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Theorem nv2 21206
Description: A vector plus itself is two times the vector. (Contributed by NM, 9-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvdi.1  |-  X  =  ( BaseSet `  U )
nvdi.2  |-  G  =  ( +v `  U
)
nvdi.4  |-  S  =  ( .s OLD `  U
)
Assertion
Ref Expression
nv2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G A )  =  ( 2 S A ) )

Proof of Theorem nv2
StepHypRef Expression
1 eqid 2296 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 21187 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVec OLD )
3 nvdi.2 . . . 4  |-  G  =  ( +v `  U
)
43vafval 21175 . . 3  |-  G  =  ( 1st `  ( 1st `  U ) )
5 nvdi.4 . . . 4  |-  S  =  ( .s OLD `  U
)
65smfval 21177 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvdi.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 21176 . . 3  |-  X  =  ran  G
94, 6, 8vc2 21127 . 2  |-  ( ( ( 1st `  U
)  e.  CVec OLD  /\  A  e.  X )  ->  ( A G A )  =  ( 2 S A ) )
102, 9sylan 457 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G A )  =  ( 2 S A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2c2 9811   CVec OLDcvc 21117   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s
OLDcns 21159
This theorem is referenced by:  ipidsq  21302  minvecolem2  21470
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-1cn 8811
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-reu 2563  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-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-1st 6138  df-2nd 6139  df-2 9820  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172
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