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Theorem hisubcomi 22596
Description: Two vector subtractions simultaneously commute in an inner product. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
hisubcom.1  |-  A  e. 
~H
hisubcom.2  |-  B  e. 
~H
hisubcom.3  |-  C  e. 
~H
hisubcom.4  |-  D  e. 
~H
Assertion
Ref Expression
hisubcomi  |-  ( ( A  -h  B ) 
.ih  ( C  -h  D ) )  =  ( ( B  -h  A )  .ih  ( D  -h  C ) )

Proof of Theorem hisubcomi
StepHypRef Expression
1 hisubcom.2 . . . 4  |-  B  e. 
~H
2 hisubcom.1 . . . 4  |-  A  e. 
~H
31, 2hvnegdii 22554 . . 3  |-  ( -u
1  .h  ( B  -h  A ) )  =  ( A  -h  B )
4 hisubcom.4 . . . 4  |-  D  e. 
~H
5 hisubcom.3 . . . 4  |-  C  e. 
~H
64, 5hvnegdii 22554 . . 3  |-  ( -u
1  .h  ( D  -h  C ) )  =  ( C  -h  D )
73, 6oveq12i 6085 . 2  |-  ( (
-u 1  .h  ( B  -h  A ) ) 
.ih  ( -u 1  .h  ( D  -h  C
) ) )  =  ( ( A  -h  B )  .ih  ( C  -h  D ) )
8 neg1cn 10057 . . . 4  |-  -u 1  e.  CC
91, 2hvsubcli 22514 . . . 4  |-  ( B  -h  A )  e. 
~H
104, 5hvsubcli 22514 . . . 4  |-  ( D  -h  C )  e. 
~H
118, 8, 9, 10his35i 22581 . . 3  |-  ( (
-u 1  .h  ( B  -h  A ) ) 
.ih  ( -u 1  .h  ( D  -h  C
) ) )  =  ( ( -u 1  x.  ( * `  -u 1
) )  x.  (
( B  -h  A
)  .ih  ( D  -h  C ) ) )
12 1re 9080 . . . . . . . 8  |-  1  e.  RR
1312renegcli 9352 . . . . . . 7  |-  -u 1  e.  RR
14 cjre 11934 . . . . . . 7  |-  ( -u
1  e.  RR  ->  ( * `  -u 1
)  =  -u 1
)
1513, 14ax-mp 8 . . . . . 6  |-  ( * `
 -u 1 )  = 
-u 1
1615oveq2i 6084 . . . . 5  |-  ( -u
1  x.  ( * `
 -u 1 ) )  =  ( -u 1  x.  -u 1 )
17 ax-1cn 9038 . . . . . 6  |-  1  e.  CC
1817, 17mul2negi 9471 . . . . 5  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
19 1t1e1 10116 . . . . 5  |-  ( 1  x.  1 )  =  1
2016, 18, 193eqtri 2459 . . . 4  |-  ( -u
1  x.  ( * `
 -u 1 ) )  =  1
2120oveq1i 6083 . . 3  |-  ( (
-u 1  x.  (
* `  -u 1 ) )  x.  ( ( B  -h  A ) 
.ih  ( D  -h  C ) ) )  =  ( 1  x.  ( ( B  -h  A )  .ih  ( D  -h  C ) ) )
229, 10hicli 22573 . . . 4  |-  ( ( B  -h  A ) 
.ih  ( D  -h  C ) )  e.  CC
2322mulid2i 9083 . . 3  |-  ( 1  x.  ( ( B  -h  A )  .ih  ( D  -h  C
) ) )  =  ( ( B  -h  A )  .ih  ( D  -h  C ) )
2411, 21, 233eqtri 2459 . 2  |-  ( (
-u 1  .h  ( B  -h  A ) ) 
.ih  ( -u 1  .h  ( D  -h  C
) ) )  =  ( ( B  -h  A )  .ih  ( D  -h  C ) )
257, 24eqtr3i 2457 1  |-  ( ( A  -h  B ) 
.ih  ( C  -h  D ) )  =  ( ( B  -h  A )  .ih  ( D  -h  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   RRcr 8979   1c1 8981    x. cmul 8985   -ucneg 9282   *ccj 11891   ~Hchil 22412    .h csm 22414    .ih csp 22415    -h cmv 22418
This theorem is referenced by:  lnophmlem2  23510
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-hfvadd 22493  ax-hvcom 22494  ax-hfvmul 22498  ax-hvmulid 22499  ax-hvmulass 22500  ax-hvdistr1 22501  ax-hfi 22571  ax-his1 22574  ax-his3 22576
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-2 10048  df-cj 11894  df-re 11895  df-im 11896  df-hvsub 22464
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