Theorem shex 9077

Description: The set of subspaces of a Hilbert space exists (is a set).

Assertion

Ref	Expression
shex	|- SH e. V

Proof of Theorem shex

Step	Hyp	Ref	Expression
1		df-sh 9076	. 2 |- SH = {h | ((h (_ H~ /\ 0h e. h) /\ (A.x e. h A.y e. h (x +h y) e. h /\ A.x e. CC A.y e. h (x .h y) e. h))}
2		df-pw 2402	. . . 4 |- P~H~ = {h | h (_ H~}
3		ax-hilex 8869	. . . . 5 |- H~ e. V
4	3	pwex 2745	. . . 4 |- P~H~ e. V
5	2, 4	eqeltrr 1545	. . 3 |- {h | h (_ H~} e. V
6		simpll 412	. . . 4 |- (((h (_ H~ /\ 0h e. h) /\ (A.x e. h A.y e. h (x +h y) e. h /\ A.x e. CC A.y e. h (x .h y) e. h)) -> h (_ H~)
7	6	ss2abi 2120	. . 3 |- {h | ((h (_ H~ /\ 0h e. h) /\ (A.x e. h A.y e. h (x +h y) e. h /\ A.x e. CC A.y e. h (x .h y) e. h))} (_ {h | h (_ H~}
8	5, 7	ssexi 2720	. 2 |- {h | ((h (_ H~ /\ 0h e. h) /\ (A.x e. h A.y e. h (x +h y) e. h /\ A.x e. CC A.y e. h (x .h y) e. h))} e. V
9	1, 8	eqeltr 1544	1 |- SH e. V

Syntax hints:   /\ wa 223   e. wcel 958  {cab 1463  A.wral 1645  Vcvv 1811   (_ wss 2047  P~cpw 2401  (class class class)co 3963  CCcc 5232  H~chil 8788   +h cva 8789   .h csm 8790  0hc0v 8791  SHcsh 8797

This theorem is referenced by:  chex 9095

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053  df-pw 2402  df-sh 9076

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