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Theorem prodvs 25571
Description: The product of a vector by a scalar is a vector. (Contributed by FL, 12-Sep-2010.)
Hypotheses
Ref Expression
prodvs.1  |-  + w  =  ( 1st `  ( 2nd `  R ) )
prodvs.2  |-  . w  =  ( 2nd `  ( 2nd `  R ) )
prodvs.3  |-  W  =  ran  + w
prodvs.4  |-  X  =  ran  + t
prodvs.5  |-  + t  =  ( 1st `  ( 1st `  R ) )
Assertion
Ref Expression
prodvs  |-  ( ( R  e.  Vec  /\  A  e.  X  /\  U  e.  W )  ->  ( A . w U )  e.  W
)

Proof of Theorem prodvs
StepHypRef Expression
1 prodvs.4 . . . 4  |-  X  =  ran  + t
2 prodvs.5 . . . . 5  |-  + t  =  ( 1st `  ( 1st `  R ) )
32rneqi 4921 . . . 4  |-  ran  + t  =  ran  ( 1st `  ( 1st `  R
) )
41, 3eqtri 2316 . . 3  |-  X  =  ran  ( 1st `  ( 1st `  R ) )
5 prodvs.2 . . 3  |-  . w  =  ( 2nd `  ( 2nd `  R ) )
6 prodvs.3 . . . 4  |-  W  =  ran  + w
7 prodvs.1 . . . . 5  |-  + w  =  ( 1st `  ( 2nd `  R ) )
87rneqi 4921 . . . 4  |-  ran  + w  =  ran  ( 1st `  ( 2nd `  R
) )
96, 8eqtri 2316 . . 3  |-  W  =  ran  ( 1st `  ( 2nd `  R ) )
104, 5, 9vecax2 25557 . 2  |-  ( R  e.  Vec  ->  . w : ( X  X.  W ) --> W )
11 fovrn 6006 . 2  |-  ( ( . w : ( X  X.  W ) --> W  /\  A  e.  X  /\  U  e.  W )  ->  ( A . w U )  e.  W )
1210, 11syl3an1 1215 1  |-  ( ( R  e.  Vec  /\  A  e.  X  /\  U  e.  W )  ->  ( A . w U )  e.  W
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    X. cxp 4703   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137    Vec cvec 25552
This theorem is referenced by:  mulveczer  25582  mulinvsca  25583  svli2  25587
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-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-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-1st 6138  df-2nd 6139  df-vec 25553
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