HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  homul12 Structured version   Unicode version

Theorem homul12 25221
Description: Swap first and second factors in a nested operator scalar product. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
homul12  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( A  .op  ( B  .op  T ) )  =  ( B  .op  ( A  .op  T ) ) )

Proof of Theorem homul12
StepHypRef Expression
1 mulcom 9380 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 6118 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  .op  T
)  =  ( ( B  x.  A ) 
.op  T ) )
323adant3 1008 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  x.  B )  .op  T
)  =  ( ( B  x.  A ) 
.op  T ) )
4 homulass 25218 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  x.  B )  .op  T
)  =  ( A 
.op  ( B  .op  T ) ) )
5 homulass 25218 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  T : ~H --> ~H )  -> 
( ( B  x.  A )  .op  T
)  =  ( B 
.op  ( A  .op  T ) ) )
653com12 1191 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( B  x.  A )  .op  T
)  =  ( B 
.op  ( A  .op  T ) ) )
73, 4, 63eqtr3d 2483 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( A  .op  ( B  .op  T ) )  =  ( B  .op  ( A  .op  T ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   -->wf 5426  (class class class)co 6103   CCcc 9292    x. cmul 9299   ~Hchil 24333    .op chot 24353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-mulcl 9356  ax-mulcom 9358  ax-hilex 24413  ax-hfvmul 24419  ax-hvmulass 24421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-map 7228  df-homul 25147
This theorem is referenced by:  hosubdi  25224
  Copyright terms: Public domain W3C validator