Theorem isvciOLD 27435

Description: Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) Obsolete as of 4-Oct-2021. Use iscvsi 22929 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
isvciOLD.1	|- G e. AbelOp
isvciOLD.2	|- dom G = ( X X. X )
isvciOLD.3	|- S : ( CC X. X ) --> X
isvciOLD.4	|- ( x e. X -> ( 1 S x ) = x )
isvciOLD.5	|- ( ( y e. CC /\ x e. X /\ z e. X ) -> ( y S ( x G z ) ) = ( ( y S x ) G ( y S z ) ) )
isvciOLD.6	|- ( ( y e. CC /\ z e. CC /\ x e. X ) -> ( ( y + z ) S x ) = ( ( y S x ) G ( z S x ) ) )
isvciOLD.7	|- ( ( y e. CC /\ z e. CC /\ x e. X ) -> ( ( y x. z ) S x ) = ( y S ( z S x ) ) )
isvciOLD.8	|- W = <. G , S >.

Assertion

Ref	Expression
isvciOLD	|- W e. CVecOLD

Distinct variable groups:   x, y, z, G   x, S, y, z   x, X, y, z

Allowed substitution hints:   W( x, y, z)

Proof of Theorem isvciOLD

Step	Hyp	Ref	Expression
1		isvciOLD.8	. 2 |- W = <. G , S >.
2		isvciOLD.1	. . 3 |- G e. AbelOp
3		isvciOLD.3	. . 3 |- S : ( CC X. X ) --> X
4		isvciOLD.4	. . . . 5 |- ( x e. X -> ( 1 S x ) = x )
5		isvciOLD.5	. . . . . . . . . 10 |- ( ( y e. CC /\ x e. X /\ z e. X ) -> ( y S ( x G z ) ) = ( ( y S x ) G ( y S z ) ) )
6	5	3com12 1269	. . . . . . . . 9 |- ( ( x e. X /\ y e. CC /\ z e. X ) -> ( y S ( x G z ) ) = ( ( y S x ) G ( y S z ) ) )
7	6	3expa 1265	. . . . . . . 8 |- ( ( ( x e. X /\ y e. CC ) /\ z e. X ) -> ( y S ( x G z ) ) = ( ( y S x ) G ( y S z ) ) )
8	7	ralrimiva 2966	. . . . . . 7 |- ( ( x e. X /\ y e. CC ) -> A. z e. X ( y S ( x G z ) ) = ( ( y S x ) G ( y S z ) ) )
9		isvciOLD.6	. . . . . . . . . . 11 |- ( ( y e. CC /\ z e. CC /\ x e. X ) -> ( ( y + z ) S x ) = ( ( y S x ) G ( z S x ) ) )
10		isvciOLD.7	. . . . . . . . . . 11 |- ( ( y e. CC /\ z e. CC /\ x e. X ) -> ( ( y x. z ) S x ) = ( y S ( z S x ) ) )
11	9, 10	jca 554	. . . . . . . . . 10 |- ( ( y e. CC /\ z e. CC /\ x e. X ) -> ( ( ( y + z ) S x ) = ( ( y S x ) G ( z S x ) ) /\ ( ( y x. z ) S x ) = ( y S ( z S x ) ) ) )
12	11	3comr 1273	. . . . . . . . 9 |- ( ( x e. X /\ y e. CC /\ z e. CC ) -> ( ( ( y + z ) S x ) = ( ( y S x ) G ( z S x ) ) /\ ( ( y x. z ) S x ) = ( y S ( z S x ) ) ) )
13	12	3expa 1265	. . . . . . . 8 |- ( ( ( x e. X /\ y e. CC ) /\ z e. CC ) -> ( ( ( y + z ) S x ) = ( ( y S x ) G ( z S x ) ) /\ ( ( y x. z ) S x ) = ( y S ( z S x ) ) ) )
14	13	ralrimiva 2966	. . . . . . 7 |- ( ( x e. X /\ y e. CC ) -> 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 ) ) ) )
15	8, 14	jca 554	. . . . . 6 |- ( ( x e. X /\ 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 ) ) ) ) )
16	15	ralrimiva 2966	. . . . 5 |- ( x e. 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 ) ) ) ) )
17	4, 16	jca 554	. . . 4 |- ( 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 ) ) ) ) ) )
18	17	rgen 2922	. . 3 |- 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 ) ) ) ) )
19		ablogrpo 27401	. . . . . 6 |- ( G e. AbelOp -> G e. GrpOp )
20	2, 19	ax-mp 5	. . . . 5 |- G e. GrpOp
21		isvciOLD.2	. . . . 5 |- dom G = ( X X. X )
22	20, 21	grporn 27375	. . . 4 |- X = ran G
23	22	isvcOLD 27434	. . 3 |- ( <. G , S >. 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 ) ) ) ) ) ) )
24	2, 3, 18, 23	mpbir3an 1244	. 2 |- <. G , S >. e. CVecOLD
25	1, 24	eqeltri 2697	1 |- W e. CVecOLD

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   X. cxp 5112   dom cdm 5114   -->wf 5884  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   GrpOpcgr 27343   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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-grpo 27347  df-ablo 27399  df-vc 27414

This theorem is referenced by:  cncvcOLD  27438  hilvc  28019  hhssnv  28121