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Theorem vc2OLD 29821
Description: A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) Obsolete theorem, use clmvs2 24610 together with cvsclm 24642 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 29817 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)
54, 4oveq12d 7427 . 2 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (𝐴𝐺𝐴))
6 df-2 12275 . . . 4 2 = (1 + 1)
76oveq1i 7419 . . 3 (2𝑆𝐴) = ((1 + 1)𝑆𝐴)
8 ax-1cn 11168 . . . 4 1 ∈ ℂ
91, 2, 3vcdir 29819 . . . . 5 ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
108, 9mp3anr1 1459 . . . 4 ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 𝐴𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
118, 10mpanr1 702 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
127, 11eqtr2id 2786 . 2 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (2𝑆𝐴))
135, 12eqtr3d 2775 1 ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  ran crn 5678  cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  cc 11108  1c1 11111   + caddc 11113  2c2 12267  CVecOLDcvc 29811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725  ax-1cn 11168
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-1st 7975  df-2nd 7976  df-2 12275  df-vc 29812
This theorem is referenced by:  nv2  29885  ipdirilem  30082
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