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Theorem eigorthi 22413
Description: A necessary and sufficient condition (that holds when  T is a Hermitian operator) for two eigenvectors  A and 
B to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Jan-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
eigorthi.1  |-  A  e. 
~H
eigorthi.2  |-  B  e. 
~H
eigorthi.3  |-  C  e.  CC
eigorthi.4  |-  D  e.  CC
Assertion
Ref Expression
eigorthi  |-  ( ( ( ( T `  A )  =  ( C  .h  A )  /\  ( T `  B )  =  ( D  .h  B ) )  /\  C  =/=  ( * `  D
) )  ->  (
( A  .ih  ( T `  B )
)  =  ( ( T `  A ) 
.ih  B )  <->  ( A  .ih  B )  =  0 ) )

Proof of Theorem eigorthi
StepHypRef Expression
1 oveq2 5828 . . . 4  |-  ( ( T `  B )  =  ( D  .h  B )  ->  ( A  .ih  ( T `  B ) )  =  ( A  .ih  ( D  .h  B )
) )
2 eigorthi.4 . . . . 5  |-  D  e.  CC
3 eigorthi.1 . . . . 5  |-  A  e. 
~H
4 eigorthi.2 . . . . 5  |-  B  e. 
~H
5 his5 21661 . . . . 5  |-  ( ( D  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( D  .h  B ) )  =  ( ( * `  D )  x.  ( A  .ih  B ) ) )
62, 3, 4, 5mp3an 1277 . . . 4  |-  ( A 
.ih  ( D  .h  B ) )  =  ( ( * `  D )  x.  ( A  .ih  B ) )
71, 6syl6eq 2332 . . 3  |-  ( ( T `  B )  =  ( D  .h  B )  ->  ( A  .ih  ( T `  B ) )  =  ( ( * `  D )  x.  ( A  .ih  B ) ) )
8 oveq1 5827 . . . 4  |-  ( ( T `  A )  =  ( C  .h  A )  ->  (
( T `  A
)  .ih  B )  =  ( ( C  .h  A )  .ih  B ) )
9 eigorthi.3 . . . . 5  |-  C  e.  CC
10 ax-his3 21659 . . . . 5  |-  ( ( C  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  (
( C  .h  A
)  .ih  B )  =  ( C  x.  ( A  .ih  B ) ) )
119, 3, 4, 10mp3an 1277 . . . 4  |-  ( ( C  .h  A ) 
.ih  B )  =  ( C  x.  ( A  .ih  B ) )
128, 11syl6eq 2332 . . 3  |-  ( ( T `  A )  =  ( C  .h  A )  ->  (
( T `  A
)  .ih  B )  =  ( C  x.  ( A  .ih  B ) ) )
137, 12eqeqan12rd 2300 . 2  |-  ( ( ( T `  A
)  =  ( C  .h  A )  /\  ( T `  B )  =  ( D  .h  B ) )  -> 
( ( A  .ih  ( T `  B ) )  =  ( ( T `  A ) 
.ih  B )  <->  ( (
* `  D )  x.  ( A  .ih  B
) )  =  ( C  x.  ( A 
.ih  B ) ) ) )
143, 4hicli 21656 . . . . . . . 8  |-  ( A 
.ih  B )  e.  CC
152cjcli 11650 . . . . . . . . 9  |-  ( * `
 D )  e.  CC
16 mulcan2 9402 . . . . . . . . 9  |-  ( ( ( * `  D
)  e.  CC  /\  C  e.  CC  /\  (
( A  .ih  B
)  e.  CC  /\  ( A  .ih  B )  =/=  0 ) )  ->  ( ( ( * `  D )  x.  ( A  .ih  B ) )  =  ( C  x.  ( A 
.ih  B ) )  <-> 
( * `  D
)  =  C ) )
1715, 9, 16mp3an12 1267 . . . . . . . 8  |-  ( ( ( A  .ih  B
)  e.  CC  /\  ( A  .ih  B )  =/=  0 )  -> 
( ( ( * `
 D )  x.  ( A  .ih  B
) )  =  ( C  x.  ( A 
.ih  B ) )  <-> 
( * `  D
)  =  C ) )
1814, 17mpan 651 . . . . . . 7  |-  ( ( A  .ih  B )  =/=  0  ->  (
( ( * `  D )  x.  ( A  .ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  <->  ( * `  D )  =  C ) )
19 eqcom 2286 . . . . . . 7  |-  ( ( * `  D )  =  C  <->  C  =  ( * `  D
) )
2018, 19syl6bb 252 . . . . . 6  |-  ( ( A  .ih  B )  =/=  0  ->  (
( ( * `  D )  x.  ( A  .ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  <->  C  =  (
* `  D )
) )
2120biimpcd 215 . . . . 5  |-  ( ( ( * `  D
)  x.  ( A 
.ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  ->  ( ( A  .ih  B )  =/=  0  ->  C  =  ( * `  D
) ) )
2221necon1d 2516 . . . 4  |-  ( ( ( * `  D
)  x.  ( A 
.ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  ->  ( C  =/=  ( * `  D
)  ->  ( A  .ih  B )  =  0 ) )
2322com12 27 . . 3  |-  ( C  =/=  ( * `  D )  ->  (
( ( * `  D )  x.  ( A  .ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  ->  ( A  .ih  B )  =  0 ) )
24 oveq2 5828 . . . 4  |-  ( ( A  .ih  B )  =  0  ->  (
( * `  D
)  x.  ( A 
.ih  B ) )  =  ( ( * `
 D )  x.  0 ) )
25 oveq2 5828 . . . . 5  |-  ( ( A  .ih  B )  =  0  ->  ( C  x.  ( A  .ih  B ) )  =  ( C  x.  0 ) )
269mul01i 8998 . . . . . 6  |-  ( C  x.  0 )  =  0
2715mul01i 8998 . . . . . 6  |-  ( ( * `  D )  x.  0 )  =  0
2826, 27eqtr4i 2307 . . . . 5  |-  ( C  x.  0 )  =  ( ( * `  D )  x.  0 )
2925, 28syl6eq 2332 . . . 4  |-  ( ( A  .ih  B )  =  0  ->  ( C  x.  ( A  .ih  B ) )  =  ( ( * `  D )  x.  0 ) )
3024, 29eqtr4d 2319 . . 3  |-  ( ( A  .ih  B )  =  0  ->  (
( * `  D
)  x.  ( A 
.ih  B ) )  =  ( C  x.  ( A  .ih  B ) ) )
3123, 30impbid1 194 . 2  |-  ( C  =/=  ( * `  D )  ->  (
( ( * `  D )  x.  ( A  .ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  <->  ( A  .ih  B )  =  0 ) )
3213, 31sylan9bb 680 1  |-  ( ( ( ( T `  A )  =  ( C  .h  A )  /\  ( T `  B )  =  ( D  .h  B ) )  /\  C  =/=  ( * `  D
) )  ->  (
( A  .ih  ( T `  B )
)  =  ( ( T `  A ) 
.ih  B )  <->  ( A  .ih  B )  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685    =/= wne 2447   ` cfv 5221  (class class class)co 5820   CCcc 8731   0cc0 8733    x. cmul 8738   *ccj 11577   ~Hchil 21495    .h csm 21497    .ih csp 21498
This theorem is referenced by:  eigorth  22414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-hfvmul 21581  ax-hfi 21654  ax-his1 21657  ax-his3 21659
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-2 9800  df-cj 11580  df-re 11581  df-im 11582
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