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Theorem homul12 26428
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 9578 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 6299 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  .op  T
)  =  ( ( B  x.  A ) 
.op  T ) )
323adant3 1016 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  x.  B )  .op  T
)  =  ( ( B  x.  A ) 
.op  T ) )
4 homulass 26425 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( A  x.  B )  .op  T
)  =  ( A 
.op  ( B  .op  T ) ) )
5 homulass 26425 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  T : ~H --> ~H )  -> 
( ( B  x.  A )  .op  T
)  =  ( B 
.op  ( A  .op  T ) ) )
653com12 1200 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  T : ~H --> ~H )  -> 
( ( B  x.  A )  .op  T
)  =  ( B 
.op  ( A  .op  T ) ) )
73, 4, 63eqtr3d 2516 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 973    = wceq 1379    e. wcel 1767   -->wf 5584  (class class class)co 6284   CCcc 9490    x. cmul 9497   ~Hchil 25540    .op chot 25560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-mulcl 9554  ax-mulcom 9556  ax-hilex 25620  ax-hfvmul 25626  ax-hvmulass 25628
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-map 7422  df-homul 26354
This theorem is referenced by:  hosubdi  26431
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