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Theorem lflvsdi2 29877
 Description: Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Hypotheses
Ref Expression
lfldi.v
lfldi.r Scalar
lfldi.k
lfldi.p
lfldi.t
lfldi.f LFnl
lfldi.w
lfldi.x
lfldi2.y
lfldi2.g
Assertion
Ref Expression
lflvsdi2

Proof of Theorem lflvsdi2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lfldi.v . . . 4
2 fvex 5742 . . . 4
31, 2eqeltri 2506 . . 3
43a1i 11 . 2
5 lfldi.w . . 3
6 lfldi2.g . . 3
7 lfldi.r . . . 4 Scalar
8 lfldi.k . . . 4
9 lfldi.f . . . 4 LFnl
107, 8, 1, 9lflf 29861 . . 3
115, 6, 10syl2anc 643 . 2
12 lfldi.x . . 3
13 fconst6g 5632 . . 3
1412, 13syl 16 . 2
15 lfldi2.y . . 3
16 fconst6g 5632 . . 3
1715, 16syl 16 . 2
187lmodrng 15958 . . . 4
195, 18syl 16 . . 3
20 lfldi.p . . . 4
21 lfldi.t . . . 4
228, 20, 21rngdi 15682 . . 3
2319, 22sylan 458 . 2
244, 11, 14, 17, 23caofdi 6340 1
 Colors of variables: wff set class Syntax hints:   wi 4   w3a 936   wceq 1652   wcel 1725  cvv 2956  csn 3814   cxp 4876  wf 5450  cfv 5454  (class class class)co 6081   cof 6303  cbs 13469   cplusg 13529  cmulr 13530  Scalarcsca 13532  crg 15660  clmod 15950  LFnlclfn 29855 This theorem is referenced by:  lflvsdi2a  29878 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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-map 7020  df-rng 15663  df-lmod 15952  df-lfl 29856
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