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Theorem hvsubval 24590
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 6210 . 2  |-  ( x  =  A  ->  (
x  +h  ( -u
1  .h  y ) )  =  ( A  +h  ( -u 1  .h  y ) ) )
2 oveq2 6211 . . 3  |-  ( y  =  B  ->  ( -u 1  .h  y )  =  ( -u 1  .h  B ) )
32oveq2d 6219 . 2  |-  ( y  =  B  ->  ( A  +h  ( -u 1  .h  y ) )  =  ( A  +h  ( -u 1  .h  B ) ) )
4 df-hvsub 24545 . 2  |-  -h  =  ( x  e.  ~H ,  y  e.  ~H  |->  ( x  +h  ( -u 1  .h  y ) ) )
5 ovex 6228 . 2  |-  ( A  +h  ( -u 1  .h  B ) )  e. 
_V
61, 3, 4, 5ovmpt2 6339 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758  (class class class)co 6203   1c1 9397   -ucneg 9710   ~Hchil 24493    +h cva 24494    .h csm 24495    -h cmv 24499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-hvsub 24545
This theorem is referenced by:  hvsubcl  24591  hvsubvali  24594  hvsubid  24600  hvnegid  24601  hv2neg  24602  hvaddsubval  24607  hvsub4  24611  hvaddsub12  24612  hvpncan  24613  hvaddsubass  24615  hvsubass  24618  hvsubdistr1  24623  hvsubdistr2  24624  hvsubcan  24648  hvsub0  24650  his2sub  24666  hhph  24752  shsubcl  24795  shsel3  24890  honegsubi  25372  lnopsubi  25550  lnfnsubi  25622  superpos  25930  cdj1i  26009
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