Theorem chocval 9166

Description: Value of the orthogonal complement of a Hilbert lattice element. The orthogonal complement of A is the set of vectors that are orthogonal to all vectors in A.

Hypothesis

Ref	Expression
chocval.1	|- A e. CH

Assertion

Ref	Expression
chocval	|- (_|_` A) = {x e. H~ | A.y e. A (x .ih y) = 0}

Distinct variable group:   x,y,A

Proof of Theorem chocval

Step	Hyp	Ref	Expression
1		chocval.1	. . 3 |- A e. CH
2	1	chssi 9096	. 2 |- A (_ H~
3		ocvalt 9148	. 2 |- (A (_ H~ -> (_|_` A) = {x e. H~ | A.y e. A (x .ih y) = 0})
4	2, 3	ax-mp 7	1 |- (_|_` A) = {x e. H~ | A.y e. A (x .ih y) = 0}

Syntax hints:   = wceq 958   e. wcel 960  A.wral 1648  {crab 1651   (_ wss 2050  ` cfv 3188  (class class class)co 3969  0cc0 5246  H~chil 8783   .ih csp 8788  CHcch 8793  _|_cort 8794

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-9 967  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  ax-hilex 8864

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-ral 1652  df-rex 1653  df-rab 1655  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-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-sh 9071  df-ch 9087  df-oc 9119

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