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Theorem hoadddir 23307
Description: Scalar product reverse distributive law for Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hoadddir  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  +  B )  .op  T
)  =  ( ( A  .op  T ) 
+op  ( B  .op  T ) ) )

Proof of Theorem hoadddir
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 addcl 9072 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
21anim1i 552 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  T : ~H --> ~H )  ->  ( ( A  +  B )  e.  CC  /\  T : ~H --> ~H ) )
323impa 1148 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  +  B )  e.  CC  /\  T : ~H --> ~H )
)
4 homval 23244 . . . . . . 7  |-  ( ( ( A  +  B
)  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( A  +  B )  .op  T ) `  x )  =  ( ( A  +  B )  .h  ( T `  x
) ) )
543expa 1153 . . . . . 6  |-  ( ( ( ( A  +  B )  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  +  B ) 
.op  T ) `  x )  =  ( ( A  +  B
)  .h  ( T `
 x ) ) )
63, 5sylan 458 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  +  B ) 
.op  T ) `  x )  =  ( ( A  +  B
)  .h  ( T `
 x ) ) )
7 homval 23244 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
873expa 1153 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
983adantl2 1114 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
10 homval 23244 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( B  .op  T ) `  x )  =  ( B  .h  ( T `  x ) ) )
11103expa 1153 . . . . . . . 8  |-  ( ( ( B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( B 
.op  T ) `  x )  =  ( B  .h  ( T `
 x ) ) )
12113adantl1 1113 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( B 
.op  T ) `  x )  =  ( B  .h  ( T `
 x ) ) )
139, 12oveq12d 6099 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  .op  T ) `
 x )  +h  ( ( B  .op  T ) `  x ) )  =  ( ( A  .h  ( T `
 x ) )  +h  ( B  .h  ( T `  x ) ) ) )
14 ffvelrn 5868 . . . . . . . . . 10  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
15 ax-hvdistr2 22512 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( T `  x )  e.  ~H )  ->  (
( A  +  B
)  .h  ( T `
 x ) )  =  ( ( A  .h  ( T `  x ) )  +h  ( B  .h  ( T `  x )
) ) )
1614, 15syl3an3 1219 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( T : ~H --> ~H  /\  x  e.  ~H )
)  ->  ( ( A  +  B )  .h  ( T `  x
) )  =  ( ( A  .h  ( T `  x )
)  +h  ( B  .h  ( T `  x ) ) ) )
17163exp 1152 . . . . . . . 8  |-  ( A  e.  CC  ->  ( B  e.  CC  ->  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  +  B )  .h  ( T `  x
) )  =  ( ( A  .h  ( T `  x )
)  +h  ( B  .h  ( T `  x ) ) ) ) ) )
1817exp4a 590 . . . . . . 7  |-  ( A  e.  CC  ->  ( B  e.  CC  ->  ( T : ~H --> ~H  ->  ( x  e.  ~H  ->  ( ( A  +  B
)  .h  ( T `
 x ) )  =  ( ( A  .h  ( T `  x ) )  +h  ( B  .h  ( T `  x )
) ) ) ) ) )
19183imp1 1166 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A  +  B )  .h  ( T `  x
) )  =  ( ( A  .h  ( T `  x )
)  +h  ( B  .h  ( T `  x ) ) ) )
2013, 19eqtr4d 2471 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  .op  T ) `
 x )  +h  ( ( B  .op  T ) `  x ) )  =  ( ( A  +  B )  .h  ( T `  x ) ) )
216, 20eqtr4d 2471 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  +  B ) 
.op  T ) `  x )  =  ( ( ( A  .op  T ) `  x )  +h  ( ( B 
.op  T ) `  x ) ) )
22 homulcl 23262 . . . . . . 7  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
