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Theorem hvmulcli 22509
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 22508 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
41, 2, 3mp2an 654 1  |-  ( A  .h  B )  e. 
~H
Colors of variables: wff set class
Syntax hints:    e. wcel 1725  (class class class)co 6073   CCcc 8980   ~Hchil 22414    .h csm 22416
This theorem is referenced by:  hvsubsub4i  22553  hvnegdii  22556  hvsubeq0i  22557  hvsubcan2i  22558  hvaddcani  22559  hvsubaddi  22560  normlem0  22603  normlem5  22608  normlem9  22612  bcseqi  22614  norm-iii-i  22633  norm3difi  22641  normpar2i  22650  polid2i  22651  polidi  22652  h1de2i  23047  pjsubii  23172  eigposi  23331  lnop0  23461  lnopunilem1  23505  lnophmlem2  23512  lnfn0i  23537
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-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-hfvmul 22500
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-csb 3244  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-op 3815  df-uni 4008  df-iun 4087  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-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076
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