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Theorem eigorthi 23345
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 6092 . . . 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 22593 . . . . 5  |-  ( ( D  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( D  .h  B ) )  =  ( ( * `  D )  x.  ( A  .ih  B ) ) )
62, 3, 4, 5mp3an 1280 . . . 4  |-  ( A 
.ih  ( D  .h  B ) )  =  ( ( * `  D )  x.  ( A  .ih  B ) )
71, 6syl6eq 2486 . . 3  |-  ( ( T `  B )  =  ( D  .h  B )  ->  ( A  .ih  ( T `  B ) )  =  ( ( * `  D )  x.  ( A  .ih  B ) ) )
8 oveq1 6091 . . . 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 22591 . . . . 5  |-  ( ( C  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  (
( C  .h  A
)  .ih  B )  =  ( C  x.  ( A  .ih  B ) ) )
119, 3, 4, 10mp3an 1280 . . . 4  |-  ( ( C  .h  A ) 
.ih  B )  =  ( C  x.  ( A  .ih  B ) )
128, 11syl6eq 2486 . . 3  |-  ( ( T `  A )  =  ( C  .h  A )  ->  (
( T `  A
)  .ih  B )  =  ( C  x.  ( A  .ih  B ) ) )
137, 12eqeqan12rd 2454 . 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 22588 . . . . . . . 8  |-  ( A 
.ih  B )  e.  CC
152cjcli 11979 . . . . . . . . 9  |-  ( * `
 D )  e.  CC
16 mulcan2 9665 . . . . . . . . 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 1270 . . . . . . . 8  |-  ( ( ( A  .ih  B
)  e.  CC  /\  ( A  .ih  B )  =/=  0 )  -> 
( ( ( * `
 D )  x.  ( A  .ih  B
) )  =  ( C  x.  ( A 
.ih  B ) )  <-> 
( * `  D
)  =  C ) )
1814, 17mpan 653 . . . . . . 7  |-  ( ( A  .ih  B )  =/=  0  ->  (
( ( * `  D )  x.  ( A  .ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  <->  ( * `  D )  =  C ) )
19 eqcom 2440 . . . . . . 7  |-  ( ( * `  D )  =  C  <->  C  =  ( * `  D
) )
2018, 19syl6bb 254 . . . . . 6  |-  ( ( A  .ih  B )  =/=  0  ->  (
( ( * `  D )  x.  ( A  .ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  <->  C  =  (
* `  D )
) )
2120biimpcd 217 . . . . 5  |-  ( ( ( * `  D
)  x.  ( A 
.ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  ->  ( ( A  .ih  B )  =/=  0  ->  C  =  ( * `  D
) ) )
2221necon1d 2675 . . . 4  |-  ( ( ( * `  D
)  x.  ( A 
.ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  ->  ( C  =/=  ( * `  D
)  ->  ( A  .ih  B )  =  0 ) )
2322com12 30 . . 3  |-  ( C  =/=  ( * `  D )  ->  (
( ( * `  D )  x.  ( A  .ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  ->  ( A  .ih  B )  =  0 ) )
24 oveq2 6092 . . . 4  |-  ( ( A  .ih  B )  =  0  ->  (
( * `  D
)  x.  ( A 
.ih  B ) )  =  ( ( * `
 D )  x.  0 ) )
25 oveq2 6092 . . . . 5  |-  ( ( A  .ih  B )  =  0  ->  ( C  x.  ( A  .ih  B ) )  =  ( C  x.  0 ) )
269mul01i 9261 . . . . . 6  |-  ( C  x.  0 )  =  0
2715mul01i 9261 . . . . . 6  |-  ( ( * `  D )  x.  0 )  =  0
2826, 27eqtr4i 2461 . . . . 5  |-  ( C  x.  0 )  =  ( ( * `  D )  x.  0 )
2925, 28syl6eq 2486 . . . 4  |-  ( ( A  .ih  B )  =  0  ->  ( C  x.  ( A  .ih  B ) )  =  ( ( * `  D )  x.  0 ) )
3024, 29eqtr4d 2473 . . 3  |-  ( ( A  .ih  B )  =  0  ->  (
( * `  D
)  x.  ( A 
.ih  B ) )  =  ( C  x.  ( A  .ih  B ) ) )
3123, 30impbid1 196 . 2  |-  ( C  =/=  ( * `  D )  ->  (
( ( * `  D )  x.  ( A  .ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  <->  ( A  .ih  B )  =  0 ) )
3213, 31sylan9bb 682 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   ` cfv 5457  (class class class)co 6084   CCcc 8993   0cc0 8995    x. cmul 9000   *ccj 11906   ~Hchil 22427    .h csm 22429    .ih csp 22430
This theorem is referenced by:  eigorth  23346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-hfvmul 22513  ax-hfi 22586  ax-his1 22589  ax-his3 22591
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-2 10063  df-cj 11909  df-re 11910  df-im 11911
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