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Theorem hoadddi 22438
Description: Scalar product distributive law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hoadddi  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) )  =  ( ( A 
.op  T )  +op  ( A  .op  U ) ) )

Proof of Theorem hoadddi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl1 958 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  A  e.  CC )
2 ffvelrn 5701 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
323ad2antl2 1118 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( T `  x )  e.  ~H )
4 ffvelrn 5701 . . . . . . 7  |-  ( ( U : ~H --> ~H  /\  x  e.  ~H )  ->  ( U `  x
)  e.  ~H )
543ad2antl3 1119 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( U `  x )  e.  ~H )
6 ax-hvdistr1 21643 . . . . . 6  |-  ( ( A  e.  CC  /\  ( T `  x )  e.  ~H  /\  ( U `  x )  e.  ~H )  ->  ( A  .h  ( ( T `  x )  +h  ( U `  x
) ) )  =  ( ( A  .h  ( T `  x ) )  +h  ( A  .h  ( U `  x ) ) ) )
71, 3, 5, 6syl3anc 1182 . . . . 5  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( T `  x )  +h  ( U `  x )
) )  =  ( ( A  .h  ( T `  x )
)  +h  ( A  .h  ( U `  x ) ) ) )
8 hosval 22375 . . . . . . . 8  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( T  +op  U ) `  x )  =  ( ( T `
 x )  +h  ( U `  x
) ) )
98oveq2d 5916 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( A  .h  (
( T  +op  U
) `  x )
)  =  ( A  .h  ( ( T `
 x )  +h  ( U `  x
) ) ) )
1093expa 1151 . . . . . 6  |-  ( ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( T  +op  U ) `  x ) )  =  ( A  .h  (
( T `  x
)  +h  ( U `
 x ) ) ) )
11103adantl1 1111 . . . . 5  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( T  +op  U ) `  x ) )  =  ( A  .h  ( ( T `
 x )  +h  ( U `  x
) ) ) )
12 homval 22376 . . . . . . . 8  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  T ) `  x )  =  ( A  .h  ( T `  x ) ) )
13123expa 1151 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
14133adantl3 1113 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  T ) `  x )  =  ( A  .h  ( T `
 x ) ) )
15 homval 22376 . . . . . . . 8  |-  ( ( A  e.  CC  /\  U : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  U ) `  x )  =  ( A  .h  ( U `  x ) ) )
16153expa 1151 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  U ) `  x )  =  ( A  .h  ( U `
 x ) ) )
17163adantl2 1112 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  U ) `  x )  =  ( A  .h  ( U `
 x ) ) )
1814, 17oveq12d 5918 . . . . 5  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  .op  T ) `
 x )  +h  ( ( A  .op  U ) `  x ) )  =  ( ( A  .h  ( T `
 x ) )  +h  ( A  .h  ( U `  x ) ) ) )
197, 11, 183eqtr4d 2358 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( T  +op  U ) `  x ) )  =  ( ( ( A  .op  T
) `  x )  +h  ( ( A  .op  U ) `  x ) ) )
20 hoaddcl 22393 . . . . . . 7  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( T  +op  U
) : ~H --> ~H )
2120anim2i 552 . . . . . 6  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)  ->  ( A  e.  CC  /\  ( T 
+op  U ) : ~H --> ~H ) )
22213impb 1147 . . . . 5  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  e.  CC  /\  ( T  +op  U
) : ~H --> ~H )
)
23 homval 22376 . . . . . 6  |-  ( ( A  e.  CC  /\  ( T  +op  U ) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  ( T  +op  U ) ) `  x )  =  ( A  .h  ( ( T  +op  U ) `  x ) ) )
24233expa 1151 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( T  +op  U
) : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  ( T  +op  U ) ) `  x
)  =  ( A  .h  ( ( T 
+op  U ) `  x ) ) )
