![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > vcrel | Structured version Visualization version GIF version |
Description: The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vcrel | β’ Rel CVecOLD |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-vc 29812 | . 2 β’ CVecOLD = {β¨π, π β© β£ (π β AbelOp β§ π :(β Γ ran π)βΆran π β§ βπ₯ β ran π((1π π₯) = π₯ β§ βπ¦ β β (βπ§ β ran π(π¦π (π₯ππ§)) = ((π¦π π₯)π(π¦π π§)) β§ βπ§ β β (((π¦ + π§)π π₯) = ((π¦π π₯)π(π§π π₯)) β§ ((π¦ Β· π§)π π₯) = (π¦π (π§π π₯))))))} | |
2 | 1 | relopabiv 5821 | 1 β’ Rel CVecOLD |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3062 Γ cxp 5675 ran crn 5678 Rel wrel 5682 βΆwf 6540 (class class class)co 7409 βcc 11108 1c1 11111 + caddc 11113 Β· cmul 11115 AbelOpcablo 29797 CVecOLDcvc 29811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-in 3956 df-ss 3966 df-opab 5212 df-xp 5683 df-rel 5684 df-vc 29812 |
This theorem is referenced by: vcex 29831 nvvop 29862 phop 30071 |
Copyright terms: Public domain | W3C validator |