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Theorem eigorthi 22342
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 5765 . . . 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 21590 . . . . 5  |-  ( ( D  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  ( D  .h  B ) )  =  ( ( * `  D )  x.  ( A  .ih  B ) ) )
62, 3, 4, 5mp3an 1282 . . . 4  |-  ( A 
.ih  ( D  .h  B ) )  =  ( ( * `  D )  x.  ( A  .ih  B ) )
71, 6syl6eq 2304 . . 3  |-  ( ( T `  B )  =  ( D  .h  B )  ->  ( A  .ih  ( T `  B ) )  =  ( ( * `  D )  x.  ( A  .ih  B ) ) )
8 oveq1 5764 . . . 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 21588 . . . . 5  |-  ( ( C  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  (
( C  .h  A
)  .ih  B )  =  ( C  x.  ( A  .ih  B ) ) )
119, 3, 4, 10mp3an 1282 . . . 4  |-  ( ( C  .h  A ) 
.ih  B )  =  ( C  x.  ( A  .ih  B ) )
128, 11syl6eq 2304 . . 3  |-  ( ( T `  A )  =  ( C  .h  A )  ->  (
( T `  A
)  .ih  B )  =  ( C  x.  ( A  .ih  B ) ) )
137, 12eqeqan12rd 2272 . 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 21585 . . . . . . . 8  |-  ( A 
.ih  B )  e.  CC
152cjcli 11584 . . . . . . . . 9  |-  ( * `
 D )  e.  CC
16 mulcan2 9339 . . . . . . . . 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 1272 . . . . . . . 8  |-  ( ( ( A  .ih  B
)  e.  CC  /\  ( A  .ih  B )  =/=  0 )  -> 
( ( ( * `
 D )  x.  ( A  .ih  B
) )  =  ( C  x.  ( A 
.ih  B ) )  <-> 
( * `  D
)  =  C ) )
1814, 17mpan 654 . . . . . . 7  |-  ( ( A  .ih  B )  =/=  0  ->  (
( ( * `  D )  x.  ( A  .ih  B ) )  =  ( C  x.  ( A  .ih  B ) )  <->  ( * `  D )  =  C ) )
19 eqcom 2258 . . . . . . 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 2488 . . . 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 5765 . . . 4  |-  ( ( A  .ih  B )  =  0  ->  (
( * `  D
)  x.  ( A 
.ih  B ) )  =  ( ( * `
 D )  x.  0 ) )
25 oveq2 5765 . . . . 5  |-  ( ( A  .ih  B )  =  0  ->  ( C  x.  ( A  .ih  B ) )  =  ( C  x.  0 ) )
269mul01i 8935 . . . . . 6  |-  ( C  x.  0 )  =  0
2715mul01i 8935 . . . . . 6  |-  ( ( * `  D )  x.  0 )  =  0
2826, 27eqtr4i 2279 . . . . 5  |-  ( C  x.  0 )  =  ( ( * `  D )  x.  0 )
2925, 28syl6eq 2304 . . . 4  |-  ( ( A  .ih  B )  =  0  ->  ( C  x.  ( A  .ih  B ) )  =  ( ( * `  D )  x.  0 ) )
3024, 29eqtr4d 2291 . . 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 683 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   ` cfv 4638  (class class class)co 5757   CCcc 8668   0cc0 8670    x. cmul 8675   *ccj 11511   ~Hchil 21424    .h csm 21426    .ih csp 21427
This theorem is referenced by:  eigorth  22343
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-hfvmul 21510  ax-hfi 21583  ax-his1 21586  ax-his3 21588
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-po 4251  df-so 4252  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-iota 6190  df-riota 6237  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-2 9737  df-cj 11514  df-re 11515  df-im 11516
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