23 homulcl 23262 . . . . . . 7  |-  ( ( B  e.  CC  /\  T : ~H --> ~H )  ->  ( B  .op  T
) : ~H --> ~H )
2422, 23anim12i 550 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( B  e.  CC  /\  T : ~H --> ~H )
)  ->  ( ( A  .op  T ) : ~H --> ~H  /\  ( B  .op  T ) : ~H --> ~H ) )
25243impdir 1240 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  .op  T ) : ~H --> ~H  /\  ( B  .op  T ) : ~H --> ~H )
)
26 hosval 23243 . . . . . 6  |-  ( ( ( A  .op  T
) : ~H --> ~H  /\  ( B  .op  T ) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( A 
.op  T )  +op  ( B  .op  T ) ) `  x )  =  ( ( ( A  .op  T ) `
 x )  +h  ( ( B  .op  T ) `  x ) ) )
27263expa 1153 . . . . 5  |-  ( ( ( ( A  .op  T ) : ~H --> ~H  /\  ( B  .op  T ) : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  .op  T ) 
+op  ( B  .op  T ) ) `  x
)  =  ( ( ( A  .op  T
) `  x )  +h  ( ( B  .op  T ) `  x ) ) )
2825, 27sylan 458 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  .op  T ) 
+op  ( B  .op  T ) ) `  x
)  =  ( ( ( A  .op  T
) `  x )  +h  ( ( B  .op  T ) `  x ) ) )
2921, 28eqtr4d 2471 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  +  B ) 
.op  T ) `  x )  =  ( ( ( A  .op  T )  +op  ( B 
.op  T ) ) `
 x ) )
3029ralrimiva 2789 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  ->  A. x  e.  ~H  ( ( ( A  +  B )  .op  T ) `  x )  =  ( ( ( A  .op  T ) 
+op  ( B  .op  T ) ) `  x
) )
31 homulcl 23262 . . . . 5  |-  ( ( ( A  +  B
)  e.  CC  /\  T : ~H --> ~H )  ->  ( ( A  +  B )  .op  T
) : ~H --> ~H )
321, 31sylan 458 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  T : ~H --> ~H )  ->  ( ( A  +  B ) 
.op  T ) : ~H --> ~H )
33323impa 1148 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  +  B )  .op  T
) : ~H --> ~H )
34 hoaddcl 23261 . . . . 5  |-  ( ( ( A  .op  T
) : ~H --> ~H  /\  ( B  .op  T ) : ~H --> ~H )  ->  ( ( A  .op  T )  +op  ( B 
.op  T ) ) : ~H --> ~H )
3522, 23, 34syl2an 464 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( B  e.  CC  /\  T : ~H --> ~H )
)  ->  ( ( A  .op  T )  +op  ( B  .op  T ) ) : ~H --> ~H )
36353impdir 1240 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  .op  T )  +op  ( B 
.op  T ) ) : ~H --> ~H )
37 hoeq 23263 . . 3  |-  ( ( ( ( A  +  B )  .op  T
) : ~H --> ~H  /\  ( ( A  .op  T )  +op  ( B 
.op  T ) ) : ~H --> ~H )  ->  ( A. x  e. 
~H  ( ( ( A  +  B ) 
.op  T ) `  x )  =  ( ( ( A  .op  T )  +op  ( B 
.op  T ) ) `
 x )  <->  ( ( A  +  B )  .op  T )  =  ( ( A  .op  T
)  +op  ( B  .op  T ) ) ) )
3833, 36, 37syl2anc 643 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( A. x  e. 
~H  ( ( ( A  +  B ) 
.op  T ) `  x )  =  ( ( ( A  .op  T )  +op  ( B 
.op  T ) ) `
 x )  <->  ( ( A  +  B )  .op  T )  =  ( ( A  .op  T
)  +op  ( B  .op  T ) ) ) )
3930, 38mpbid 202 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  +  B )  .op  T
)  =  ( ( A  .op  T ) 
+op  ( B  .op  T ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988    + caddc 8993   ~Hchil 22422    +h cva 22423    .h csm 22424    +op chos 22441    .op chot 22442
This theorem is referenced by:  ho2times  23322
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-addcl 9050  ax-hilex 22502  ax-hfvadd 22503  ax-hfvmul 22508  ax-hvdistr2 22512
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-hosum 23233  df-homul 23234
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