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Theorem his2sub 22599
Description: Distributive law for inner product of vector subtraction. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
Assertion
Ref Expression
his2sub  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  .ih  C )  =  ( ( A 
.ih  C )  -  ( B  .ih  C ) ) )

Proof of Theorem his2sub
StepHypRef Expression
1 hvsubval 22524 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
21oveq1d 6099 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  -h  B )  .ih  C
)  =  ( ( A  +h  ( -u
1  .h  B ) )  .ih  C ) )
323adant3 978 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  .ih  C )  =  ( ( A  +h  ( -u 1  .h  B ) )  .ih  C ) )
4 neg1cn 10072 . . . . 5  |-  -u 1  e.  CC
5 hvmulcl 22521 . . . . 5  |-  ( (
-u 1  e.  CC  /\  B  e.  ~H )  ->  ( -u 1  .h  B )  e.  ~H )
64, 5mpan 653 . . . 4  |-  ( B  e.  ~H  ->  ( -u 1  .h  B )  e.  ~H )
7 ax-his2 22590 . . . 4  |-  ( ( A  e.  ~H  /\  ( -u 1  .h  B
)  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  ( -u 1  .h  B
) )  .ih  C
)  =  ( ( A  .ih  C )  +  ( ( -u
1  .h  B ) 
.ih  C ) ) )
86, 7syl3an2 1219 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  ( -u 1  .h  B ) )  .ih  C )  =  ( ( A 
.ih  C )  +  ( ( -u 1  .h  B )  .ih  C
) ) )
9 ax-his3 22591 . . . . . . 7  |-  ( (
-u 1  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( -u 1  .h  B )  .ih  C
)  =  ( -u
1  x.  ( B 
.ih  C ) ) )
104, 9mp3an1 1267 . . . . . 6  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( ( -u 1  .h  B )  .ih  C
)  =  ( -u
1  x.  ( B 
.ih  C ) ) )
11 hicl 22587 . . . . . . 7  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  C
)  e.  CC )
1211mulm1d 9490 . . . . . 6  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( -u 1  x.  ( B  .ih  C
) )  =  -u ( B  .ih  C ) )
1310, 12eqtrd 2470 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( ( -u 1  .h  B )  .ih  C
)  =  -u ( B  .ih  C ) )
1413oveq2d 6100 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  .ih  C )  +  ( (
-u 1  .h  B
)  .ih  C )
)  =  ( ( A  .ih  C )  +  -u ( B  .ih  C ) ) )
15143adant1 976 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  .ih  C
)  +  ( (
-u 1  .h  B
)  .ih  C )
)  =  ( ( A  .ih  C )  +  -u ( B  .ih  C ) ) )
168, 15eqtrd 2470 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  ( -u 1  .h  B ) )  .ih  C )  =  ( ( A 
.ih  C )  + 
-u ( B  .ih  C ) ) )
17 hicl 22587 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  C
)  e.  CC )
18173adant2 977 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  C )  e.  CC )
19113adant1 976 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  C )  e.  CC )
2018, 19negsubd 9422 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  .ih  C
)  +  -u ( B  .ih  C ) )  =  ( ( A 
.ih  C )  -  ( B  .ih  C ) ) )
213, 16, 203eqtrd 2474 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  .ih  C )  =  ( ( A 
.ih  C )  -  ( B  .ih  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726  (class class class)co 6084   CCcc 8993   1c1 8996    + caddc 8998    x. cmul 9000    - cmin 9296   -ucneg 9297   ~Hchil 22427    +h cva 22428    .h csm 22429    .ih csp 22430    -h cmv 22433
This theorem is referenced by:  his2sub2  22600  hi2eq  22612  pjhthlem1  22898  h1de2i  23060  pjdifnormii  23190  lnopeqi  23516  riesz3i  23570  leop2  23632  hmopidmpji  23660  pjssposi  23680  pjclem4  23707  pj3si  23715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-hfvmul 22513  ax-hfi 22586  ax-his2 22590  ax-his3 22591
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-ltxr 9130  df-sub 9298  df-neg 9299  df-hvsub 22479
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