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Theorem homulass 26385
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 9567 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
2 homval 26324 . . . . . . . . 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 1256 . . . . . . . 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 1193 . . . . . . 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 1186 . . . . . 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 26324 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( B  .op  T ) `  x )  =  ( B  .h  ( T `  x ) ) )
87oveq2d 6293 . . . . . . . 8  |-  ( ( B  e.  CC  /\  T : ~H --> ~H  /\  x  e.  ~H )  ->  ( A  .h  (
( B  .op  T
) `  x )
)  =  ( A  .h  ( B  .h  ( T `  x ) ) ) )
983expa 1191 . . . . . . 7  |-  ( ( ( B  e.  CC  /\  T : ~H --> ~H )  /\  x  e.  ~H )  ->  ( A  .h  ( ( B  .op  T ) `  x ) )  =  ( A  .h  ( B  .h  ( T `  x ) ) ) )
1093adantl1 1147 . . . . . 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 6012 . . . . . . . . . 10  |-  ( ( T : ~H --> ~H  /\  x  e.  ~H )  ->  ( T `  x
)  e.  ~H )
12 ax-hvmulass 25588 . . . . . . . . . 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 1258 . . . . . . . . 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 1191 . . . . . . . 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 1204 . . . . . 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 2506 . . . . 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 2506 . . . 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 26342 . . . . . . . 8  |-  ( ( B  e.  CC  /\  T : ~H --> ~H )  ->  ( B  .op  T
) : ~H --> ~H )
20 homval 26324 . . . . . . . 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 1257 . . . . . . 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 1193 . . . . . 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 1187 . . . . 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 2506 . . 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 2873 . 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 26342 . . . . 5  |-  ( ( ( A  x.  B
)  e.  CC  /\  T : ~H --> ~H )  ->  ( ( A  x.  B )  .op  T
) : ~H --> ~H )
281, 27sylan 471 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  T : ~H --> ~H )  ->  ( ( A  x.  B ) 
.op  T ) : ~H --> ~H )
29283impa 1186 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  x.  B )  .op  T
) : ~H --> ~H )
30 homulcl 26342 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  .op  T ) : ~H --> ~H )  ->  ( A  .op  ( B  .op  T ) ) : ~H --> ~H )
3119, 30sylan2 474 . . . 4  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  T : ~H --> ~H )
)  ->  ( A  .op  ( B  .op  T
) ) : ~H --> ~H )
32313impb 1187 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( A  .op  ( B  .op  T ) ) : ~H --> ~H )
33 hoeq 26343 . . 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 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 ) ) ) )
3526, 34mpbid 210 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 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2809   -->wf 5577   ` cfv 5581  (class class class)co 6277   CCcc 9481    x. cmul 9488   ~Hchil 25500    .h csm 25502    .op chot 25520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-mulcl 9545  ax-hilex 25580  ax-hfvmul 25586  ax-hvmulass 25588
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7414  df-homul 26314
This theorem is referenced by:  homul12  26388  honegneg  26389  leopmul  26717  nmopleid  26722  opsqrlem1  26723  opsqrlem6  26728
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