Theorem chocval 9087

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 9022	. 2 |- A (_ H~
3		ocvalt 9069	. 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 953   e. wcel 955  A.wral 1637  {crab 1640   (_ wss 2037  ` cfv 3172  (class class class)co 3948  0cc0 5206  H~chil 8727   .ih csp 8732  CHcch 8737  _|_cort 8738

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-hilex 8790

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188  df-sh 8997  df-ch 9013  df-oc 9045

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