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Theorem bramul 23297
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 22434 . . 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 22364 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  ~H )  ->  ( B  .h  C
)  e.  ~H )
4 braval 23295 . . . 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 23295 . . . 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 6036 . 2  |-  ( ( A  e.  ~H  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( B  x.  ( ( bra `  A ) `  C ) )  =  ( B  x.  ( C  .ih  A ) ) )
102, 6, 93eqtr4d 2429 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 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   CCcc 8921    x. cmul 8928   ~Hchil 22270    .h csm 22272    .ih csp 22273   bracbr 22307
This theorem is referenced by:  bralnfn  23299
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-hilex 22350  ax-hfvmul 22356  ax-his3 22434
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-bra 23201
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