Theorem vcidOLD 27419

Description: Identity element for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) Obsolete as of 21-Sep-2021. Use clmvs1 22893 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
vcidOLD	|- ( ( W e. CVecOLD /\ A e. X ) -> ( 1 S A ) = A )

Proof of Theorem vcidOLD

Dummy variables x y z are mutually distinct and distinct from all other variables.

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	vciOLD 27416	. . 3 |- ( W e. CVecOLD -> ( G e. AbelOp /\ S : ( CC X. X ) --> X /\ A. x e. X ( ( 1 S x ) = x /\ A. y e. CC ( A. z e. X ( y S ( x G z ) ) = ( ( y S x ) G ( y S z ) ) /\ A. z e. CC ( ( ( y + z ) S x ) = ( ( y S x ) G ( z S x ) ) /\ ( ( y x. z ) S x ) = ( y S ( z S x ) ) ) ) ) ) )
5		simpl 473	. . . . 5 |- ( ( ( 1 S x ) = x /\ A. y e. CC ( A. z e. X ( y S ( x G z ) ) = ( ( y S x ) G ( y S z ) ) /\ A. z e. CC ( ( ( y + z ) S x ) = ( ( y S x ) G ( z S x ) ) /\ ( ( y x. z ) S x ) = ( y S ( z S x ) ) ) ) ) -> ( 1 S x ) = x )
6	5	ralimi 2952	. . . 4 |- ( A. x e. X ( ( 1 S x ) = x /\ A. y e. CC ( A. z e. X ( y S ( x G z ) ) = ( ( y S x ) G ( y S z ) ) /\ A. z e. CC ( ( ( y + z ) S x ) = ( ( y S x ) G ( z S x ) ) /\ ( ( y x. z ) S x ) = ( y S ( z S x ) ) ) ) ) -> A. x e. X ( 1 S x ) = x )
7	6	3ad2ant3 1084	. . 3 |- ( ( G e. AbelOp /\ S : ( CC X. X ) --> X /\ A. x e. X ( ( 1 S x ) = x /\ A. y e. CC ( A. z e. X ( y S ( x G z ) ) = ( ( y S x ) G ( y S z ) ) /\ A. z e. CC ( ( ( y + z ) S x ) = ( ( y S x ) G ( z S x ) ) /\ ( ( y x. z ) S x ) = ( y S ( z S x ) ) ) ) ) ) -> A. x e. X ( 1 S x ) = x )
8	4, 7	syl 17	. 2 |- ( W e. CVecOLD -> A. x e. X ( 1 S x ) = x )
9		oveq2 6658	. . . 4 |- ( x = A -> ( 1 S x ) = ( 1 S A ) )
10		id 22	. . . 4 |- ( x = A -> x = A )
11	9, 10	eqeq12d 2637	. . 3 |- ( x = A -> ( ( 1 S x ) = x <-> ( 1 S A ) = A ) )
12	11	rspccva 3308	. 2 |- ( ( A. x e. X ( 1 S x ) = x /\ A e. X ) -> ( 1 S A ) = A )
13	8, 12	sylan 488	1 |- ( ( W e. CVecOLD /\ A e. X ) -> ( 1 S A ) = A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   AbelOpcablo 27398   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

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-vc 27414

This theorem is referenced by:  vc2OLD  27423  vc0  27429  vcm  27431  nvsid  27482