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Theorem hmops 27140
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 26994 . . 3  |-  ( T  e.  HrmOp  ->  T : ~H
--> ~H )
2 hmopf 26994 . . 3  |-  ( U  e.  HrmOp  ->  U : ~H
--> ~H )
3 hoaddcl 26878 . . 3  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T  +op  U
) : ~H --> ~H )
41, 2, 3syl2an 475 . 2  |-  ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( T  +op  U ) : ~H --> ~H )
5 hmop 27042 . . . . . . 7  |-  ( ( T  e.  HrmOp  /\  x  e.  ~H  /\  y  e. 
~H )  ->  (
x  .ih  ( T `  y ) )  =  ( ( T `  x )  .ih  y
) )
653expb 1195 . . . . . 6  |-  ( ( T  e.  HrmOp  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( T `  y
) )  =  ( ( T `  x
)  .ih  y )
)
7 hmop 27042 . . . . . . 7  |-  ( ( U  e.  HrmOp  /\  x  e.  ~H  /\  y  e. 
~H )  ->  (
x  .ih  ( U `  y ) )  =  ( ( U `  x )  .ih  y
) )
873expb 1195 . . . . . 6  |-  ( ( U  e.  HrmOp  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( x  .ih  ( U `  y
) )  =  ( ( U `  x
)  .ih  y )
)
96, 8oveqan12d 6289 . . . . 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 829 . . . 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 564 . . . . 5  |-  ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)
12 hosval 26860 . . . . . . . . 9  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  y  e.  ~H )  ->  ( ( T  +op  U ) `  y )  =  ( ( T `
 y )  +h  ( U `  y
) ) )
1312oveq2d 6286 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  y  e.  ~H )  ->  ( x  .ih  (
( T  +op  U
) `  y )
)  =  ( x 
.ih  ( ( T `
 y )  +h  ( U `  y
) ) ) )
14133expa 1194 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  y  e.  ~H )  ->  (
x  .ih  ( ( T  +op  U ) `  y ) )  =  ( x  .ih  (
( T `  y
)  +h  ( U `
 y ) ) ) )
1514adantrl 713 . . . . . 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 754 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  x  e.  ~H )
17 ffvelrn 6005 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  y  e.  ~H )  ->  ( T `  y
)  e.  ~H )
1817ad2ant2rl 746 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( T `  y )  e.  ~H )
19 ffvelrn 6005 . . . . . . . 8  |-  ( ( U : ~H --> ~H  /\  y  e.  ~H )  ->  ( U `  y
)  e.  ~H )
2019ad2ant2l 743 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( U `  y )  e.  ~H )
21 his7 26208 . . . . . . 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 1226 . . . . . 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 2495 . . . . 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 469 . . . 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 26860 . . . . . . . . 9  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  +op  U ) `  x )  =  ( ( T `
 x )  +h  ( U `  x
) ) )
2625oveq1d 6285 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( T 
+op  U ) `  x )  .ih  y
)  =  ( ( ( T `  x
)  +h  ( U `
 x ) ) 
.ih  y ) )
27263expa 1194 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  (
( ( T  +op  U ) `  x ) 
.ih  y )  =  ( ( ( T `
 x )  +h  ( U `  x
) )  .ih  y
) )
2827adantrr 714 . . . . . 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 6005 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
3029ad2ant2r 744 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( T `  x )  e.  ~H )
31 ffvelrn 6005 . . . . . . . 8  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
3231ad2ant2lr 745 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  ( U `  x )  e.  ~H )
33 simprr 755 . . . . . . 7  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  (
x  e.  ~H  /\  y  e.  ~H )
)  ->  y  e.  ~H )
34 ax-his2 26201 . . . . . . 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 1226 . . . . . 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 2495 . . . . 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 469 . . . 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 2505 . . 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 2875 . 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 26993 . 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 662 1  |-  ( ( T  e.  HrmOp  /\  U  e.  HrmOp )  ->  ( T  +op  U )  e. 
HrmOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   -->wf 5566   ` cfv 5570  (class class class)co 6270    + caddc 9484   ~Hchil 26037    +h cva 26038    .ih csp 26040    +op chos 26056   HrmOpcho 26068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-hilex 26117  ax-hfvadd 26118  ax-hfi 26197  ax-his1 26200  ax-his2 26201
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-2 10590  df-cj 13017  df-re 13018  df-im 13019  df-hosum 26850  df-hmop 26964
This theorem is referenced by:  hmopd  27142  leopadd  27252  opsqrlem4  27263
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