Theorem shex 9016

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 9015	. 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 2398	. . . 4 |- P~H~ = {h | h (_ H~}
3		ax-hilex 8808	. . . . 5 |- H~ e. V
4	3	pwex 2740	. . . 4 |- P~H~ e. V
5	2, 4	eqeltrr 1542	. . 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 2116	. . 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 2715	. 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 1541	1 |- SH e. V

Syntax hints:   /\ wa 223   e. wcel 956  {cab 1461  A.wral 1642  Vcvv 1807   (_ wss 2043  P~cpw 2397  (class class class)co 3954  CCcc 5212  H~chil 8727   +h cva 8728   .h csm 8729  0hc0v 8730  SHcsh 8736

This theorem is referenced by:  chex 9034

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-hilex 8808

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-in 2047  df-ss 2049  df-pw 2398  df-sh 9015

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