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Theorem hv2times 22524
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 10022 . . . 4  |-  2  =  ( 1  +  1 )
21oveq1i 6058 . . 3  |-  ( 2  .h  A )  =  ( ( 1  +  1 )  .h  A
)
3 ax-1cn 9012 . . . 4  |-  1  e.  CC
4 ax-hvdistr2 22473 . . . 4  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  A  e.  ~H )  ->  (
( 1  +  1 )  .h  A )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
53, 3, 4mp3an12 1269 . . 3  |-  ( A  e.  ~H  ->  (
( 1  +  1 )  .h  A )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
62, 5syl5eq 2456 . 2  |-  ( A  e.  ~H  ->  (
2  .h  A )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
7 ax-hvdistr1 22472 . . . 4  |-  ( ( 1  e.  CC  /\  A  e.  ~H  /\  A  e.  ~H )  ->  (
1  .h  ( A  +h  A ) )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
83, 7mp3an1 1266 . . 3  |-  ( ( A  e.  ~H  /\  A  e.  ~H )  ->  ( 1  .h  ( A  +h  A ) )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
98anidms 627 . 2  |-  ( A  e.  ~H  ->  (
1  .h  ( A  +h  A ) )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
10 hvaddcl 22476 . . . 4  |-  ( ( A  e.  ~H  /\  A  e.  ~H )  ->  ( A  +h  A
)  e.  ~H )
1110anidms 627 . . 3  |-  ( A  e.  ~H  ->  ( A  +h  A )  e. 
~H )
12 ax-hvmulid 22470 . . 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 2450 1  |-  ( A  e.  ~H  ->  (
2  .h  A )  =  ( A  +h  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721  (class class class)co 6048   CCcc 8952   1c1 8955    + caddc 8957   2c2 10013   ~Hchil 22383    +h cva 22384    .h csm 22385
This theorem is referenced by:  hvsubcan2i  22527  mayete3i  23191  mayete3iOLD  23192
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-1cn 9012  ax-hfvadd 22464  ax-hvmulid 22470  ax-hvdistr1 22472  ax-hvdistr2 22473
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-2 10022
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