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Theorem homulass 22398
Description: Scalar product associative law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
homulass  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  x.  B )  .op  T
)  =  ( A 
.op  ( B  .op  T ) ) )

Proof of Theorem homulass
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mulcl 8837 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
2 homval 22337 . . . . . . . . 9  |-  ( ( ( A  x.  B
)  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( A  x.  B )  .op  T ) `  x )  =  ( ( A  x.  B )  .h  ( T `  x
) ) )
31, 2syl3an1 1215 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( A  x.  B ) 
.op  T ) `  x )  =  ( ( A  x.  B
)  .h  ( T `
 x ) ) )
433expia 1153 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  T : ~H --> ~H )  ->  ( x  e.  ~H  ->  (
( ( A  x.  B )  .op  T
) `  x )  =  ( ( A  x.  B )  .h  ( T `  x
) ) ) )
543impa 1146 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( x  e.  ~H  ->  ( ( ( A  x.  B )  .op  T ) `  x )  =  ( ( A  x.  B )  .h  ( T `  x
) ) ) )
65imp 418 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  x.  B ) 
.op  T ) `  x )  =  ( ( A  x.  B
)  .h  ( T `
 x ) ) )
7 homval 22337 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( B  .op  T ) `  x )  =  ( B  .h  ( T `  x ) ) )
87oveq2d 5890 . . . . . . . 8  |-  ( ( B  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( A  .h  (
( B  .op  T
) `  x )
)  =  ( A  .h  ( B  .h  ( T `  x ) ) ) )
983expa 1151 . . . . . . 7  |-  ( ( ( B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( B  .op  T ) `  x ) )  =  ( A  .h  ( B  .h  ( T `  x ) ) ) )
1093adantl1 1111 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( B  .op  T ) `  x ) )  =  ( A  .h  ( B  .h  ( T `  x ) ) ) )
11 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
12 ax-hvmulass 21603 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( T `  x )  e.  ~H )  ->  (
( A  x.  B
)  .h  ( T `
 x ) )  =  ( A  .h  ( B  .h  ( T `  x )
) ) )
1311, 12syl3an3 1217 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( T : ~H --> ~H  /\  x  e.  ~H )
)  ->  ( ( A  x.  B )  .h  ( T `  x
) )  =  ( A  .h  ( B  .h  ( T `  x ) ) ) )
14133expa 1151 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( T : ~H
--> ~H  /\  x  e. 
~H ) )  -> 
( ( A  x.  B )  .h  ( T `  x )
)  =  ( A  .h  ( B  .h  ( T `  x ) ) ) )
1514exp43 595 . . . . . . 7  |-  ( A  e.  CC  ->  ( B  e.  CC  ->  ( T : ~H --> ~H  ->  ( x  e.  ~H  ->  ( ( A  x.  B
)  .h  ( T `
 x ) )  =  ( A  .h  ( B  .h  ( T `  x )
) ) ) ) ) )
16153imp1 1164 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A  x.  B )  .h  ( T `  x
) )  =  ( A  .h  ( B  .h  ( T `  x ) ) ) )
1710, 16eqtr4d 2331 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( B  .op  T ) `  x ) )  =  ( ( A  x.  B )  .h  ( T `  x ) ) )
186, 17eqtr4d 2331 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  x.  B ) 
.op  T ) `  x )  =  ( A  .h  ( ( B  .op  T ) `
 x ) ) )
19 homulcl 22355 . . . . . . . 8  |-  ( ( B  e.  CC  /\  T : ~H --> ~H )  ->  ( B  .op  T
) : ~H --> ~H )
20 homval 22337 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( B  .op  T ) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( A  .op  ( B  .op  T ) ) `  x )  =  ( A  .h  ( ( B  .op  T ) `  x ) ) )
2119, 20syl3an2 1216 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  ( B  .op  T ) ) `  x
)  =  ( A  .h  ( ( B 
.op  T ) `  x ) ) )
22213expia 1153 . . . . . 6  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  T : ~H --> ~H )
)  ->  ( x  e.  ~H  ->  ( ( A  .op  ( B  .op  T ) ) `  x
)  =  ( A  .h  ( ( B 
.op  T ) `  x ) ) ) )
23223impb 1147 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( x  e.  ~H  ->  ( ( A  .op  ( B  .op  T ) ) `  x )  =  ( A  .h  ( ( B  .op  T ) `  x ) ) ) )
2423imp 418 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( A 
.op  ( B  .op  T ) ) `  x
)  =  ( A  .h  ( ( B 
.op  T ) `  x ) ) )
2518, 24eqtr4d 2331 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( ( ( A  x.  B ) 
.op  T ) `  x )  =  ( ( A  .op  ( B  .op  T ) ) `
 x ) )
2625ralrimiva 2639 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  ->  A. x  e.  ~H  ( ( ( A  x.  B )  .op  T ) `  x )  =  ( ( A 
.op  ( B  .op  T ) ) `  x
) )
27 homulcl 22355 . . . . 5  |-  ( ( ( A  x.  B
)  e.  CC  /\  T : ~H --> ~H )  ->  ( ( A  x.  B )  .op  T
) : ~H --> ~H )
281, 27sylan 457 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  T : ~H --> ~H )  ->  ( ( A  x.  B ) 
.op  T ) : ~H --> ~H )
29283impa 1146 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  x.  B )  .op  T
) : ~H --> ~H )
30 homulcl 22355 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  .op  T ) : ~H --> ~H )  ->  ( A  .op  ( B  .op  T ) ) : ~H --> ~H )
3119, 30sylan2 460 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  T : ~H --> ~H )
)  ->  ( A  .op  ( B  .op  T
) ) : ~H --> ~H )
32313impb 1147 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( A  .op  ( B  .op  T ) ) : ~H --> ~H )
33 hoeq 22356 . . 3  |-  ( ( ( ( A  x.  B )  .op  T
) : ~H --> ~H  /\  ( A  .op  ( B 
.op  T ) ) : ~H --> ~H )  ->  ( A. x  e. 
~H  ( ( ( A  x.  B ) 
.op  T ) `  x )  =  ( ( A  .op  ( B  .op  T ) ) `
 x )  <->  ( ( A  x.  B )  .op  T )  =  ( A  .op  ( B 
.op  T ) ) ) )
3429, 32, 33syl2anc 642 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( A. x  e. 
~H  ( ( ( A  x.  B ) 
.op  T ) `  x )  =  ( ( A  .op  ( B  .op  T ) ) `
 x )  <->  ( ( A  x.  B )  .op  T )  =  ( A  .op  ( B 
.op  T ) ) ) )
3526, 34mpbid 201 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  x.  B )  .op  T
)  =  ( A 
.op  ( B  .op  T ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751    x. cmul 8758   ~Hchil 21515    .h csm 21517    .op chot 21535
This theorem is referenced by:  homul12  22401  honegneg  22402  leopmul  22730  nmopleid  22735  opsqrlem1  22736  opsqrlem6  22741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-mulcl 8815  ax-hilex 21595  ax-hfvmul 21601  ax-hvmulass 21603
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-homul 22327
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