HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hvsubval Unicode version

Theorem hvsubval 21542
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 5785 . 2  |-  ( x  =  A  ->  (
x  +h  ( -u
1  .h  y ) )  =  ( A  +h  ( -u 1  .h  y ) ) )
2 oveq2 5786 . . 3  |-  ( y  =  B  ->  ( -u 1  .h  y )  =  ( -u 1  .h  B ) )
32oveq2d 5794 . 2  |-  ( y  =  B  ->  ( A  +h  ( -u 1  .h  y ) )  =  ( A  +h  ( -u 1  .h  B ) ) )
4 df-hvsub 21497 . 2  |-  -h  =  ( x  e.  ~H ,  y  e.  ~H  |->  ( x  +h  ( -u 1  .h  y ) ) )
5 ovex 5803 . 2  |-  ( A  +h  ( -u 1  .h  B ) )  e. 
_V
61, 3, 4, 5ovmpt2 5903 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 1619    e. wcel 1621  (class class class)co 5778   1c1 8692   -ucneg 8992   ~Hchil 21445    +h cva 21446    .h csm 21447    -h cmv 21451
This theorem is referenced by:  hvsubcl  21543  hvsubvali  21546  hvsubid  21551  hvnegid  21552  hv2neg  21553  hvaddsubval  21558  hvsub4  21562  hvaddsub12  21563  hvpncan  21564  hvaddsubass  21566  hvsubass  21569  hvsubdistr1  21574  hvsubdistr2  21575  hvsubcan  21599  hvsub0  21601  his2sub  21617  hhph  21703  shsubcl  21746  shsel3  21840  honegsubi  22322  lnopsubi  22500  lnfnsubi  22572  superpos  22880  cdj1i  22959
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-hvsub 21497
  Copyright terms: Public domain W3C validator