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Theorem hmopm 22601
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 8827 . . 3  |-  ( A  e.  RR  ->  A  e.  CC )
2 hmopf 22454 . . 3  |-  ( T  e.  HrmOp  ->  T : ~H
--> ~H )
3 homulcl 22339 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
41, 2, 3syl2an 463 . 2  |-  ( ( A  e.  RR  /\  T  e.  HrmOp )  -> 
( A  .op  T
) : ~H --> ~H )
5 cjre 11624 . . . . . 6  |-  ( A  e.  RR  ->  (
* `  A )  =  A )
6 hmop 22502 . . . . . . 7  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H  /\  y  e. 
~H )  ->  (
x  .ih  ( T `  y ) )  =  ( ( T `  x )  .ih  y
) )
763expb 1152 . . . . . 6  |-  ( ( T  e.  HrmOp  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( T `  y
) )  =  ( ( T `  x
)  .ih  y )
)
85, 7oveqan12d 5877 . . . . 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 629 . . . 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 549 . . . . 5  |-  ( ( A  e.  RR  /\  T  e.  HrmOp )  -> 
( A  e.  CC  /\  T : ~H --> ~H )
)
11 homval 22321 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  y  e.  ~H )  ->  ( ( A  .op  T ) `  y )  =  ( A  .h  ( T `  y ) ) )
12113expa 1151 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  y  e.  ~H )  ->  ( ( A 
.op  T ) `  y )  =  ( A  .h  ( T `
 y ) ) )
1312adantrl 696 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( A  .op  T ) `  y )  =  ( A  .h  ( T `
 y ) ) )
1413oveq2d 5874 . . . . . 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 730 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  A  e.  CC )
16 simprl 732 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  x  e.  ~H )
17 ffvelrn 5663 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  y  e.  ~H )  ->  ( T `  y
)  e.  ~H )
1817ad2ant2l 726 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( T `  y )  e.  ~H )
19 his5 21665 . . . . . . 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 1182 . . . . . 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 2315 . . . . 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 457 . . . 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 22321 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
24233expa 1151 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
2524adantrr 697 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
2625oveq1d 5873 . . . . . 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 5663 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
2827ad2ant2lr 728 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( T `  x )  e.  ~H )
29 simprr 733 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  y  e.  ~H )
30 ax-his3 21663 . . . . . . 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 1182 . . . . . 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 2315 . . . . 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 457 . . . 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 2325 . . 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 2635 . 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 22453 . 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 645 1  |-  ( ( A  e.  RR  /\  T  e.  HrmOp )  -> 
( A  .op  T
)  e.  HrmOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736    x. cmul 8742   *ccj 11581   ~Hchil 21499    .h csm 21501    .ih csp 21502    .op chot 21519   HrmOpcho 21530
This theorem is referenced by:  hmopd  22602  leopmuli  22713  leopmul  22714  leopmul2i  22715  leopnmid  22718  nmopleid  22719  opsqrlem1  22720  opsqrlem4  22723
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-hilex 21579  ax-hfvmul 21585  ax-hfi 21658  ax-his1 21661  ax-his3 21663
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-cj 11584  df-re 11585  df-im 11586  df-homul 22311  df-hmop 22424
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