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Theorem hvmulcli 22358
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 22357 . 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 1717  (class class class)co 6013   CCcc 8914   ~Hchil 22263    .h csm 22265
This theorem is referenced by:  hvsubsub4i  22402  hvnegdii  22405  hvsubeq0i  22406  hvsubcan2i  22407  hvaddcani  22408  hvsubaddi  22409  normlem0  22452  normlem5  22457  normlem9  22461  bcseqi  22463  norm-iii-i  22482  norm3difi  22490  normpar2i  22499  polid2i  22500  polidi  22501  h1de2i  22896  pjsubii  23021  eigposi  23180  lnop0  23310  lnopunilem1  23354  lnophmlem2  23361  lnfn0i  23386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-hfvmul 22349
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-ov 6016
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