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Theorem eigorthi 22247
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 5718 . . . 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 21495 . . . . 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 2301 . . 3  |-  ( ( T `  B )  =  ( D  .h  B )  ->  ( A  .ih  ( T `  B ) )  =  ( ( * `  D )  x.  ( A  .ih  B ) ) )
8 oveq1 5717 . . . 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 21493 . . . . 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 2301 . . 3  |-  ( ( T `  A )  =  ( C  .h  A )  ->  (
( T `  A
)  .ih  B )  =  ( C  x.  ( A  .ih  B ) ) )
137, 12eqeqan12rd 2269 . 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 21490 . . . . . . . 8  |-  ( A 
.ih  B )  e.  CC
152cjcli 11531 . . . . . . . . 9  |-  ( * `
 D )  e.  CC
16 mulcan2 9286 . . . . . . . . 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 2255 . . . . . . 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 2481 . . . 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 5718 . . . 4  |-  ( ( A  .ih  B )  =  0  ->  (
( * `  D
)  x.  ( A 
.ih  B ) )  =  ( ( * `
 D )  x.  0 ) )
25 oveq2 5718 . . . . 5  |-  ( ( A  .ih  B )  =  0  ->  ( C  x.  ( A  .ih  B ) )  =  ( C  x.  0 ) )
269mul01i 8882 . . . . . 6  |-  ( C  x.  0 )  =  0
2715mul01i 8882 . . . . . 6  |-  ( ( * `  D )  x.  0 )  =  0
2826, 27eqtr4i 2276 . . . . 5  |-  ( C  x.  0 )  =  ( ( * `  D )  x.  0 )
2925, 28syl6eq 2301 . . . 4  |-  ( ( A  .ih  B )  =  0  ->  ( C  x.  ( A  .ih  B ) )  =  ( ( * `  D )  x.  0 ) )
3024, 29eqtr4d 2288 . . 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 2412   ` cfv 4592  (class class class)co 5710   CCcc 8615   0cc0 8617    x. cmul 8622   *ccj 11458   ~Hchil 21329    .h csm 21331    .ih csp 21332
This theorem is referenced by:  eigorth  22248
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-hfvmul 21415  ax-hfi 21488  ax-his1 21491  ax-his3 21493
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-2 9684  df-cj 11461  df-re 11462  df-im 11463
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