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Theorem nv2 25231
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 2467 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 25212 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
3 nvdi.2 . . . 4  |-  G  =  ( +v `  U
)
43vafval 25200 . . 3  |-  G  =  ( 1st `  ( 1st `  U ) )
5 nvdi.4 . . . 4  |-  S  =  ( .sOLD `  U )
65smfval 25202 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvdi.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 25201 . . 3  |-  X  =  ran  G
94, 6, 8vc2 25152 . 2  |-  ( ( ( 1st `  U
)  e.  CVecOLD  /\  A  e.  X )  ->  ( A G A )  =  ( 2 S A ) )
102, 9sylan 471 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G A )  =  ( 2 S A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5588  (class class class)co 6284   1stc1st 6782   2c2 10585   CVecOLDcvc 25142   NrmCVeccnv 25181   +vcpv 25182   BaseSetcba 25183   .sOLDcns 25184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-1cn 9550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-1st 6784  df-2nd 6785  df-2 10594  df-vc 25143  df-nv 25189  df-va 25192  df-ba 25193  df-sm 25194  df-0v 25195  df-nmcv 25197
This theorem is referenced by:  ipidsq  25327  minvecolem2  25495
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