Theorem vcrel 8162

Description: The class of all complex vector spaces is a relation.

Assertion

Ref	Expression
vcrel	|- Rel CVec

Proof of Theorem vcrel

Step	Hyp	Ref	Expression
1		relopab 3272	. 2 |- Rel {<.g, s>. | (g e. Abel /\ s:(CC X. ran g)-->ran g /\ A.x e. ran g((1sx) = x /\ A.y e. CC (A.z e. ran g(ys(xgz)) = ((ysx)g(ysz)) /\ A.z e. CC (((y + z)sx) = ((ysx)g(zsx)) /\ ((y x. z)sx) = (ys(zsx))))))}
2		df-vc 8161	. . 3 |- CVec = {<.g, s>. | (g e. Abel /\ s:(CC X. ran g)-->ran g /\ A.x e. ran g((1sx) = x /\ A.y e. CC (A.z e. ran g(ys(xgz)) = ((ysx)g(ysz)) /\ A.z e. CC (((y + z)sx) = ((ysx)g(zsx)) /\ ((y x. z)sx) = (ys(zsx))))))}
3	2	releqi 3250	. 2 |- (Rel CVec <-> Rel {<.g, s>. | (g e. Abel /\ s:(CC X. ran g)-->ran g /\ A.x e. ran g((1sx) = x /\ A.y e. CC (A.z e. ran g(ys(xgz)) = ((ysx)g(ysz)) /\ A.z e. CC (((y + z)sx) = ((ysx)g(zsx)) /\ ((y x. z)sx) = (ys(zsx))))))})
4	1, 3	mpbir 190	1 |- Rel CVec

Syntax hints:   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  {copab 2671   X. cxp 3174  ran crn 3177  Rel wrel 3181  -->wf 3184  (class class class)co 3969  CCcc 5244  1c1 5247   + caddc 5249   x. cmul 5251  Abelcabl 8095  CVeccvc 8160

This theorem is referenced by:  vcoprnelem 8193  vcex 8195  phop 8473

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-opab 2672  df-xp 3190  df-rel 3191  df-vc 8161

Copyright terms: Public domain