HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hvsubdistr1 Unicode version

Theorem hvsubdistr1 22392
Description: Scalar multiplication distributive law for subtraction. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
Assertion
Ref Expression
hvsubdistr1  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  ( B  -h  C ) )  =  ( ( A  .h  B )  -h  ( A  .h  C )
) )

Proof of Theorem hvsubdistr1
StepHypRef Expression
1 neg1cn 9992 . . . . 5  |-  -u 1  e.  CC
2 hvmulcl 22357 . . . . 5  |-  ( (
-u 1  e.  CC  /\  C  e.  ~H )  ->  ( -u 1  .h  C )  e.  ~H )
31, 2mpan 652 . . . 4  |-  ( C  e.  ~H  ->  ( -u 1  .h  C )  e.  ~H )
4 ax-hvdistr1 22352 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  ( -u 1  .h  C )  e.  ~H )  -> 
( A  .h  ( B  +h  ( -u 1  .h  C ) ) )  =  ( ( A  .h  B )  +h  ( A  .h  ( -u 1  .h  C ) ) ) )
53, 4syl3an3 1219 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  ( B  +h  ( -u 1  .h  C ) ) )  =  ( ( A  .h  B )  +h  ( A  .h  ( -u 1  .h  C ) ) ) )
6 hvmulcom 22386 . . . . . 6  |-  ( ( A  e.  CC  /\  -u 1  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  ( -u 1  .h  C ) )  =  ( -u
1  .h  ( A  .h  C ) ) )
71, 6mp3an2 1267 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  ( -u 1  .h  C ) )  =  ( -u
1  .h  ( A  .h  C ) ) )
87oveq2d 6029 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( ( A  .h  B )  +h  ( A  .h  ( -u 1  .h  C ) ) )  =  ( ( A  .h  B )  +h  ( -u 1  .h  ( A  .h  C
) ) ) )
983adant2 976 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  .h  B
)  +h  ( A  .h  ( -u 1  .h  C ) ) )  =  ( ( A  .h  B )  +h  ( -u 1  .h  ( A  .h  C
) ) ) )
105, 9eqtrd 2412 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  ( B  +h  ( -u 1  .h  C ) ) )  =  ( ( A  .h  B )  +h  ( -u 1  .h  ( A  .h  C
) ) ) )
11 hvsubval 22360 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  -h  C
)  =  ( B  +h  ( -u 1  .h  C ) ) )
12113adant1 975 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  -h  C )  =  ( B  +h  ( -u 1  .h  C ) ) )
1312oveq2d 6029 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  ( B  -h  C ) )  =  ( A  .h  ( B  +h  ( -u 1  .h  C ) ) ) )
14 hvmulcl 22357 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
15143adant3 977 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  B )  e.  ~H )
16 hvmulcl 22357 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  C
)  e.  ~H )
17163adant2 976 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  C )  e.  ~H )
18 hvsubval 22360 . . 3  |-  ( ( ( A  .h  B
)  e.  ~H  /\  ( A  .h  C
)  e.  ~H )  ->  ( ( A  .h  B )  -h  ( A  .h  C )
)  =  ( ( A  .h  B )  +h  ( -u 1  .h  ( A  .h  C
) ) ) )
1915, 17, 18syl2anc 643 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  .h  B
)  -h  ( A  .h  C ) )  =  ( ( A  .h  B )  +h  ( -u 1  .h  ( A  .h  C
) ) ) )
2010, 13, 193eqtr4d 2422 1  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  ( B  -h  C ) )  =  ( ( A  .h  B )  -h  ( A  .h  C )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717  (class class class)co 6013   CCcc 8914   1c1 8917   -ucneg 9217   ~Hchil 22263    +h cva 22264    .h csm 22265    -h cmv 22269
This theorem is referenced by:  hvsubdistr1i  22395  hvmulcan  22415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-hfvmul 22349  ax-hvmulass 22351  ax-hvdistr1 22352
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-ltxr 9051  df-sub 9218  df-neg 9219  df-hvsub 22315
  Copyright terms: Public domain W3C validator