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Theorem vc2OLD 29808
Description: A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) Obsolete theorem, use clmvs2 24601 together with cvsclm 24633 instead. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
vciOLD.1 𝐺 = (1st𝑊)
vciOLD.2 𝑆 = (2nd𝑊)
vciOLD.3 𝑋 = ran 𝐺
Assertion
Ref Expression
vc2OLD ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))

Proof of Theorem vc2OLD
StepHypRef Expression
1 vciOLD.1 . . . 4 𝐺 = (1st𝑊)
2 vciOLD.2 . . . 4 𝑆 = (2nd𝑊)
3 vciOLD.3 . . . 4 𝑋 = ran 𝐺
41, 2, 3vcidOLD 29804 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)
54, 4oveq12d 7423 . 2 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (𝐴𝐺𝐴))
6 df-2 12271 . . . 4 2 = (1 + 1)
76oveq1i 7415 . . 3 (2𝑆𝐴) = ((1 + 1)𝑆𝐴)
8 ax-1cn 11164 . . . 4 1 ∈ ℂ
91, 2, 3vcdir 29806 . . . . 5 ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
108, 9mp3anr1 1458 . . . 4 ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 𝐴𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
118, 10mpanr1 701 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
127, 11eqtr2id 2785 . 2 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (2𝑆𝐴))
135, 12eqtr3d 2774 1 ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  ran crn 5676  cfv 6540  (class class class)co 7405  1st c1st 7969  2nd c2nd 7970  cc 11104  1c1 11107   + caddc 11109  2c2 12263  CVecOLDcvc 29798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721  ax-1cn 11164
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-1st 7971  df-2nd 7972  df-2 12271  df-vc 29799
This theorem is referenced by:  nv2  29872  ipdirilem  30069
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