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Theorem nv2 26871
 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 = ( +𝑣𝑈)
nvdi.4 𝑆 = ( ·𝑠OLD𝑈)
Assertion
Ref Expression
nv2 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))

Proof of Theorem nv2
StepHypRef Expression
1 eqid 2610 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 26854 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 nvdi.2 . . . 4 𝐺 = ( +𝑣𝑈)
43vafval 26842 . . 3 𝐺 = (1st ‘(1st𝑈))
5 nvdi.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 26844 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvdi.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 26843 . . 3 𝑋 = ran 𝐺
94, 6, 8vc2OLD 26807 . 2 (((1st𝑈) ∈ CVecOLD𝐴𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))
102, 9sylan 487 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2c2 10947  CVecOLDcvc 26797  NrmCVeccnv 26823   +𝑣 cpv 26824  BaseSetcba 26825   ·𝑠OLD cns 26826 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-1cn 9873 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-1st 7059  df-2nd 7060  df-2 10956  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-nmcv 26839 This theorem is referenced by:  ipidsq  26949  minvecolem2  27115
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