HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hv2times Unicode version

Theorem hv2times 22516
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 10014 . . . 4  |-  2  =  ( 1  +  1 )
21oveq1i 6050 . . 3  |-  ( 2  .h  A )  =  ( ( 1  +  1 )  .h  A
)
3 ax-1cn 9004 . . . 4  |-  1  e.  CC
4 ax-hvdistr2 22465 . . . 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 2448 . 2  |-  ( A  e.  ~H  ->  (
2  .h  A )  =  ( ( 1  .h  A )  +h  ( 1  .h  A
) ) )
7 ax-hvdistr1 22464 . . . 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 22468 . . . 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 22462 . . 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 2442 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 6040   CCcc 8944   1c1 8947    + caddc 8949   2c2 10005   ~Hchil 22375    +h cva 22376    .h csm 22377
This theorem is referenced by:  hvsubcan2i  22519  mayete3i  23183  mayete3iOLD  23184
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 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-1cn 9004  ax-hfvadd 22456  ax-hvmulid 22462  ax-hvdistr1 22464  ax-hvdistr2 22465
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-2 10014
  Copyright terms: Public domain W3C validator