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Theorem hoadddir 22439
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 8864 . . . . . . . 8  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
21anim1i 551 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  T : ~H --> ~H )  ->  ( ( A  +  B )  e.  CC  /\  T : ~H --> ~H ) )
323impa 1146 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  +  B )  e.  CC  /\  T : ~H --> ~H )
)
4 homval 22376 . . . . . . 7  |-  ( ( ( A  +  B
)  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( A  +  B )  .op  T ) `  x )  =  ( ( A  +  B )  .h  ( T `  x
) ) )
543expa 1151 . . . . . 6  |-  ( ( ( ( A  +  B )  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  +  B ) 
.op  T ) `  x )  =  ( ( A  +  B
)  .h  ( T `
 x ) ) )
63, 5sylan 457 . . . . 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 22376 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
873expa 1151 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
983adantl2 1112 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
10 homval 22376 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( B  .op  T ) `  x )  =  ( B  .h  ( T `  x ) ) )
11103expa 1151 . . . . . . . 8  |-  ( ( ( B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( B 
.op  T ) `  x )  =  ( B  .h  ( T `
 x ) ) )
12113adantl1 1111 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( B 
.op  T ) `  x )  =  ( B  .h  ( T `
 x ) ) )
139, 12oveq12d 5918 . . . . . 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 5701 . . . . . . . . . 10  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
15 ax-hvdistr2 21644 . . . . . . . . . 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 1217 . . . . . . . . 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 1150 . . . . . . . 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 589 . . . . . . 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 1164 . . . . . 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 2351 . . . . 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 2351 . . . 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 22394 . . . . . . 7  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
23 homulcl 22394 . . . . . . 7  |-  ( ( B  e.  CC  /\  T : ~H --> ~H )  ->  ( B  .op  T
) : ~H --> ~H )
2422, 23anim12i 549 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( B  e.  CC  /\  T : ~H --> ~H )
)  ->  ( ( A  .op  T ) : ~H --> ~H  /\  ( B  .op  T ) : ~H --> ~H ) )
25243impdir 1238 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  .op  T ) : ~H --> ~H  /\  ( B  .op  T ) : ~H --> ~H )
)
26 hosval 22375 . . . . . 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 1151 . . . . 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 457 . . . 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 2351 . . 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 2660 . 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 22394 . . . . 5  |-  ( ( ( A  +  B
)  e.  CC  /\  T : ~H --> ~H )  ->  ( ( A  +  B )  .op  T
) : ~H --> ~H )
321, 31sylan 457 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  T : ~H --> ~H )  ->  ( ( A  +  B ) 
.op  T ) : ~H --> ~H )
33323impa 1146 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  +  B )  .op  T
) : ~H --> ~H )
34 hoaddcl 22393 . . . . 5  |-  ( ( ( A  .op  T
) : ~H --> ~H  /\  ( B  .op  T ) : ~H --> ~H )  ->  ( ( A  .op  T )  +op  ( B 
.op  T ) ) : ~H --> ~H )
3522, 23, 34syl2an 463 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( B  e.  CC  /\  T : ~H --> ~H )
)  ->  ( ( A  .op  T )  +op  ( B  .op  T ) ) : ~H --> ~H )
36353impdir 1238 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  .op  T )  +op  ( B 
.op  T ) ) : ~H --> ~H )
37 hoeq 22395 . . 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 642 . 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 201 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 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   A.wral 2577   -->wf 5288   ` cfv 5292  (class class class)co 5900   CCcc 8780    + caddc 8785   ~Hchil 21554    +h cva 21555    .h csm 21556    +op chos 21573    .op chot 21574
This theorem is referenced by:  ho2times  22454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-addcl 8842  ax-hilex 21634  ax-hfvadd 21635  ax-hfvmul 21640  ax-hvdistr2 21644
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-map 6817  df-hosum 22365  df-homul 22366
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