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Theorem hvsubval 25805
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 6288 . 2  |-  ( x  =  A  ->  (
x  +h  ( -u
1  .h  y ) )  =  ( A  +h  ( -u 1  .h  y ) ) )
2 oveq2 6289 . . 3  |-  ( y  =  B  ->  ( -u 1  .h  y )  =  ( -u 1  .h  B ) )
32oveq2d 6297 . 2  |-  ( y  =  B  ->  ( A  +h  ( -u 1  .h  y ) )  =  ( A  +h  ( -u 1  .h  B ) ) )
4 df-hvsub 25760 . 2  |-  -h  =  ( x  e.  ~H ,  y  e.  ~H  |->  ( x  +h  ( -u 1  .h  y ) ) )
5 ovex 6309 . 2  |-  ( A  +h  ( -u 1  .h  B ) )  e. 
_V
61, 3, 4, 5ovmpt2 6423 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 1383    e. wcel 1804  (class class class)co 6281   1c1 9496   -ucneg 9811   ~Hchil 25708    +h cva 25709    .h csm 25710    -h cmv 25714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-hvsub 25760
This theorem is referenced by:  hvsubcl  25806  hvsubvali  25809  hvsubid  25815  hvnegid  25816  hv2neg  25817  hvaddsubval  25822  hvsub4  25826  hvaddsub12  25827  hvpncan  25828  hvaddsubass  25830  hvsubass  25833  hvsubdistr1  25838  hvsubdistr2  25839  hvsubcan  25863  hvsub0  25865  his2sub  25881  hhph  25967  shsubcl  26010  shsel3  26105  honegsubi  26587  lnopsubi  26765  lnfnsubi  26837  superpos  27145  cdj1i  27224
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