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Theorem his2sub 22547
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 22472 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
21oveq1d 6055 . . 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 10023 . . . . 5  |-  -u 1  e.  CC
5 hvmulcl 22469 . . . . 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 22538 . . . 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 22539 . . . . . . 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 22535 . . . . . . 7  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  C
)  e.  CC )
1211mulm1d 9441 . . . . . 6  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( -u 1  x.  ( B  .ih  C
) )  =  -u ( B  .ih  C ) )
1310, 12eqtrd 2436 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( ( -u 1  .h  B )  .ih  C
)  =  -u ( B  .ih  C ) )
1413oveq2d 6056 . . . 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 2436 . 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 22535 . . . 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 9373 . 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 2440 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 6040   CCcc 8944   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247   -ucneg 9248   ~Hchil 22375    +h cva 22376    .h csm 22377    .ih csp 22378    -h cmv 22381
This theorem is referenced by:  his2sub2  22548  hi2eq  22560  pjhthlem1  22846  h1de2i  23008  pjdifnormii  23138  lnopeqi  23464  riesz3i  23518  leop2  23580  hmopidmpji  23608  pjssposi  23628  pjclem4  23655  pj3si  23663
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-hfvmul 22461  ax-hfi 22534  ax-his2 22538  ax-his3 22539
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-neg 9250  df-hvsub 22427
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