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Theorem hmopm 23477
Description: The scalar product of a Hermitian operator with a real is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hmopm  |-  ( ( A  e.  RR  /\  T  e.  HrmOp )  -> 
( A  .op  T
)  e.  HrmOp )

Proof of Theorem hmopm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 recn 9036 . . 3  |-  ( A  e.  RR  ->  A  e.  CC )
2 hmopf 23330 . . 3  |-  ( T  e.  HrmOp  ->  T : ~H
--> ~H )
3 homulcl 23215 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
41, 2, 3syl2an 464 . 2  |-  ( ( A  e.  RR  /\  T  e.  HrmOp )  -> 
( A  .op  T
) : ~H --> ~H )
5 cjre 11899 . . . . . 6  |-  ( A  e.  RR  ->  (
* `  A )  =  A )
6 hmop 23378 . . . . . . 7  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H  /\  y  e. 
~H )  ->  (
x  .ih  ( T `  y ) )  =  ( ( T `  x )  .ih  y
) )
763expb 1154 . . . . . 6  |-  ( ( T  e.  HrmOp  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( T `  y
) )  =  ( ( T `  x
)  .ih  y )
)
85, 7oveqan12d 6059 . . . . 5  |-  ( ( A  e.  RR  /\  ( T  e.  HrmOp  /\  (
x  e.  ~H  /\  y  e.  ~H )
) )  ->  (
( * `  A
)  x.  ( x 
.ih  ( T `  y ) ) )  =  ( A  x.  ( ( T `  x )  .ih  y
) ) )
98anassrs 630 . . . 4  |-  ( ( ( A  e.  RR  /\  T  e.  HrmOp )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( (
* `  A )  x.  ( x  .ih  ( T `  y )
) )  =  ( A  x.  ( ( T `  x ) 
.ih  y ) ) )
101, 2anim12i 550 . . . . 5  |-  ( ( A  e.  RR  /\  T  e.  HrmOp )  -> 
( A  e.  CC  /\  T : ~H --> ~H )
)
11 homval 23197 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  y  e.  ~H )  ->  ( ( A  .op  T ) `  y )  =  ( A  .h  ( T `  y ) ) )
12113expa 1153 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  y  e.  ~H )  ->  ( ( A 
.op  T ) `  y )  =  ( A  .h  ( T `
 y ) ) )
1312adantrl 697 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( A  .op  T ) `  y )  =  ( A  .h  ( T `
 y ) ) )
1413oveq2d 6056 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( ( A  .op  T ) `  y ) )  =  ( x 
.ih  ( A  .h  ( T `  y ) ) ) )
15 simpll 731 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  A  e.  CC )
16 simprl 733 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  x  e.  ~H )
17 ffvelrn 5827 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  y  e.  ~H )  ->  ( T `  y
)  e.  ~H )
1817ad2ant2l 727 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( T `  y )  e.  ~H )
19 his5 22541 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  ~H  /\  ( T `  y )  e.  ~H )  ->  (
x  .ih  ( A  .h  ( T `  y
) ) )  =  ( ( * `  A )  x.  (
x  .ih  ( T `  y ) ) ) )
2015, 16, 18, 19syl3anc 1184 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( A  .h  ( T `  y )
) )  =  ( ( * `  A
)  x.  ( x 
.ih  ( T `  y ) ) ) )
2114, 20eqtrd 2436 . . . . 5  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( ( A  .op  T ) `  y ) )  =  ( ( * `  A )  x.  ( x  .ih  ( T `  y ) ) ) )
2210, 21sylan 458 . . . 4  |-  ( ( ( A  e.  RR  /\  T  e.  HrmOp )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( ( A  .op  T ) `  y ) )  =  ( ( * `  A )  x.  ( x  .ih  ( T `  y ) ) ) )
23 homval 23197 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
24233expa 1153 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
2524adantrr 698 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
2625oveq1d 6055 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( (
( A  .op  T
) `  x )  .ih  y )  =  ( ( A  .h  ( T `  x )
)  .ih  y )
)
27 ffvelrn 5827 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
2827ad2ant2lr 729 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( T `  x )  e.  ~H )
29 simprr 734 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  y  e.  ~H )
30 ax-his3 22539 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( T `  x )  e.  ~H  /\  y  e.  ~H )  ->  (
( A  .h  ( T `  x )
)  .ih  y )  =  ( A  x.  ( ( T `  x )  .ih  y
) ) )
3115, 28, 29, 30syl3anc 1184 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( A  .h  ( T `  x ) )  .ih  y )  =  ( A  x.  ( ( T `  x ) 
.ih  y ) ) )
3226, 31eqtrd 2436 . . . . 5  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( (
( A  .op  T
) `  x )  .ih  y )  =  ( A  x.  ( ( T `  x ) 
.ih  y ) ) )
3310, 32sylan 458 . . . 4  |-  ( ( ( A  e.  RR  /\  T  e.  HrmOp )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( (
( A  .op  T
) `  x )  .ih  y )  =  ( A  x.  ( ( T `  x ) 
.ih  y ) ) )
349, 22, 333eqtr4d 2446 . . 3  |-  ( ( ( A  e.  RR  /\  T  e.  HrmOp )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( ( A  .op  T ) `  y ) )  =  ( ( ( A  .op  T
) `  x )  .ih  y ) )
3534ralrimivva 2758 . 2  |-  ( ( A  e.  RR  /\  T  e.  HrmOp )  ->  A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( ( A  .op  T ) `  y ) )  =  ( ( ( A 
.op  T ) `  x )  .ih  y
) )
36 elhmop 23329 . 2  |-  ( ( A  .op  T )  e.  HrmOp 
<->  ( ( A  .op  T ) : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( ( A  .op  T ) `  y ) )  =  ( ( ( A 
.op  T ) `  x )  .ih  y
) ) )
374, 35, 36sylanbrc 646 1  |-  ( ( A  e.  RR  /\  T  e.  HrmOp )  -> 
( A  .op  T
)  e.  HrmOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945    x. cmul 8951   *ccj 11856   ~Hchil 22375    .h csm 22377    .ih csp 22378    .op chot 22395   HrmOpcho 22406
This theorem is referenced by:  hmopd  23478  leopmuli  23589  leopmul  23590  leopmul2i  23591  leopnmid  23594  nmopleid  23595  opsqrlem1  23596  opsqrlem4  23599
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-hilex 22455  ax-hfvmul 22461  ax-hfi 22534  ax-his1 22537  ax-his3 22539
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-cj 11859  df-re 11860  df-im 11861  df-homul 23187  df-hmop 23300
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