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Theorem hv2times 21656
Description: Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hv2times  |-  ( A  e.  ~H  ->  (
2  .h  A )  =  ( A  +h  A ) )

Proof of Theorem hv2times
StepHypRef Expression
1 df-2 9820 . . . 4  |-  2  =  ( 1  +  1 )
21oveq1i 5884 . . 3  |-  ( 2  .h  A )  =  ( ( 1  +  1 )  .h  A
)
3 ax-1cn 8811 . . . 4  |-  1  e.  CC
4 ax-hvdistr2 21605 . . . 4  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  A  e.  ~H )  ->  (
( 1  +  1 )  .h  A )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
53, 3, 4mp3an12 1267 . . 3  |-  ( A  e.  ~H  ->  (
( 1  +  1 )  .h  A )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
62, 5syl5eq 2340 . 2  |-  ( A  e.  ~H  ->  (
2  .h  A )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
7 ax-hvdistr1 21604 . . . 4  |-  ( ( 1  e.  CC  /\  A  e.  ~H  /\  A  e.  ~H )  ->  (
1  .h  ( A  +h  A ) )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
83, 7mp3an1 1264 . . 3  |-  ( ( A  e.  ~H  /\  A  e.  ~H )  ->  ( 1  .h  ( A  +h  A ) )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
98anidms 626 . 2  |-  ( A  e.  ~H  ->  (
1  .h  ( A  +h  A ) )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
10 hvaddcl 21608 . . . 4  |-  ( ( A  e.  ~H  /\  A  e.  ~H )  ->  ( A  +h  A
)  e.  ~H )
1110anidms 626 . . 3  |-  ( A  e.  ~H  ->  ( A  +h  A )  e. 
~H )
12 ax-hvmulid 21602 . . 3  |-  ( ( A  +h  A )  e.  ~H  ->  (
1  .h  ( A  +h  A ) )  =  ( A  +h  A ) )
1311, 12syl 15 . 2  |-  ( A  e.  ~H  ->  (
1  .h  ( A  +h  A ) )  =  ( A  +h  A ) )
146, 9, 133eqtr2d 2334 1  |-  ( A  e.  ~H  ->  (
2  .h  A )  =  ( A  +h  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696  (class class class)co 5874   CCcc 8751   1c1 8754    + caddc 8756   2c2 9811   ~Hchil 21515    +h cva 21516    .h csm 21517
This theorem is referenced by:  hvsubcan2i  21659  mayete3i  22323  mayete3iOLD  22324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-1cn 8811  ax-hfvadd 21596  ax-hvmulid 21602  ax-hvdistr1 21604  ax-hvdistr2 21605
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-2 9820
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