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

Theorem hvsubvali 22523
Description: Value of vector subtraction definition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvaddcl.1  |-  A  e. 
~H
hvaddcl.2  |-  B  e. 
~H
Assertion
Ref Expression
hvsubvali  |-  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) )

Proof of Theorem hvsubvali
StepHypRef Expression
1 hvaddcl.1 . 2  |-  A  e. 
~H
2 hvaddcl.2 . 2  |-  B  e. 
~H
3 hvsubval 22519 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
41, 2, 3mp2an 654 1  |-  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725  (class class class)co 6081   1c1 8991   -ucneg 9292   ~Hchil 22422    +h cva 22423    .h csm 22424    -h cmv 22428
This theorem is referenced by:  hvsubsub4i  22561  hvnegdii  22564  hvsubeq0i  22565  hvsubcan2i  22566  hvsubaddi  22568  normlem0  22611  normlem9  22620  norm3difi  22649  normpar2i  22658  pjsubii  23180  pjssmii  23183  pjcji  23186  lnophmlem2  23520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-hvsub 22474
  Copyright terms: Public domain W3C validator