Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcoval Structured version   Unicode version

Theorem lcoval 31055
 Description: The value of a linear combination. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
Hypotheses
Ref Expression
lcoop.b
lcoop.s Scalar
lcoop.r
Assertion
Ref Expression
lcoval LinCo finSupp linC
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem lcoval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 lcoop.b . . . 4
2 lcoop.s . . . 4 Scalar
3 lcoop.r . . . 4
41, 2, 3lcoop 31054 . . 3 LinCo finSupp linC
54eleq2d 2521 . 2 LinCo finSupp linC
6 eqeq1 2455 . . . . 5 linC linC
76anbi2d 703 . . . 4 finSupp linC finSupp linC
87rexbidv 2848 . . 3 finSupp linC finSupp linC
98elrab 3216 . 2 finSupp linC finSupp linC
105, 9syl6bb 261 1 LinCo finSupp linC
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1370   wcel 1758  wrex 2796  crab 2799  cpw 3960   class class class wbr 4392  cfv 5518  (class class class)co 6192   cmap 7316   finSupp cfsupp 7723  cbs 14278  Scalarcsca 14345  c0g 14482   linC clinc 31047   LinCo clinco 31048 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-lco 31050 This theorem is referenced by:  lcoel0  31071  lincsumcl  31074  lincscmcl  31075  lincolss  31077  ellcoellss  31078  lcoss  31079  lindslinindsimp1  31100  lindslinindsimp2  31106
 Copyright terms: Public domain W3C validator