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Theorem homul12 27293
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 9624 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 6320 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  .op  T
)  =  ( ( B  x.  A ) 
.op  T ) )
323adant3 1025 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  x.  B )  .op  T
)  =  ( ( B  x.  A ) 
.op  T ) )
4 homulass 27290 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  x.  B )  .op  T
)  =  ( A 
.op  ( B  .op  T ) ) )
5 homulass 27290 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  T : ~H --> ~H )  -> 
( ( B  x.  A )  .op  T
)  =  ( B 
.op  ( A  .op  T ) ) )
653com12 1209 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( B  x.  A )  .op  T
)  =  ( B 
.op  ( A  .op  T ) ) )
73, 4, 63eqtr3d 2478 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 370    /\ w3a 982    = wceq 1437    e. wcel 1870   -->wf 5597  (class class class)co 6305   CCcc 9536    x. cmul 9543   ~Hchil 26407    .op chot 26427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-mulcl 9600  ax-mulcom 9602  ax-hilex 26487  ax-hfvmul 26493  ax-hvmulass 26495
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7482  df-homul 27219
This theorem is referenced by:  hosubdi  27296
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