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Theorem hvsubvali 22476
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 22472 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
41, 2, 3mp2an 654 1  |-  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721  (class class class)co 6040   1c1 8947   -ucneg 9248   ~Hchil 22375    +h cva 22376    .h csm 22377    -h cmv 22381
This theorem is referenced by:  hvsubsub4i  22514  hvnegdii  22517  hvsubeq0i  22518  hvsubcan2i  22519  hvsubaddi  22521  normlem0  22564  normlem9  22573  norm3difi  22602  normpar2i  22611  pjsubii  23133  pjssmii  23136  pjcji  23139  lnophmlem2  23473
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-hvsub 22427
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