Theorem isvcOLD 27434

Description: The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) Obsolete as of 4-Oct-2021. Use iscvsp 22928 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
isvcOLD.1	|- X = ran G

Assertion

Ref	Expression
isvcOLD	|- ( <. 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 ) ) ) ) ) ) )

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

Allowed substitution hint:   X( y)

Proof of Theorem isvcOLD

Step	Hyp	Ref	Expression
1		vcex 27433	. 2 |- ( <. G , S >. e. CVecOLD -> ( G e. _V /\ S e. _V ) )
2		elex 3212	. . . . 5 |- ( G e. AbelOp -> G e. _V )
3	2	adantr 481	. . . 4 |- ( ( G e. AbelOp /\ S : ( CC X. X ) --> X ) -> G e. _V )
4		cnex 10017	. . . . . . 7 |- CC e. _V
5		ablogrpo 27401	. . . . . . . 8 |- ( G e. AbelOp -> G e. GrpOp )
6		isvcOLD.1	. . . . . . . . 9 |- X = ran G
7		rnexg 7098	. . . . . . . . 9 |- ( G e. GrpOp -> ran G e. _V )
8	6, 7	syl5eqel 2705	. . . . . . . 8 |- ( G e. GrpOp -> X e. _V )
9	5, 8	syl 17	. . . . . . 7 |- ( G e. AbelOp -> X e. _V )
10		xpexg 6960	. . . . . . 7 |- ( ( CC e. _V /\ X e. _V ) -> ( CC X. X ) e. _V )
11	4, 9, 10	sylancr 695	. . . . . 6 |- ( G e. AbelOp -> ( CC X. X ) e. _V )
12		fex 6490	. . . . . 6 |- ( ( S : ( CC X. X ) --> X /\ ( CC X. X ) e. _V ) -> S e. _V )
13	11, 12	sylan2 491	. . . . 5 |- ( ( S : ( CC X. X ) --> X /\ G e. AbelOp ) -> S e. _V )
14	13	ancoms 469	. . . 4 |- ( ( G e. AbelOp /\ S : ( CC X. X ) --> X ) -> S e. _V )
15	3, 14	jca 554	. . 3 |- ( ( G e. AbelOp /\ S : ( CC X. X ) --> X ) -> ( G e. _V /\ S e. _V ) )
16	15	3adant3 1081	. 2 |- ( ( 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 ) ) ) ) ) ) -> ( G e. _V /\ S e. _V ) )
17	6	isvclem 27432	. 2 |- ( ( G e. _V /\ S e. _V ) -> ( <. 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 ) ) ) ) ) ) ) )
18	1, 16, 17	pm5.21nii 368	1 |- ( <. 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 ) ) ) ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   <.cop 4183   X. cxp 5112   ran crn 5115   -->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-ablo 27399  df-vc 27414

This theorem is referenced by:  isvciOLD  27435