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Theorem his2sub 22555
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 22480 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
21oveq1d 6063 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  -h  B )  .ih  C
)  =  ( ( A  +h  ( -u
1  .h  B ) )  .ih  C ) )
323adant3 977 . 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 10031 . . . . 5  |-  -u 1  e.  CC
5 hvmulcl 22477 . . . . 5  |-  ( (
-u 1  e.  CC  /\  B  e.  ~H )  ->  ( -u 1  .h  B )  e.  ~H )
64, 5mpan 652 . . . 4  |-  ( B  e.  ~H  ->  ( -u 1  .h  B )  e.  ~H )
7 ax-his2 22546 . . . 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 1218 . . 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 22547 . . . . . . 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 1266 . . . . . 6  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( ( -u 1  .h  B )  .ih  C
)  =  ( -u
1  x.  ( B 
.ih  C ) ) )
11 hicl 22543 . . . . . . 7  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  C
)  e.  CC )
1211mulm1d 9449 . . . . . 6  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( -u 1  x.  ( B  .ih  C
) )  =  -u ( B  .ih  C ) )
1310, 12eqtrd 2444 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( ( -u 1  .h  B )  .ih  C
)  =  -u ( B  .ih  C ) )
1413oveq2d 6064 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  .ih  C )  +  ( (
-u 1  .h  B
)  .ih  C )
)  =  ( ( A  .ih  C )  +  -u ( B  .ih  C ) ) )
15143adant1 975 . . 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 2444 . 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 22543 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  C
)  e.  CC )
18173adant2 976 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  C )  e.  CC )
19113adant1 975 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  C )  e.  CC )
2018, 19negsubd 9381 . 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 2448 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721  (class class class)co 6048   CCcc 8952   1c1 8955    + caddc 8957    x. cmul 8959    - cmin 9255   -ucneg 9256   ~Hchil 22383    +h cva 22384    .h csm 22385    .ih csp 22386    -h cmv 22389
This theorem is referenced by:  his2sub2  22556  hi2eq  22568  pjhthlem1  22854  h1de2i  23016  pjdifnormii  23146  lnopeqi  23472  riesz3i  23526  leop2  23588  hmopidmpji  23616  pjssposi  23636  pjclem4  23663  pj3si  23671
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-hfvmul 22469  ax-hfi 22542  ax-his2 22546  ax-his3 22547
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-ltxr 9089  df-sub 9257  df-neg 9258  df-hvsub 22435
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