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Theorem homulass 27147
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 9608 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
2 homval 27086 . . . . . . . . 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 1265 . . . . . . . 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 1201 . . . . . . 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 1194 . . . . . 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 429 . . . . 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 27086 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( B  .op  T ) `  x )  =  ( B  .h  ( T `  x ) ) )
87oveq2d 6296 . . . . . . . 8  |-  ( ( B  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( A  .h  (
( B  .op  T
) `  x )
)  =  ( A  .h  ( B  .h  ( T `  x ) ) ) )
983expa 1199 . . . . . . 7  |-  ( ( ( B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( B  .op  T ) `  x ) )  =  ( A  .h  ( B  .h  ( T `  x ) ) ) )
1093adantl1 1155 . . . . . 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 6009 . . . . . . . . . 10  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
12 ax-hvmulass 26351 . . . . . . . . . 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 1267 . . . . . . . . 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 1199 . . . . . . . 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 612 . . . . . . 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 1212 . . . . . 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 2448 . . . . 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 2448 . . . 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 27104 . . . . . . . 8  |-  ( ( B  e.  CC  /\  T : ~H --> ~H )  ->  ( B  .op  T
) : ~H --> ~H )
20 homval 27086 . . . . . . . 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 1266 . . . . . . 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 1201 . . . . . 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 1195 . . . . 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 429 . . . 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 2448 . . 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 2820 . 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 27104 . . . 4  |-  ( ( ( A  x.  B
)  e.  CC  /\  T : ~H --> ~H )  ->  ( ( A  x.  B )  .op  T
) : ~H --> ~H )
281, 27stoic3 1632 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  x.  B )  .op  T
) : ~H --> ~H )
29 homulcl 27104 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  .op  T ) : ~H --> ~H )  ->  ( A  .op  ( B  .op  T ) ) : ~H --> ~H )
3019, 29sylan2 474 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  T : ~H --> ~H )
)  ->  ( A  .op  ( B  .op  T
) ) : ~H --> ~H )
31303impb 1195 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( A  .op  ( B  .op  T ) ) : ~H --> ~H )
32 hoeq 27105 . . 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 ) ) ) )
3328, 31, 32syl2anc 661 . 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 ) ) ) )
3426, 33mpbid 212 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  x.  B )  .op  T
)  =  ( A 
.op  ( B  .op  T ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   A.wral 2756   -->wf 5567   ` cfv 5571  (class class class)co 6280   CCcc 9522    x. cmul 9529   ~Hchil 26263    .h csm 26265    .op chot 26283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-mulcl 9586  ax-hilex 26343  ax-hfvmul 26349  ax-hvmulass 26351
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-map 7461  df-homul 27076
This theorem is referenced by:  homul12  27150  honegneg  27151  leopmul  27479  nmopleid  27484  opsqrlem1  27485  opsqrlem6  27490
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