Theorem vc2OLD 27423

Description: A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) Obsolete as of 21-Sep-2021. Use clmvs2 22894 together with cvsclm 22926 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 e. CVecOLD /\ A e. X ) -> ( A G A ) = ( 2 S A ) )

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 27419	. . 3 |- ( ( W e. CVecOLD /\ A e. X ) -> ( 1 S A ) = A )
5	4, 4	oveq12d 6668	. 2 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( 1 S A ) G ( 1 S A ) ) = ( A G A ) )
6		df-2 11079	. . . 4 |- 2 = ( 1 + 1 )
7	6	oveq1i 6660	. . 3 |- ( 2 S A ) = ( ( 1 + 1 ) S A )
8		ax-1cn 9994	. . . 4 |- 1 e. CC
9	1, 2, 3	vcdir 27421	. . . . 5 |- ( ( W e. CVecOLD /\ ( 1 e. CC /\ 1 e. CC /\ A e. X ) ) -> ( ( 1 + 1 ) S A ) = ( ( 1 S A ) G ( 1 S A ) ) )
10	8, 9	mp3anr1 1421	. . . 4 |- ( ( W e. CVecOLD /\ ( 1 e. CC /\ A e. X ) ) -> ( ( 1 + 1 ) S A ) = ( ( 1 S A ) G ( 1 S A ) ) )
11	8, 10	mpanr1 719	. . 3 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( 1 + 1 ) S A ) = ( ( 1 S A ) G ( 1 S A ) ) )
12	7, 11	syl5req 2669	. 2 |- ( ( W e. CVecOLD /\ A e. X ) -> ( ( 1 S A ) G ( 1 S A ) ) = ( 2 S A ) )
13	5, 12	eqtr3d 2658	1 |- ( ( W e. CVecOLD /\ A e. X ) -> ( A G A ) = ( 2 S A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   CCcc 9934   1c1 9937   + caddc 9939   2c2 11070   CVecOLDcvc 27413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-1cn 9994

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-1st 7168  df-2nd 7169  df-2 11079  df-vc 27414

This theorem is referenced by:  nv2  27487  ipdirilem  27684