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Theorem hvsubval 21612
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 5881 . 2  |-  ( x  =  A  ->  (
x  +h  ( -u
1  .h  y ) )  =  ( A  +h  ( -u 1  .h  y ) ) )
2 oveq2 5882 . . 3  |-  ( y  =  B  ->  ( -u 1  .h  y )  =  ( -u 1  .h  B ) )
32oveq2d 5890 . 2  |-  ( y  =  B  ->  ( A  +h  ( -u 1  .h  y ) )  =  ( A  +h  ( -u 1  .h  B ) ) )
4 df-hvsub 21567 . 2  |-  -h  =  ( x  e.  ~H ,  y  e.  ~H  |->  ( x  +h  ( -u 1  .h  y ) ) )
5 ovex 5899 . 2  |-  ( A  +h  ( -u 1  .h  B ) )  e. 
_V
61, 3, 4, 5ovmpt2 5999 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 4    /\ wa 358    = wceq 1632    e. wcel 1696  (class class class)co 5874   1c1 8754   -ucneg 9054   ~Hchil 21515    +h cva 21516    .h csm 21517    -h cmv 21521
This theorem is referenced by:  hvsubcl  21613  hvsubvali  21616  hvsubid  21621  hvnegid  21622  hv2neg  21623  hvaddsubval  21628  hvsub4  21632  hvaddsub12  21633  hvpncan  21634  hvaddsubass  21636  hvsubass  21639  hvsubdistr1  21644  hvsubdistr2  21645  hvsubcan  21669  hvsub0  21671  his2sub  21687  hhph  21773  shsubcl  21816  shsel3  21910  honegsubi  22392  lnopsubi  22570  lnfnsubi  22642  superpos  22950  cdj1i  23029
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
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-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-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-hvsub 21567
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