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Theorem hmopm 27353
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 9612 . . 3  |-  ( A  e.  RR  ->  A  e.  CC )
2 hmopf 27206 . . 3  |-  ( T  e.  HrmOp  ->  T : ~H
--> ~H )
3 homulcl 27091 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
41, 2, 3syl2an 475 . 2  |-  ( ( A  e.  RR  /\  T  e.  HrmOp )  -> 
( A  .op  T
) : ~H --> ~H )
5 cjre 13121 . . . . . 6  |-  ( A  e.  RR  ->  (
* `  A )  =  A )
6 hmop 27254 . . . . . . 7  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H  /\  y  e. 
~H )  ->  (
x  .ih  ( T `  y ) )  =  ( ( T `  x )  .ih  y
) )
763expb 1198 . . . . . 6  |-  ( ( T  e.  HrmOp  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( T `  y
) )  =  ( ( T `  x
)  .ih  y )
)
85, 7oveqan12d 6297 . . . . 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 646 . . . 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 564 . . . . 5  |-  ( ( A  e.  RR  /\  T  e.  HrmOp )  -> 
( A  e.  CC  /\  T : ~H --> ~H )
)
11 homval 27073 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  y  e.  ~H )  ->  ( ( A  .op  T ) `  y )  =  ( A  .h  ( T `  y ) ) )
12113expa 1197 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  y  e.  ~H )  ->  ( ( A 
.op  T ) `  y )  =  ( A  .h  ( T `
 y ) ) )
1312adantrl 714 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( A  .op  T ) `  y )  =  ( A  .h  ( T `
 y ) ) )
1413oveq2d 6294 . . . . . 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 752 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  A  e.  CC )
16 simprl 756 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  x  e.  ~H )
17 ffvelrn 6007 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  y  e.  ~H )  ->  ( T `  y
)  e.  ~H )
1817ad2ant2l 744 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( T `  y )  e.  ~H )
19 his5 26417 . . . . . . 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 1230 . . . . . 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 2443 . . . . 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 469 . . . 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 27073 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
24233expa 1197 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
2524adantrr 715 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
2625oveq1d 6293 . . . . . 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 6007 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
2827ad2ant2lr 746 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( T `  x )  e.  ~H )
29 simprr 758 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  y  e.  ~H )
30 ax-his3 26415 . . . . . . 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 1230 . . . . . 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 2443 . . . . 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 469 . . . 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 2453 . . 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 2825 . 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 27205 . 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 662 1  |-  ( ( A  e.  RR  /\  T  e.  HrmOp )  -> 
( A  .op  T
)  e.  HrmOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   -->wf 5565   ` cfv 5569  (class class class)co 6278   CCcc 9520   RRcr 9521    x. cmul 9527   *ccj 13078   ~Hchil 26250    .h csm 26252    .ih csp 26253    .op chot 26270   HrmOpcho 26281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-hilex 26330  ax-hfvmul 26336  ax-hfi 26410  ax-his1 26413  ax-his3 26415
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-2 10635  df-cj 13081  df-re 13082  df-im 13083  df-homul 27063  df-hmop 27176
This theorem is referenced by:  hmopd  27354  leopmuli  27465  leopmul  27466  leopmul2i  27467  leopnmid  27470  nmopleid  27471  opsqrlem1  27472  opsqrlem4  27475
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