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Theorem hv2times 22568
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 10063 . . . 4  |-  2  =  ( 1  +  1 )
21oveq1i 6094 . . 3  |-  ( 2  .h  A )  =  ( ( 1  +  1 )  .h  A
)
3 ax-1cn 9053 . . . 4  |-  1  e.  CC
4 ax-hvdistr2 22517 . . . 4  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  A  e.  ~H )  ->  (
( 1  +  1 )  .h  A )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
53, 3, 4mp3an12 1270 . . 3  |-  ( A  e.  ~H  ->  (
( 1  +  1 )  .h  A )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
62, 5syl5eq 2482 . 2  |-  ( A  e.  ~H  ->  (
2  .h  A )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
7 ax-hvdistr1 22516 . . . 4  |-  ( ( 1  e.  CC  /\  A  e.  ~H  /\  A  e.  ~H )  ->  (
1  .h  ( A  +h  A ) )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
83, 7mp3an1 1267 . . 3  |-  ( ( A  e.  ~H  /\  A  e.  ~H )  ->  ( 1  .h  ( A  +h  A ) )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
98anidms 628 . 2  |-  ( A  e.  ~H  ->  (
1  .h  ( A  +h  A ) )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
10 hvaddcl 22520 . . . 4  |-  ( ( A  e.  ~H  /\  A  e.  ~H )  ->  ( A  +h  A
)  e.  ~H )
1110anidms 628 . . 3  |-  ( A  e.  ~H  ->  ( A  +h  A )  e. 
~H )
12 ax-hvmulid 22514 . . 3  |-  ( ( A  +h  A )  e.  ~H  ->  (
1  .h  ( A  +h  A ) )  =  ( A  +h  A ) )
1311, 12syl 16 . 2  |-  ( A  e.  ~H  ->  (
1  .h  ( A  +h  A ) )  =  ( A  +h  A ) )
146, 9, 133eqtr2d 2476 1  |-  ( A  e.  ~H  ->  (
2  .h  A )  =  ( A  +h  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726  (class class class)co 6084   CCcc 8993   1c1 8996    + caddc 8998   2c2 10054   ~Hchil 22427    +h cva 22428    .h csm 22429
This theorem is referenced by:  hvsubcan2i  22571  mayete3i  23235  mayete3iOLD  23236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-1cn 9053  ax-hfvadd 22508  ax-hvmulid 22514  ax-hvdistr1 22516  ax-hvdistr2 22517
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-2 10063
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