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

Proof of Theorem braadd
StepHypRef Expression
1 ax-his2 22578 . . 3  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  A  e.  ~H )  ->  (
( B  +h  C
)  .ih  A )  =  ( ( B 
.ih  A )  +  ( C  .ih  A
) ) )
213comr 1161 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( B  +h  C
)  .ih  A )  =  ( ( B 
.ih  A )  +  ( C  .ih  A
) ) )
3 hvaddcl 22508 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C
)  e.  ~H )
4 braval 23440 . . . 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.  ~H  /\  C  e.  ~H )
)  ->  ( ( bra `  A ) `  ( B  +h  C
) )  =  ( ( B  +h  C
)  .ih  A )
)
653impb 1149 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( bra `  A
) `  ( B  +h  C ) )  =  ( ( B  +h  C )  .ih  A
) )
7 braval 23440 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  =  ( B  .ih  A ) )
873adant3 977 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( bra `  A
) `  B )  =  ( B  .ih  A ) )
9 braval 23440 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( ( bra `  A
) `  C )  =  ( C  .ih  A ) )
1093adant2 976 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( bra `  A
) `  C )  =  ( C  .ih  A ) )
118, 10oveq12d 6092 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( ( bra `  A
) `  B )  +  ( ( bra `  A ) `  C
) )  =  ( ( B  .ih  A
)  +  ( C 
.ih  A ) ) )
122, 6, 113eqtr4d 2478 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( bra `  A
) `  ( B  +h  C ) )  =  ( ( ( bra `  A ) `  B
)  +  ( ( bra `  A ) `
 C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ` cfv 5447  (class class class)co 6074    + caddc 8986   ~Hchil 22415    +h cva 22416    .ih csp 22418   bracbr 22452
This theorem is referenced by:  bralnfn  23444
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 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pr 4396  ax-hilex 22495  ax-hfvadd 22496  ax-his2 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-bra 23346
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