Theorem vcrel 27415

Description: The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
vcrel	|- Rel CVecOLD

Proof of Theorem vcrel

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

Step	Hyp	Ref	Expression
1		df-vc 27414	. 2 |- CVecOLD = { <. g , s >. | ( g e. AbelOp /\ s : ( CC X. ran g ) --> ran g /\ A. x e. ran g ( ( 1 s x ) = x /\ A. y e. CC ( A. z e. ran g ( 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 ) ) ) ) ) ) }
2	1	relopabi 5245	1 |- Rel CVecOLD

Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   X. cxp 5112   ran crn 5115   Rel wrel 5119   -->wf 5884  (class class class)co 6650   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  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-opab 4713  df-xp 5120  df-rel 5121  df-vc 27414

This theorem is referenced by:  vcex  27433  nvvop  27464  phop  27673