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Theorem hmops 23515
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 23369 . . 3  |-  ( T  e.  HrmOp  ->  T : ~H
--> ~H )
2 hmopf 23369 . . 3  |-  ( U  e.  HrmOp  ->  U : ~H
--> ~H )
3 hoaddcl 23253 . . 3  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T  +op  U
) : ~H --> ~H )
41, 2, 3syl2an 464 . 2  |-  ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( T  +op  U ) : ~H --> ~H )
5 hmop 23417 . . . . . . 7  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H  /\  y  e. 
~H )  ->  (
x  .ih  ( T `  y ) )  =  ( ( T `  x )  .ih  y
) )
653expb 1154 . . . . . 6  |-  ( ( T  e.  HrmOp  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( T `  y
) )  =  ( ( T `  x
)  .ih  y )
)
7 hmop 23417 . . . . . . 7  |-  ( ( U  e.  HrmOp  /\  x  e.  ~H  /\  y  e. 
~H )  ->  (
x  .ih  ( U `  y ) )  =  ( ( U `  x )  .ih  y
) )
873expb 1154 . . . . . 6  |-  ( ( U  e.  HrmOp  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( U `  y
) )  =  ( ( U `  x
)  .ih  y )
)
96, 8oveqan12d 6092 . . . . 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 805 . . . 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 550 . . . . 5  |-  ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)
12 hosval 23235 . . . . . . . . 9  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  y  e.  ~H )  ->  ( ( T  +op  U ) `  y )  =  ( ( T `
 y )  +h  ( U `  y
) ) )
1312oveq2d 6089 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  y  e.  ~H )  ->  ( x  .ih  (
( T  +op  U
) `  y )
)  =  ( x 
.ih  ( ( T `
 y )  +h  ( U `  y
) ) ) )
14133expa 1153 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  y  e.  ~H )  ->  (
x  .ih  ( ( T  +op  U ) `  y ) )  =  ( x  .ih  (
( T `  y
)  +h  ( U `
 y ) ) ) )
1514adantrl 697 . . . . . 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 733 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  x  e.  ~H )
17 ffvelrn 5860 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  y  e.  ~H )  ->  ( T `  y
)  e.  ~H )
1817ad2ant2rl 730 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( T `  y )  e.  ~H )
19 ffvelrn 5860 . . . . . . . 8  |-  ( ( U : ~H --> ~H  /\  y  e.  ~H )  ->  ( U `  y
)  e.  ~H )
2019ad2ant2l 727 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( U `  y )  e.  ~H )
21 his7 22584 . . . . . . 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 1184 . . . . . 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 2467 . . . . 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 458 . . . 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 23235 . . . . . . . . 9  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  +op  U ) `  x )  =  ( ( T `
 x )  +h  ( U `  x
) ) )
2625oveq1d 6088 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( T 
+op  U ) `  x )  .ih  y
)  =  ( ( ( T `  x
)  +h  ( U `
 x ) ) 
.ih  y ) )
27263expa 1153 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  (
( ( T  +op  U ) `  x ) 
.ih  y )  =  ( ( ( T `
 x )  +h  ( U `  x
) )  .ih  y
) )
2827adantrr 698 . . . . . 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 5860 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
3029ad2ant2r 728 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( T `  x )  e.  ~H )
31 ffvelrn 5860 . . . . . . . 8  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
3231ad2ant2lr 729 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( U `  x )  e.  ~H )
33 simprr 734 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  y  e.  ~H )
34 ax-his2 22577 . . . . . . 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 1184 . . . . . 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 2467 . . . . 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 458 . . . 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 2477 . . 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 2790 . 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 23368 . 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 646 1  |-  ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( T  +op  U )  e. 
HrmOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   -->wf 5442   ` cfv 5446  (class class class)co 6073    + caddc 8985   ~Hchil 22414    +h cva 22415    .ih csp 22417    +op chos 22433   HrmOpcho 22445
This theorem is referenced by:  hmopd  23517  leopadd  23627  opsqrlem4  23638
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-hilex 22494  ax-hfvadd 22495  ax-hfi 22573  ax-his1 22576  ax-his2 22577
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-2 10050  df-cj 11896  df-re 11897  df-im 11898  df-hosum 23225  df-hmop 23339
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