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Theorem hmops 22600
Description: The sum of two Hermitian operators is Hermitian. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hmops  |-  ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( T  +op  U )  e. 
HrmOp )

Proof of Theorem hmops
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hmopf 22454 . . 3  |-  ( T  e.  HrmOp  ->  T : ~H
--> ~H )
2 hmopf 22454 . . 3  |-  ( U  e.  HrmOp  ->  U : ~H
--> ~H )
3 hoaddcl 22338 . . 3  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T  +op  U
) : ~H --> ~H )
41, 2, 3syl2an 463 . 2  |-  ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( T  +op  U ) : ~H --> ~H )
5 hmop 22502 . . . . . . 7  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H  /\  y  e. 
~H )  ->  (
x  .ih  ( T `  y ) )  =  ( ( T `  x )  .ih  y
) )
653expb 1152 . . . . . 6  |-  ( ( T  e.  HrmOp  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( T `  y
) )  =  ( ( T `  x
)  .ih  y )
)
7 hmop 22502 . . . . . . 7  |-  ( ( U  e.  HrmOp  /\  x  e.  ~H  /\  y  e. 
~H )  ->  (
x  .ih  ( U `  y ) )  =  ( ( U `  x )  .ih  y
) )
873expb 1152 . . . . . 6  |-  ( ( U  e.  HrmOp  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( U `  y
) )  =  ( ( U `  x
)  .ih  y )
)
96, 8oveqan12d 5877 . . . . 5  |-  ( ( ( T  e.  HrmOp  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  /\  ( U  e.  HrmOp  /\  ( x  e.  ~H  /\  y  e. 
~H ) ) )  ->  ( ( x 
.ih  ( T `  y ) )  +  ( x  .ih  ( U `  y )
) )  =  ( ( ( T `  x )  .ih  y
)  +  ( ( U `  x ) 
.ih  y ) ) )
109anandirs 804 . . . 4  |-  ( ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( (
x  .ih  ( T `  y ) )  +  ( x  .ih  ( U `  y )
) )  =  ( ( ( T `  x )  .ih  y
)  +  ( ( U `  x ) 
.ih  y ) ) )
111, 2anim12i 549 . . . . 5  |-  ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)
12 hosval 22320 . . . . . . . . 9  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  y  e.  ~H )  ->  ( ( T  +op  U ) `  y )  =  ( ( T `
 y )  +h  ( U `  y
) ) )
1312oveq2d 5874 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  y  e.  ~H )  ->  ( x  .ih  (
( T  +op  U
) `  y )
)  =  ( x 
.ih  ( ( T `
 y )  +h  ( U `  y
) ) ) )
14133expa 1151 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  y  e.  ~H )  ->  (
x  .ih  ( ( T  +op  U ) `  y ) )  =  ( x  .ih  (
( T `  y
)  +h  ( U `
 y ) ) ) )
1514adantrl 696 . . . . . 6  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( ( T  +op  U ) `  y ) )  =  ( x 
.ih  ( ( T `
 y )  +h  ( U `  y
) ) ) )
16 simprl 732 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~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 )
1817ad2ant2rl 729 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( T `  y )  e.  ~H )
19 ffvelrn 5663 . . . . . . . 8  |-  ( ( U : ~H --> ~H  /\  y  e.  ~H )  ->  ( U `  y
)  e.  ~H )
2019ad2ant2l 726 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( U `  y )  e.  ~H )
21 his7 21669 . . . . . . 7  |-  ( ( x  e.  ~H  /\  ( T `  y )  e.  ~H  /\  ( U `  y )  e.  ~H )  ->  (
x  .ih  ( ( T `  y )  +h  ( U `  y
) ) )  =  ( ( x  .ih  ( T `  y ) )  +  ( x 
.ih  ( U `  y ) ) ) )
2216, 18, 20, 21syl3anc 1182 . . . . . 6  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( ( T `  y )  +h  ( U `  y )
) )  =  ( ( x  .ih  ( T `  y )
)  +  ( x 
.ih  ( U `  y ) ) ) )
