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Theorem hoadddir 22384
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 8819 . . . . . . . 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 22321 . . . . . . 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 22321 . . . . . . . . 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 22321 . . . . . . . . 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 5876 . . . . . 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 5663 . . . . . . . . . 10  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
15 ax-hvdistr2 21589 . . . . . . . . . 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 2318 . . . . 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 2318 . . . 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 22339 . . . . . . 7  |-  ( ( A  e.  CC  /\  T : ~H --> ~H )  ->  ( A  .op  T
) : ~H --> ~H )
23 homulcl 22339 . . . . . . 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 22320 . . . . . 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 2318 . . 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 2626 . 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 22339 . . . . 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 22338 . . . . 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 22340 . . 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 1623    e. wcel 1684   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735    + caddc 8740   ~Hchil 21499    +h cva 21500    .h csm 21501    +op chos 21518    .op chot 21519
This theorem is referenced by:  ho2times  22399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-addcl 8797  ax-hilex 21579  ax-hfvadd 21580  ax-hfvmul 21585  ax-hvdistr2 21589
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-hosum 22310  df-homul 22311
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