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Theorem vc2OLD 30587
Description: A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) Obsolete theorem, use clmvs2 25127 together with cvsclm 25159 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 30583 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)
54, 4oveq12d 7449 . 2 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (𝐴𝐺𝐴))
6 df-2 12329 . . . 4 2 = (1 + 1)
76oveq1i 7441 . . 3 (2𝑆𝐴) = ((1 + 1)𝑆𝐴)
8 ax-1cn 11213 . . . 4 1 ∈ ℂ
91, 2, 3vcdir 30585 . . . . 5 ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
108, 9mp3anr1 1460 . . . 4 ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 𝐴𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
118, 10mpanr1 703 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
127, 11eqtr2id 2790 . 2 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (2𝑆𝐴))
135, 12eqtr3d 2779 1 ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  ran crn 5686  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  cc 11153  1c1 11156   + caddc 11158  2c2 12321  CVecOLDcvc 30577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-1cn 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-1st 8014  df-2nd 8015  df-2 12329  df-vc 30578
This theorem is referenced by:  nv2  30651  ipdirilem  30848
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