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Theorem bramul 25301
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 24437 . . 3  |-  ( ( B  e.  CC  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
( B  .h  C
)  .ih  A )  =  ( B  x.  ( C  .ih  A ) ) )
213comr 1195 . 2  |-  ( ( A  e.  ~H  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( B  .h  C
)  .ih  A )  =  ( B  x.  ( C  .ih  A ) ) )
3 hvmulcl 24366 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  ~H )  ->  ( B  .h  C
)  e.  ~H )
4 braval 25299 . . . 4  |-  ( ( A  e.  ~H  /\  ( B  .h  C
)  e.  ~H )  ->  ( ( bra `  A
) `  ( B  .h  C ) )  =  ( ( B  .h  C )  .ih  A
) )
53, 4sylan2 474 . . 3  |-  ( ( A  e.  ~H  /\  ( B  e.  CC  /\  C  e.  ~H )
)  ->  ( ( bra `  A ) `  ( B  .h  C
) )  =  ( ( B  .h  C
)  .ih  A )
)
653impb 1183 . 2  |-  ( ( A  e.  ~H  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( bra `  A
) `  ( B  .h  C ) )  =  ( ( B  .h  C )  .ih  A
) )
7 braval 25299 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( ( bra `  A
) `  C )  =  ( C  .ih  A ) )
873adant2 1007 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( bra `  A
) `  C )  =  ( C  .ih  A ) )
98oveq2d 6102 . 2  |-  ( ( A  e.  ~H  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( B  x.  ( ( bra `  A ) `  C ) )  =  ( B  x.  ( C  .ih  A ) ) )
102, 6, 93eqtr4d 2480 1  |-  ( ( A  e.  ~H  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( bra `  A
) `  ( B  .h  C ) )  =  ( B  x.  (
( bra `  A
) `  C )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5413  (class class class)co 6086   CCcc 9272    x. cmul 9279   ~Hchil 24272    .h csm 24274    .ih csp 24275   bracbr 24309
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-hilex 24352  ax-hfvmul 24358  ax-his3 24437
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-bra 25205
This theorem is referenced by:  bralnfn  25303
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