mmtheorems96 - Metamath Proof Explorer

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Metamath Proof Explorer (1-8743)

Hilbert Space Explorer (8744-10691)

**Statement List for Metamath Proof Explorer -** 9501-9600 - Page 96 of 107
Type	Label	Description
Statement

Theorem	cmbr3 9501	Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23.


Theorem	cmbr4 9502	Alternate definition for the commutes relation.


Theorem	lecm 9503	Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40.


Theorem	lecmi 9504	Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40.


Theorem	cmj1 9505	A Hilbert lattice element commutes with its join.


Theorem	cmj2 9506	A Hilbert lattice element commutes with its join.


Theorem	cmm1 9507	A Hilbert lattice element commutes with its meet.


Theorem	cmm2 9508	A Hilbert lattice element commutes with its meet.


Theorem	cmbr3t 9509	Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23.
$/\$

Theorem	cm0t 9510	The zero Hilbert lattice element commutes with every element.


Theorem	cmid 9511	The commutes relation is reflexive.


Theorem	pjoml2t 9512	Variation of orthomodular law. Definition in [Kalmbach] p. 22.
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Theorem	pjoml3t 9513	Variation of orthomodular law.
$/\$

Theorem	pjoml5t 9514	The orthomodular law. Remark in [Kalmbach] p. 22.
$/\$

Theorem	cmcmt 9515	Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22.
$/\$

Theorem	cmcm3t 9516	Commutation with orthocomplement. Remark in [Kalmbach] p. 23.
$/\$

Theorem	cmcm2t 9517	Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39.
$/\$

Theorem	lecmt 9518	Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40.
$/\$ $/\$

Foulis-Holland theorem

Theorem	fh1t 9519	Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. First of two parts. Theorem 5 of [Kalmbach] p. 25.
$/\$ $/\$ $/\$ $/\$

Theorem	fh2t 9520	Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Second of two parts. Theorem 5 of [Kalmbach] p. 25.
$/\$ $/\$ $/\$ $/\$

Theorem	cm2jt 9521	A lattice element that commutes with two others also commutes with their join. Theorem 4.2 of [Beran] p. 49.
$/\$ $/\$ $/\$ $/\$

Theorem	fh1 9522	Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. First of two parts. Theorem 5 of [Kalmbach] p. 25.


Theorem	fh2 9523	Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Second of two parts. Theorem 5 of [Kalmbach] p. 25.


Theorem	fh3 9524	Variation of the Foulis-Holland Theorem.


Theorem	fh4 9525	Variation of the Foulis-Holland Theorem.


Quantum Logic Explorer axioms

Theorem	qlax1 9526	One of the equations showing is an ortholattice. (This corresponds to axiom "ax-1" in the Quantum Logic Explorer.)


Theorem	qlax2 9527	One of the equations showing is an ortholattice. (This corresponds to axiom "ax-2" in the Quantum Logic Explorer.)


Theorem	qlax3 9528	One of the equations showing is an ortholattice. (This corresponds to axiom "ax-3" in the Quantum Logic Explorer.)


Theorem	qlax4 9529	One of the equations showing is an ortholattice. (This corresponds to axiom "ax-4" in the Quantum Logic Explorer.)


Theorem	qlax5 9530	One of the equations showing is an ortholattice. (This corresponds to axiom "ax-5" in the Quantum Logic Explorer.)


Theorem	qlaxr1 9531	One of the conditions showing is an ortholattice. (This corresponds to axiom "ax-r1" in the Quantum Logic Explorer.)


Theorem	qlaxr2 9532	One of the conditions showing is an ortholattice. (This corresponds to axiom "ax-r2" in the Quantum Logic Explorer.)


Theorem	qlaxr4 9533	One of the conditions showing is an ortholattice. (This corresponds to axiom "ax-r4" in the Quantum Logic Explorer.)


Theorem	qlaxr5 9534	One of the conditions showing is an ortholattice. (This corresponds to axiom "ax-r5" in the Quantum Logic Explorer.)


Theorem	qlaxr3 9535	A variation of the orthomodular law, showing is an orthomodular lattice. (This corresponds to axiom "ax-r3" in the Quantum Logic Explorer.)


Orthogonal subspaces

Theorem	osumlem1 9536	Lemma for osum 9544.

Theorem	osumlem2 9537	Lemma for osum 9544.

Theorem	osumlem3 9538	Lemma for osum 9544.

Theorem	osumlem4 9539	Lemma for osum 9544.

Theorem	osumlem5 9540	Lemma for osum 9544.

Theorem	osumlem6 9541	Lemma for osum 9544.

Theorem	osumlem7 9542	Lemma for osum 9544.

Theorem	osumlem8 9543	Lemma for osum 9544.

Theorem	osum 9544	If two closed subspaces of a Hilbert space are orthogonal, their subspace sum equals their subspace join. Lemma 3 of [Kalmbach] p. 67. Note that the Axiom of Choice is used for this proof (in osumlem5 9540 and via pjpjtht 9214 in osumlem7 9542).


Theorem	osumcor 9545	Corollary of osum 9544.


Theorem	osumt 9546	If two closed subspaces of a Hilbert space are orthogonal, their subspace sum equals their subspace join. Lemma 3 of [Kalmbach] p. 67.
$/\$ $/\$

Theorem	chsot 9547	The subspace sum of a closed subspace and its complement is all of Hilbert space.


Theorem	osumcor2 9548	Corollary of osum 9544, showing it holds under the weaker hypothesis that and commute.


Theorem	spansnj 9549	The subspace sum of a closed subspace and a one-dimensional subspace equals their join. (Proof suggested by Eric Schechter 1-Jun-2004.)


Theorem	spansnjt 9550	The subspace sum of a closed subspace and a one-dimensional subspace equals their join.
$/\$

Theorem	spansnsclt 9551	The subspace sum of a closed subspace and a one-dimensional subspace is closed.
$/\$

Theorem	sumspansnt 9552	The sum of two vectors belong to the span of one of them iff the other vector also belongs.
$/\$

Theorem	spansnm0 9553	The meet of different one-dimensional subspaces is the zero subspace.


Theorem	nonbool 9554	A Hilbert lattice with two or more dimensions fails the distributive law and therefore cannot be a Boolean algebra. This counterexample demonstrates a condition where but . The antecedent specifies that the vectors and are nonzero and non-colinear. The last three hypotheses assign one-dimensional subspaces to , , and .