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Theorem hvsubvali 21600
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 21596 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
41, 2, 3mp2an 653 1  |-  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684  (class class class)co 5858   1c1 8738   -ucneg 9038   ~Hchil 21499    +h cva 21500    .h csm 21501    -h cmv 21505
This theorem is referenced by:  hvsubsub4i  21638  hvnegdii  21641  hvsubeq0i  21642  hvsubcan2i  21643  hvsubaddi  21645  normlem0  21688  normlem9  21697  norm3difi  21726  normpar2i  21735  pjsubii  22257  pjssmii  22260  pjcji  22263  lnophmlem2  22597
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-hvsub 21551
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