Theorem vc2OLD 30957

Description: A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) Obsolete theorem, use clmvs2 25290 together with cvsclm 25322 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
vciOLD.1	⊢ 𝐺 = (1st ‘𝑊)
vciOLD.2	⊢ 𝑆 = (2nd ‘𝑊)
vciOLD.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
vc2OLD	⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))

Proof of Theorem vc2OLD

Step	Hyp	Ref	Expression
1		vciOLD.1	. . . 4 ⊢ 𝐺 = (1st ‘𝑊)
2		vciOLD.2	. . . 4 ⊢ 𝑆 = (2nd ‘𝑊)
3		vciOLD.3	. . . 4 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	vcidOLD 30953	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
5	4, 4	oveq12d 7441	. 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (𝐴𝐺𝐴))
6		df-2 12321	. . . 4 ⊢ 2 = (1 + 1)
7	6	oveq1i 7433	. . 3 ⊢ (2𝑆𝐴) = ((1 + 1)𝑆𝐴)
8		ax-1cn 11176	. . . 4 ⊢ 1 ∈ ℂ
9	1, 2, 3	vcdir 30955	. . . . 5 ⊢ ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
10	8, 9	mp3anr1 1487	. . . 4 ⊢ ((𝑊 ∈ CVecOLD ∧ (1 ∈ ℂ ∧ 𝐴 ∈ 𝑋)) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
11	8, 10	mpanr1 716	. . 3 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1 + 1)𝑆𝐴) = ((1𝑆𝐴)𝐺(1𝑆𝐴)))
12	7, 11	eqtr2id 2814	. 2 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → ((1𝑆𝐴)𝐺(1𝑆𝐴)) = (2𝑆𝐴))
13	5, 12	eqtr3d 2803	1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ran crn 5667  ‘cfv 6543  (class class class)co 7423  1^st c1st 7993  2^nd c2nd 7994  ℂcc 11116  1c1 11119   + caddc 11121  2c2 12313  CVec_OLDcvc 30947

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745  ax-1cn 11176

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7426  df-1st 7995  df-2nd 7996  df-2 12321  df-vc 30948

This theorem is used by:  nv2  31021  ipdirilem  31218