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Theorem hvsubval 21588
Description: Value of vector subtraction. (Contributed by NM, 5-Sep-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
hvsubval  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )

Proof of Theorem hvsubval
StepHypRef Expression
1 oveq1 5826 . 2  |-  ( x  =  A  ->  (
x  +h  ( -u
1  .h  y ) )  =  ( A  +h  ( -u 1  .h  y ) ) )
2 oveq2 5827 . . 3  |-  ( y  =  B  ->  ( -u 1  .h  y )  =  ( -u 1  .h  B ) )
32oveq2d 5835 . 2  |-  ( y  =  B  ->  ( A  +h  ( -u 1  .h  y ) )  =  ( A  +h  ( -u 1  .h  B ) ) )
4 df-hvsub 21543 . 2  |-  -h  =  ( x  e.  ~H ,  y  e.  ~H  |->  ( x  +h  ( -u 1  .h  y ) ) )
5 ovex 5844 . 2  |-  ( A  +h  ( -u 1  .h  B ) )  e. 
_V
61, 3, 4, 5ovmpt2 5944 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1628    e. wcel 1688  (class class class)co 5819   1c1 8733   -ucneg 9033   ~Hchil 21491    +h cva 21492    .h csm 21493    -h cmv 21497
This theorem is referenced by:  hvsubcl  21589  hvsubvali  21592  hvsubid  21597  hvnegid  21598  hv2neg  21599  hvaddsubval  21604  hvsub4  21608  hvaddsub12  21609  hvpncan  21610  hvaddsubass  21612  hvsubass  21615  hvsubdistr1  21620  hvsubdistr2  21621  hvsubcan  21645  hvsub0  21647  his2sub  21663  hhph  21749  shsubcl  21792  shsel3  21886  honegsubi  22368  lnopsubi  22546  lnfnsubi  22618  superpos  22926  cdj1i  23005
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-hvsub 21543
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