2315, 22eqtrd 2315 . . . . 5  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( ( T  +op  U ) `  y ) )  =  ( ( x  .ih  ( T `
 y ) )  +  ( x  .ih  ( U `  y ) ) ) )
2411, 23sylan 457 . . . 4  |-  ( ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( ( T  +op  U ) `  y ) )  =  ( ( x  .ih  ( T `
 y ) )  +  ( x  .ih  ( U `  y ) ) ) )
25 hosval 22320 . . . . . . . . 9  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  +op  U ) `  x )  =  ( ( T `
 x )  +h  ( U `  x
) ) )
2625oveq1d 5873 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( T 
+op  U ) `  x )  .ih  y
)  =  ( ( ( T `  x
)  +h  ( U `
 x ) ) 
.ih  y ) )
27263expa 1151 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  (
( ( T  +op  U ) `  x ) 
.ih  y )  =  ( ( ( T `
 x )  +h  ( U `  x
) )  .ih  y
) )
2827adantrr 697 . . . . . 6  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( (
( T  +op  U
) `  x )  .ih  y )  =  ( ( ( T `  x )  +h  ( U `  x )
)  .ih  y )
)
29 ffvelrn 5663 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
3029ad2ant2r 727 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( T `  x )  e.  ~H )
31 ffvelrn 5663 . . . . . . . 8  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
3231ad2ant2lr 728 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( U `  x )  e.  ~H )
33 simprr 733 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  y  e.  ~H )
34 ax-his2 21662 . . . . . . 7  |-  ( ( ( T `  x
)  e.  ~H  /\  ( U `  x )  e.  ~H  /\  y  e.  ~H )  ->  (
( ( T `  x )  +h  ( U `  x )
)  .ih  y )  =  ( ( ( T `  x ) 
.ih  y )  +  ( ( U `  x )  .ih  y
) ) )
3530, 32, 33, 34syl3anc 1182 . . . . . 6  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( (
( T `  x
)  +h  ( U `
 x ) ) 
.ih  y )  =  ( ( ( T `
 x )  .ih  y )  +  ( ( U `  x
)  .ih  y )
) )
3628, 35eqtrd 2315 . . . . 5  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( (
( T  +op  U
) `  x )  .ih  y )  =  ( ( ( T `  x )  .ih  y
)  +  ( ( U `  x ) 
.ih  y ) ) )
3711, 36sylan 457 . . . 4  |-  ( ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( (
( T  +op  U
) `  x )  .ih  y )  =  ( ( ( T `  x )  .ih  y
)  +  ( ( U `  x ) 
.ih  y ) ) )
3810, 24, 373eqtr4d 2325 . . 3  |-  ( ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  /\  ( x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( ( T  +op  U ) `  y ) )  =  ( ( ( T  +op  U
) `  x )  .ih  y ) )
3938ralrimivva 2635 . 2  |-  ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  A. x  e.  ~H  A. y  e. 
~H  ( x  .ih  ( ( T  +op  U ) `  y ) )  =  ( ( ( T  +op  U
) `  x )  .ih  y ) )
40 elhmop 22453 . 2  |-  ( ( T  +op  U )  e.  HrmOp 
<->  ( ( T  +op  U ) : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  ~H  (
x  .ih  ( ( T  +op  U ) `  y ) )  =  ( ( ( T 
+op  U ) `  x )  .ih  y
) ) )
414, 39, 40sylanbrc 645 1  |-  ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( T  +op  U )  e. 
HrmOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858    + caddc 8740   ~Hchil 21499    +h cva 21500    .ih csp 21502    +op chos 21518   HrmOpcho 21530
This theorem is referenced by:  hmopd  22602  leopadd  22712  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-hfvadd 21580  ax-hfi 21658  ax-his1 21661  ax-his2 21662
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-hosum 22310  df-hmop 22424
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