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Theorem vc2OLD 30391
Description: A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) Obsolete theorem, use clmvs2 25034 together with cvsclm 25066 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 30387 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)
54, 4oveq12d 7438 . 2 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (𝐴𝐺𝐴))
6 df-2 12306 . . . 4 2 = (1 + 1)
76oveq1i 7430 . . 3 (2𝑆𝐴) = ((1 + 1)𝑆𝐴)
8 ax-1cn 11197 . . . 4 1 ∈ ℂ
91, 2, 3vcdir 30389 . . . . 5 ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
108, 9mp3anr1 1455 . . . 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 2781 . 2 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (2𝑆𝐴))
135, 12eqtr3d 2770 1 ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  ran crn 5679  cfv 6548  (class class class)co 7420  1st c1st 7991  2nd c2nd 7992  cc 11137  1c1 11140   + caddc 11142  2c2 12298  CVecOLDcvc 30381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740  ax-1cn 11197
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-1st 7993  df-2nd 7994  df-2 12306  df-vc 30382
This theorem is referenced by:  nv2  30455  ipdirilem  30652
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