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Theorem hmopm 26616
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 9578 . . 3  |-  ( A  e.  RR  ->  A  e.  CC )
2 hmopf 26469 . . 3  |-  ( T  e.  HrmOp  ->  T : ~H
--> ~H )
3 homulcl 26354 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
41, 2, 3syl2an 477 . 2  |-  ( ( A  e.  RR  /\  T  e.  HrmOp )  -> 
( A  .op  T
) : ~H --> ~H )
5 cjre 12931 . . . . . 6  |-  ( A  e.  RR  ->  (
* `  A )  =  A )
6 hmop 26517 . . . . . . 7  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H  /\  y  e. 
~H )  ->  (
x  .ih  ( T `  y ) )  =  ( ( T `  x )  .ih  y
) )
763expb 1197 . . . . . 6  |-  ( ( T  e.  HrmOp  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( T `  y
) )  =  ( ( T `  x
)  .ih  y )
)
85, 7oveqan12d 6301 . . . . 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 648 . . . 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 566 . . . . 5  |-  ( ( A  e.  RR  /\  T  e.  HrmOp )  -> 
( A  e.  CC  /\  T : ~H --> ~H )
)
11 homval 26336 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  y  e.  ~H )  ->  ( ( A  .op  T ) `  y )  =  ( A  .h  ( T `  y ) ) )
12113expa 1196 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  y  e.  ~H )  ->  ( ( A 
.op  T ) `  y )  =  ( A  .h  ( T `
 y ) ) )
1312adantrl 715 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( A  .op  T ) `  y )  =  ( A  .h  ( T `
 y ) ) )
1413oveq2d 6298 . . . . . 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 753 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  A  e.  CC )
16 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  x  e.  ~H )
17 ffvelrn 6017 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  y  e.  ~H )  ->  ( T `  y
)  e.  ~H )
1817ad2ant2l 745 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( T `  y )  e.  ~H )
19 his5 25679 . . . . . . 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 1228 . . . . . 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 2508 . . . . 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 471 . . . 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 26336 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
24233expa 1196 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
2524adantrr 716 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
2625oveq1d 6297 . . . . . 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 6017 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
2827ad2ant2lr 747 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( T `  x )  e.  ~H )
29 simprr 756 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  y  e.  ~H )
30 ax-his3 25677 . . . . . . 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 1228 . . . . . 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 2508 . . . . 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 471 . . . 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 2518 . . 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 2885 . 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 26468 . 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 664 1  |-  ( ( A  e.  RR  /\  T  e.  HrmOp )  -> 
( A  .op  T
)  e.  HrmOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   -->wf 5582   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487    x. cmul 9493   *ccj 12888   ~Hchil 25512    .h csm 25514    .ih csp 25515    .op chot 25532   HrmOpcho 25543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-hilex 25592  ax-hfvmul 25598  ax-hfi 25672  ax-his1 25675  ax-his3 25677
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-2 10590  df-cj 12891  df-re 12892  df-im 12893  df-homul 26326  df-hmop 26439
This theorem is referenced by:  hmopd  26617  leopmuli  26728  leopmul  26729  leopmul2i  26730  leopnmid  26733  nmopleid  26734  opsqrlem1  26735  opsqrlem4  26738
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