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**Statement List for Metamath Proof Explorer -** 9101-9200 - Page 92 of 107
Type	Label	Description
Statement

Theorem	shocss 9101	An orthogonal complement is a subset of Hilbert space.


Theorem	occont 9102	Contraposition law for orthogonal complement.
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Theorem	occon2t 9103	Double contraposition for orthogonal complement.
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Theorem	occon2 9104	Double contraposition for orthogonal complement.


Theorem	oc0 9105	The zero vector belongs to an orthogonal complement of a Hilbert subspace.


Theorem	ocorth 9106	Members of a subset and its complement are orthogonal.
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Theorem	shocorth 9107	Members of a subspace and its complement are orthogonal.
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Theorem	ococss 9108	Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65.


Theorem	shococss 9109	Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65.


Theorem	shorth 9110	Members of orthogonal subspaces are orthogonal.
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Theorem	ocin 9111	Intersection of a Hilbert subspace and its complement. Part of Proposition 1 of [Kalmbach] p. 65.


Theorem	ocnelt 9112	A nonzero vector in the complement of a subspace does not belong to the subspace.
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Theorem	chocval 9113	Value of the orthogonal complement of a Hilbert lattice element. The orthogonal complement of is the set of vectors that are orthogonal to all vectors in .


Theorem	chocuni 9114	Lemma for uniqueness part of Projection Theorem. Theorem 3.7(i) of [Beran] p. 102 (uniqueness part).

Theorem	occllem1 9115	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem2 9116	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem3 9117	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem4 9118	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem5 9119	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem6 9120	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem7 9121	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occllem8 9122	Lemma for closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.

Theorem	occl 9123	Closure of complement of Hilbert subset. Part of Remark 3.12 of [Beran] p. 107.


Theorem	occlt 9124	Closure of complement of Hilbert subset. Part of Remark 3.12 of [Beran] p. 107.


Theorem	shocclt 9125	Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.


Theorem	chocclt 9126	Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107.


Theorem	choccl 9127	Closure of orthocomplement.


Projection theorem

Theorem	projlem1 9128	Part of Lemma 3.6 of [Beran] p. 100: "Choose e > 0. Let n₀ be a natural number satisfying the inequality n₀ > 4(2i₀ + 1) e -1." Used by projlem2 9129.


Theorem	projlem2 9129	Part of Lemma 3.6 of [Beran] p. 100. We need the square root for the norm limit. Used by projlem28 9155.


Theorem	projlem3 9130	Part of Lemma 3.6 of [Beran] p. 100, bottom inequality. Used by projlem6 9133.


Theorem	projlem4 9131	Part of Lemma 3.6 of [Beran] p. 101, top. Used by projlem6 9133.
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Theorem	projlem5 9132	Part of Lemma 3.6 of [Beran] p. 100, bottom. Used by projlem6 9133.


Theorem	projlem6 9133	Part of Lemma 3.6 of [Beran] p. 101. Used by projlem7 9134.
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Theorem	projlem7 9134	Part of Lemma 3.6 of [Beran] p. 101. Applies weak deduction theorem to projlem6 9133. Used by projlem19 9146.
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Theorem	projlem8 9135	Part of Lemma 3.6 of [Beran] p. 100. The set is a non-empty set of reals with an upper bound. Part of Lemma 3.6 of [Beran] p. 100. Used by projlem9 9136 projlem12 9139 projlem13 9140 projlem15 9142. Note we use 'supremum'; its negative is the infimum.
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Theorem	projlem9 9136	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Real closure of the infimum of norms. Used by projlem11 9138 projlem12 9139 projlem13 9140 projlem15 9142.


Theorem	projlem10 9137	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). A member of the infimum set. Used by projlem12 9139.


Theorem	projlem11 9138	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). is the infimum of the set of norms. Show it is real. Used by projlem12 9139 projlem13 9140 projlem14 9141 projlem15 9142 projlem18 9145 projlem19 9146 projlem26 9153 projlem28 9155 projlem31 9158.


Theorem	projlem12 9139	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). The infimum is less than any norm in the set of norms. Used by projlem14 9141 projlem18 9145 projlem31 9158.


Theorem	projlem13 9140	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). The infimum of the set of norms is nonnegative. Used by projlem18 9145 projlem19 9146 projlem28 9155.


Theorem	projlem14 9141	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Used by projlem16 9143.