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Theorem vc2OLD 27393
 Description: A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) Obsolete as of 21-Sep-2021. Use clmvs2 22875 together with cvsclm 22907 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 27389 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)
54, 4oveq12d 6653 . 2 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (𝐴𝐺𝐴))
6 df-2 11064 . . . 4 2 = (1 + 1)
76oveq1i 6645 . . 3 (2𝑆𝐴) = ((1 + 1)𝑆𝐴)
8 ax-1cn 9979 . . . 4 1 ∈ ℂ
91, 2, 3vcdir 27391 . . . . 5 ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
108, 9mp3anr1 1419 . . . 4 ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 𝐴𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
118, 10mpanr1 718 . . 3 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
127, 11syl5req 2667 . 2 ((𝑊 ∈ CVecOLD𝐴𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (2𝑆𝐴))
135, 12eqtr3d 2656 1 ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1481   ∈ wcel 1988  ran crn 5105  ‘cfv 5876  (class class class)co 6635  1st c1st 7151  2nd c2nd 7152  ℂcc 9919  1c1 9922   + caddc 9924  2c2 11055  CVecOLDcvc 27383 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-1cn 9979 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-1st 7153  df-2nd 7154  df-2 11064  df-vc 27384 This theorem is referenced by:  nv2  27457  ipdirilem  27654
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