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

Theorem hisubcomi 22454
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 22412 . . 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 22412 . . 3  |-  ( -u
1  .h  ( D  -h  C ) )  =  ( C  -h  D )
73, 6oveq12i 6032 . 2  |-  ( (
-u 1  .h  ( B  -h  A ) ) 
.ih  ( -u 1  .h  ( D  -h  C
) ) )  =  ( ( A  -h  B )  .ih  ( C  -h  D ) )
8 neg1cn 9999 . . . 4  |-  -u 1  e.  CC
91, 2hvsubcli 22372 . . . 4  |-  ( B  -h  A )  e. 
~H
104, 5hvsubcli 22372 . . . 4  |-  ( D  -h  C )  e. 
~H
118, 8, 9, 10his35i 22439 . . 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 9023 . . . . . . . 8  |-  1  e.  RR
1312renegcli 9294 . . . . . . 7  |-  -u 1  e.  RR
14 cjre 11871 . . . . . . 7  |-  ( -u
1  e.  RR  ->  ( * `  -u 1
)  =  -u 1
)
1513, 14ax-mp 8 . . . . . 6  |-  ( * `
 -u 1 )  = 
-u 1
1615oveq2i 6031 . . . . 5  |-  ( -u
1  x.  ( * `
 -u 1 ) )  =  ( -u 1  x.  -u 1 )
17 ax-1cn 8981 . . . . . 6  |-  1  e.  CC
1817, 17mul2negi 9413 . . . . 5  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
19 1t1e1 10058 . . . . 5  |-  ( 1  x.  1 )  =  1
2016, 18, 193eqtri 2411 . . . 4  |-  ( -u
1  x.  ( * `
 -u 1 ) )  =  1
2120oveq1i 6030 . . 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 22431 . . . 4  |-  ( ( B  -h  A ) 
.ih  ( D  -h  C ) )  e.  CC
2322mulid2i 9026 . . 3  |-  ( 1  x.  ( ( B  -h  A )  .ih  ( D  -h  C
) ) )  =  ( ( B  -h  A )  .ih  ( D  -h  C ) )
2411, 21, 233eqtri 2411 . 2  |-  ( (
-u 1  .h  ( B  -h  A ) ) 
.ih  ( -u 1  .h  ( D  -h  C
) ) )  =  ( ( B  -h  A )  .ih  ( D  -h  C ) )
257, 24eqtr3i 2409 1  |-  ( ( A  -h  B ) 
.ih  ( C  -h  D ) )  =  ( ( B  -h  A )  .ih  ( D  -h  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   RRcr 8922   1c1 8924    x. cmul 8928   -ucneg 9224   *ccj 11828   ~Hchil 22270    .h csm 22272    .ih csp 22273    -h cmv 22276
This theorem is referenced by:  lnophmlem2  23368
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-hfvadd 22351  ax-hvcom 22352  ax-hfvmul 22356  ax-hvmulid 22357  ax-hvmulass 22358  ax-hvdistr1 22359  ax-hfi 22429  ax-his1 22432  ax-his3 22434
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-2 9990  df-cj 11831  df-re 11832  df-im 11833  df-hvsub 22322
  Copyright terms: Public domain W3C validator