqr 2485 . . . . . . . . . . 11 Scalar
32 ofeq 6299 . . . . . . . . . . 11 Scalar Scalar
3331, 32syl 16 . . . . . . . . . 10 Scalar
34 eqidd 2436 . . . . . . . . . 10
35 fveq2 5720 . . . . . . . . . . . 12
36 ldualset.v . . . . . . . . . . . 12
3735, 36syl6eqr 2485 . . . . . . . . . . 11
3837xpeq1d 4893 . . . . . . . . . 10
3933, 34, 38oveq123d 6094 . . . . . . . . 9 Scalar
4028, 6, 39mpt2eq123dv 6128 . . . . . . . 8 Scalar LFnl Scalar
41 ldualset.s . . . . . . . 8
4240, 41syl6eqr 2485 . . . . . . 7 Scalar LFnl Scalar
4342opeq2d 3983 . . . . . 6 Scalar LFnl Scalar
4443sneqd 3819 . . . . 5 Scalar LFnl Scalar
4525, 44uneq12d 3494 . . . 4 LFnl Scalar LFnl LFnl Scalar opprScalar Scalar LFnl Scalar Scalar
46 df-ldual 29859 . . . 4 LDual LFnl Scalar LFnl LFnl Scalar opprScalar Scalar LFnl Scalar
47 tpex 4700 . . . . 5 Scalar
48 snex 4397 . . . . 5
4947, 48unex 4699 . . . 4 Scalar
5045, 46, 49fvmpt 5798 . . 3 LDual Scalar
513, 50syl5eq 2479 . 2 Scalar
521, 2, 513syl 19 1 Scalar
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  cvv 2948   cun 3310  csn 3806  ctp 3808  cop 3809   cxp 4868   cres 4872  cfv 5446  (class class class)co 6073   cmpt2 6075   cof 6295  cnx 13458  cbs 13461   cplusg 13521  cmulr 13522  Scalarcsca 13524  cvsca 13525  opprcoppr 15719  LFnlclfn 29792  LDualcld 29858 This theorem is referenced by:  ldualvbase  29861  ldualfvadd  29863  ldualsca  29867  ldualfvs  29871 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-ldual 29859
 Copyright terms: Public domain W3C validator