HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hvmulcli Structured version   Unicode version

Theorem hvmulcli 24553
Description: Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvmulcl.1  |-  A  e.  CC
hvmulcl.2  |-  B  e. 
~H
Assertion
Ref Expression
hvmulcli  |-  ( A  .h  B )  e. 
~H

Proof of Theorem hvmulcli
StepHypRef Expression
1 hvmulcl.1 . 2  |-  A  e.  CC
2 hvmulcl.2 . 2  |-  B  e. 
~H
3 hvmulcl 24552 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
41, 2, 3mp2an 672 1  |-  ( A  .h  B )  e. 
~H
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1758  (class class class)co 6192   CCcc 9383   ~Hchil 24458    .h csm 24460
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  ax-hfvmul 24544
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-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-ov 6195
This theorem is referenced by:  hvsubsub4i  24598  hvnegdii  24601  hvsubeq0i  24602  hvsubcan2i  24603  hvaddcani  24604  hvsubaddi  24605  normlem0  24648  normlem5  24653  normlem9  24657  bcseqi  24659  norm-iii-i  24678  norm3difi  24686  normpar2i  24695  polid2i  24696  polidi  24697  h1de2i  25093  pjsubii  25218  eigposi  25377  lnop0  25507  lnopunilem1  25551  lnophmlem2  25558  lnfn0i  25583
  Copyright terms: Public domain W3C validator