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Theorem nv2 26098
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  =  ( .sOLD `  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 2429 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 26079 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
3 nvdi.2 . . . 4  |-  G  =  ( +v `  U
)
43vafval 26067 . . 3  |-  G  =  ( 1st `  ( 1st `  U ) )
5 nvdi.4 . . . 4  |-  S  =  ( .sOLD `  U )
65smfval 26069 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvdi.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 26068 . . 3  |-  X  =  ran  G
94, 6, 8vc2 26019 . 2  |-  ( ( ( 1st `  U
)  e.  CVecOLD  /\  A  e.  X )  ->  ( A G A )  =  ( 2 S A ) )
102, 9sylan 473 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G A )  =  ( 2 S A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   ` cfv 5601  (class class class)co 6305   1stc1st 6805   2c2 10659   CVecOLDcvc 26009   NrmCVeccnv 26048   +vcpv 26049   BaseSetcba 26050   .sOLDcns 26051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-1cn 9596
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-1st 6807  df-2nd 6808  df-2 10668  df-vc 26010  df-nv 26056  df-va 26059  df-ba 26060  df-sm 26061  df-0v 26062  df-nmcv 26064
This theorem is referenced by:  ipidsq  26194  minvecolem2  26362
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