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Theorem hvmulcli 25803
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 25802 . 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 1804  (class class class)co 6281   CCcc 9493   ~Hchil 25708    .h csm 25710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-hfvmul 25794
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284
This theorem is referenced by:  hvsubsub4i  25848  hvnegdii  25851  hvsubeq0i  25852  hvsubcan2i  25853  hvaddcani  25854  hvsubaddi  25855  normlem0  25898  normlem5  25903  normlem9  25907  bcseqi  25909  norm-iii-i  25928  norm3difi  25936  normpar2i  25945  polid2i  25946  polidi  25947  h1de2i  26343  pjsubii  26468  eigposi  26627  lnop0  26757  lnopunilem1  26801  lnophmlem2  26808  lnfn0i  26833
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