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Theorem hvmulcli 21610
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 21609 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
41, 2, 3mp2an 653 1  |-  ( A  .h  B )  e. 
~H
Colors of variables: wff set class
Syntax hints:    e. wcel 1696  (class class class)co 5874   CCcc 8751   ~Hchil 21515    .h csm 21517
This theorem is referenced by:  hvsubsub4i  21654  hvnegdii  21657  hvsubeq0i  21658  hvsubcan2i  21659  hvaddcani  21660  hvsubaddi  21661  normlem0  21704  normlem5  21709  normlem9  21713  bcseqi  21715  norm-iii-i  21734  norm3difi  21742  normpar2i  21751  polid2i  21752  polidi  21753  h1de2i  22148  pjsubii  22273  eigposi  22432  lnop0  22562  lnopunilem1  22606  lnophmlem2  22613  lnfn0i  22638
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-hfvmul 21601
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877
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