2522, 24sylan 457 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  ( T  +op  U ) ) `  x
)  =  ( A  .h  ( ( T 
+op  U ) `  x ) ) )
26 homulcl 22394 . . . . . . 7  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
27 homulcl 22394 . . . . . . 7  |-  ( ( A  e.  CC  /\  U : ~H --> ~H )  ->  ( A  .op  U
) : ~H --> ~H )
2826, 27anim12i 549 . . . . . 6  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( A  e.  CC  /\  U : ~H --> ~H )
)  ->  ( ( A  .op  T ) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H ) )
29283impdi 1237 . . . . 5  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T ) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H )
)
30 hosval 22375 . . . . . 6  |-  ( ( ( A  .op  T
) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( A 
.op  T )  +op  ( A  .op  U ) ) `  x )  =  ( ( ( A  .op  T ) `
 x )  +h  ( ( A  .op  U ) `  x ) ) )
31303expa 1151 . . . . 5  |-  ( ( ( ( A  .op  T ) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  .op  T ) 
+op  ( A  .op  U ) ) `  x
)  =  ( ( ( A  .op  T
) `  x )  +h  ( ( A  .op  U ) `  x ) ) )
3229, 31sylan 457 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  .op  T ) 
+op  ( A  .op  U ) ) `  x
)  =  ( ( ( A  .op  T
) `  x )  +h  ( ( A  .op  U ) `  x ) ) )
3319, 25, 323eqtr4d 2358 . . 3  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  ( T  +op  U ) ) `  x
)  =  ( ( ( A  .op  T
)  +op  ( A  .op  U ) ) `  x ) )
3433ralrimiva 2660 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  A. x  e.  ~H  ( ( A  .op  ( T  +op  U ) ) `  x )  =  ( ( ( A  .op  T ) 
+op  ( A  .op  U ) ) `  x
) )
35 homulcl 22394 . . . . 5  |-  ( ( A  e.  CC  /\  ( T  +op  U ) : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) ) : ~H --> ~H )
3620, 35sylan2 460 . . . 4  |-  ( ( A  e.  CC  /\  ( T : ~H --> ~H  /\  U : ~H --> ~H )
)  ->  ( A  .op  ( T  +op  U
) ) : ~H --> ~H )
37363impb 1147 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) ) : ~H --> ~H )
38 hoaddcl 22393 . . . . 5  |-  ( ( ( A  .op  T
) : ~H --> ~H  /\  ( A  .op  U ) : ~H --> ~H )  ->  ( ( A  .op  T )  +op  ( A 
.op  U ) ) : ~H --> ~H )
3926, 27, 38syl2an 463 . . . 4  |-  ( ( ( A  e.  CC  /\  T : ~H --> ~H )  /\  ( A  e.  CC  /\  U : ~H --> ~H )
)  ->  ( ( A  .op  T )  +op  ( A  .op  U ) ) : ~H --> ~H )
40393impdi 1237 . . 3  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( ( A  .op  T )  +op  ( A 
.op  U ) ) : ~H --> ~H )
41 hoeq 22395 . . 3  |-  ( ( ( A  .op  ( T  +op  U ) ) : ~H --> ~H  /\  ( ( A  .op  T )  +op  ( A 
.op  U ) ) : ~H --> ~H )  ->  ( A. x  e. 
~H  ( ( A 
.op  ( T  +op  U ) ) `  x
)  =  ( ( ( A  .op  T
)  +op  ( A  .op  U ) ) `  x )  <->  ( A  .op  ( T  +op  U
) )  =  ( ( A  .op  T
)  +op  ( A  .op  U ) ) ) )
4237, 40, 41syl2anc 642 . 2  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A. x  e. 
~H  ( ( A 
.op  ( T  +op  U ) ) `  x
)  =  ( ( ( A  .op  T
)  +op  ( A  .op  U ) ) `  x )  <->  ( A  .op  ( T  +op  U
) )  =  ( ( A  .op  T
)  +op  ( A  .op  U ) ) ) )
4334, 42mpbid 201 1  |-  ( ( A  e.  CC  /\  T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A  .op  ( T  +op  U ) )  =  ( ( A 
.op  T )  +op  ( A  .op  U ) ) )
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   ~Hchil 21554    +h cva 21555    .h csm 21556    +op chos 21573    .op chot 21574
This theorem is referenced by:  hosubdi  22443  honegdi  22444  ho2times  22454  opsqrlem6  22780
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-hilex 21634  ax-hfvadd 21635  ax-hfvmul 21640  ax-hvdistr1 21643
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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