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Theorem hvsubvali 24357
Description: Value of vector subtraction definition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvaddcl.1  |-  A  e. 
~H
hvaddcl.2  |-  B  e. 
~H
Assertion
Ref Expression
hvsubvali  |-  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) )

Proof of Theorem hvsubvali
StepHypRef Expression
1 hvaddcl.1 . 2  |-  A  e. 
~H
2 hvaddcl.2 . 2  |-  B  e. 
~H
3 hvsubval 24353 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
41, 2, 3mp2an 667 1  |-  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1364    e. wcel 1761  (class class class)co 6090   1c1 9279   -ucneg 9592   ~Hchil 24256    +h cva 24257    .h csm 24258    -h cmv 24262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-hvsub 24308
This theorem is referenced by:  hvsubsub4i  24396  hvnegdii  24399  hvsubeq0i  24400  hvsubcan2i  24401  hvsubaddi  24403  normlem0  24446  normlem9  24455  norm3difi  24484  normpar2i  24493  pjsubii  25016  pjssmii  25019  pjcji  25022  lnophmlem2  25356
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