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Theorem eigorthi 23301
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 6056 . . . 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 22549 . . . . 5  |-  ( ( D  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( D  .h  B ) )  =  ( ( * `  D )  x.  ( A  .ih  B ) ) )
62, 3, 4, 5mp3an 1279 . . . 4  |-  ( A 
.ih  ( D  .h  B ) )  =  ( ( * `  D )  x.  ( A  .ih  B ) )
71, 6syl6eq 2460 . . 3  |-  ( ( T `  B )  =  ( D  .h  B )  ->  ( A  .ih  ( T `  B ) )  =  ( ( * `  D )  x.  ( A  .ih  B ) ) )
8 oveq1 6055 . . . 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 22547 . . . . 5  |-  ( ( C  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  (
( C  .h  A
)  .ih  B )  =  ( C  x.  ( A  .ih  B ) ) )
119, 3, 4, 10mp3an 1279 . . . 4  |-  ( ( C  .h  A ) 
.ih  B )  =  ( C  x.  ( A  .ih  B ) )
128, 11syl6eq 2460 . . 3  |-  ( ( T `  A )  =  ( C  .h  A )  ->  (
( T `  A
)  .ih  B )  =  ( C  x.  ( A  .ih  B ) ) )
137, 12eqeqan12rd 2428 . 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 22544 . . . . . . . 8  |-  ( A 
.ih  B )  e.  CC
152cjcli 11937 . . . . . . . . 9  |-  ( * `
 D )  e.  CC
16 mulcan2 9624 . . . . . . . . 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 1269 . . . . . . . 8  |-  ( ( ( A  .ih  B
)  e.  CC  /\  ( A  .ih  B )  =/=  0 )  -> 
( ( ( * `
 D )  x.  ( A  .ih  B
) )  =  ( C  x.  ( A 
.ih  B ) )  <-> 
( * `  D
)  =  C ) )
1814, 17mpan 652 . . . . . . 7  |-  ( ( A  .ih  B )  =/=  0  ->  (
( ( * `  D )  x.  ( A  .ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  <->  ( * `  D )  =  C ) )
19 eqcom 2414 . . . . . . 7  |-  ( ( * `  D )  =  C  <->  C  =  ( * `  D
) )
2018, 19syl6bb 253 . . . . . 6  |-  ( ( A  .ih  B )  =/=  0  ->  (
( ( * `  D )  x.  ( A  .ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  <->  C  =  (
* `  D )
) )
2120biimpcd 216 . . . . 5  |-  ( ( ( * `  D
)  x.  ( A 
.ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  ->  ( ( A  .ih  B )  =/=  0  ->  C  =  ( * `  D
) ) )
2221necon1d 2644 . . . 4  |-  ( ( ( * `  D
)  x.  ( A 
.ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  ->  ( C  =/=  ( * `  D
)  ->  ( A  .ih  B )  =  0 ) )
2322com12 29 . . 3  |-  ( C  =/=  ( * `  D )  ->  (
( ( * `  D )  x.  ( A  .ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  ->  ( A  .ih  B )  =  0 ) )
24 oveq2 6056 . . . 4  |-  ( ( A  .ih  B )  =  0  ->  (
( * `  D
)  x.  ( A 
.ih  B ) )  =  ( ( * `
 D )  x.  0 ) )
25 oveq2 6056 . . . . 5  |-  ( ( A  .ih  B )  =  0  ->  ( C  x.  ( A  .ih  B ) )  =  ( C  x.  0 ) )
269mul01i 9220 . . . . . 6  |-  ( C  x.  0 )  =  0
2715mul01i 9220 . . . . . 6  |-  ( ( * `  D )  x.  0 )  =  0
2826, 27eqtr4i 2435 . . . . 5  |-  ( C  x.  0 )  =  ( ( * `  D )  x.  0 )
2925, 28syl6eq 2460 . . . 4  |-  ( ( A  .ih  B )  =  0  ->  ( C  x.  ( A  .ih  B ) )  =  ( ( * `  D )  x.  0 ) )
3024, 29eqtr4d 2447 . . 3  |-  ( ( A  .ih  B )  =  0  ->  (
( * `  D
)  x.  ( A 
.ih  B ) )  =  ( C  x.  ( A  .ih  B ) ) )
3123, 30impbid1 195 . 2  |-  ( C  =/=  ( * `  D )  ->  (
( ( * `  D )  x.  ( A  .ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  <->  ( A  .ih  B )  =  0 ) )
3213, 31sylan9bb 681 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   ` cfv 5421  (class class class)co 6048   CCcc 8952   0cc0 8954    x. cmul 8959   *ccj 11864   ~Hchil 22383    .h csm 22385    .ih csp 22386
This theorem is referenced by:  eigorth  23302
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-pre-mulgt0 9031  ax-hfvmul 22469  ax-hfi 22542  ax-his1 22545  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-rmo 2682  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-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-2 10022  df-cj 11867  df-re 11868  df-im 11869
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