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Theorem hmops 22616
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 22470 . . 3  |-  ( T  e.  HrmOp  ->  T : ~H
--> ~H )
2 hmopf 22470 . . 3  |-  ( U  e.  HrmOp  ->  U : ~H
--> ~H )
3 hoaddcl 22354 . . 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 22518 . . . . . . 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 22518 . . . . . . 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 5893 . . . . 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 22336 . . . . . . . . 9  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  y  e.  ~H )  ->  ( ( T  +op  U ) `  y )  =  ( ( T `
 y )  +h  ( U `  y
) ) )
1312oveq2d 5890 . . . . . . . 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 5679 . . . . . . . 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 5679 . . . . . . . 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 21685 . . . . . . 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 2328 . . . . 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 22336 . . . . . . . . 9  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  +op  U ) `  x )  =  ( ( T `
 x )  +h  ( U `  x
) ) )
2625oveq1d 5889 . . . . . . . 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 5679 . . . . . . . 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 5679 . . . . . . . 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 21678 . . . . . . 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 2328 . . . . 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 2338 . . 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 2648 . 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 22469 . 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 1632    e. wcel 1696   A.wral 2556   -->wf 5267   ` cfv 5271  (class class class)co 5874    + caddc 8756   ~Hchil 21515    +h cva 21516    .ih csp 21518    +op chos 21534   HrmOpcho 21546
This theorem is referenced by:  hmopd  22618  leopadd  22728  opsqrlem4  22739
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-hilex 21595  ax-hfvadd 21596  ax-hfi 21674  ax-his1 21677  ax-his2 21678
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-cj 11600  df-re 11601  df-im 11602  df-hosum 22326  df-hmop 22440
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