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Theorem bramul 23441
Description: Linearity property of bra for multiplication. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
bramul  |-  ( ( A  e.  ~H  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( bra `  A
) `  ( B  .h  C ) )  =  ( B  x.  (
( bra `  A
) `  C )
) )

Proof of Theorem bramul
StepHypRef Expression
1 ax-his3 22578 . . 3  |-  ( ( B  e.  CC  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
( B  .h  C
)  .ih  A )  =  ( B  x.  ( C  .ih  A ) ) )
213comr 1161 . 2  |-  ( ( A  e.  ~H  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( B  .h  C
)  .ih  A )  =  ( B  x.  ( C  .ih  A ) ) )
3 hvmulcl 22508 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  ~H )  ->  ( B  .h  C
)  e.  ~H )
4 braval 23439 . . . 4  |-  ( ( A  e.  ~H  /\  ( B  .h  C
)  e.  ~H )  ->  ( ( bra `  A
) `  ( B  .h  C ) )  =  ( ( B  .h  C )  .ih  A
) )
53, 4sylan2 461 . . 3  |-  ( ( A  e.  ~H  /\  ( B  e.  CC  /\  C  e.  ~H )
)  ->  ( ( bra `  A ) `  ( B  .h  C
) )  =  ( ( B  .h  C
)  .ih  A )
)
653impb 1149 . 2  |-  ( ( A  e.  ~H  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( bra `  A
) `  ( B  .h  C ) )  =  ( ( B  .h  C )  .ih  A
) )
7 braval 23439 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( ( bra `  A
) `  C )  =  ( C  .ih  A ) )
873adant2 976 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( bra `  A
) `  C )  =  ( C  .ih  A ) )
98oveq2d 6089 . 2  |-  ( ( A  e.  ~H  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( B  x.  ( ( bra `  A ) `  C ) )  =  ( B  x.  ( C  .ih  A ) ) )
102, 6, 93eqtr4d 2477 1  |-  ( ( A  e.  ~H  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( bra `  A
) `  ( B  .h  C ) )  =  ( B  x.  (
( bra `  A
) `  C )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980    x. cmul 8987   ~Hchil 22414    .h csm 22416    .ih csp 22417   bracbr 22451
This theorem is referenced by:  bralnfn  23443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-hilex 22494  ax-hfvmul 22500  ax-his3 22578
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-bra 23345
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