mmtheorems96 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 9501-9600 - Page 96 of 107

Type	Label	Description
Statement
Theorem	pjoml4 9501	Variation of orthomodular law.
|- A e. CH   &   |- B e. CH   =>   |- (A vH (B i^i ((_|_` A) vH (_|_` B)))) = (A vH B)
Theorem	pjoml5 9502	The orthomodular law. Remark in [Kalmbach] p. 22.
|- A e. CH   &   |- B e. CH   =>   |- (A vH ((_|_` A) i^i (A vH B))) = (A vH B)
Theorem	pjoml6 9503	An equivalent of the orthomodular law. Theorem 29.13(e) of [MaedaMaeda] p. 132.
|- A e. CH   &   |- B e. CH   =>   |- (A (_ B -> E.x e. CH (A (_ (_|_` x) /\ (A vH x) = B))
Theorem	cmbr 9504	Binary relation expressing the commutes relation. Definition of commutes in [Kalmbach] p. 20.
|- A e. CH   &   |- B e. CH   =>   |- (A C_H B <-> A = ((A i^i B) vH (A i^i (_|_` B))))
Theorem	cmcmlem 9505	Commutation is symmetric. Theorem 3.4 of [Beran] p. 45.
|- A e. CH   &   |- B e. CH   =>   |- (A C_H B -> B C_H A)
Theorem	cmcm 9506	Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22.
|- A e. CH   &   |- B e. CH   =>   |- (A C_H B <-> B C_H A)
Theorem	cmcm2 9507	Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39.
|- A e. CH   &   |- B e. CH   =>   |- (A C_H B <-> A C_H (_|_` B))
Theorem	cmcm3 9508	Commutation with orthocomplement. Remark in [Kalmbach] p. 23.
|- A e. CH   &   |- B e. CH   =>   |- (A C_H B <-> (_|_` A) C_H B)
Theorem	cmcm4 9509	Commutation with orthocomplement. Remark in [Kalmbach] p. 23.
|- A e. CH   &   |- B e. CH   =>   |- (A C_H B <-> (_|_` A) C_H (_|_` B))
Theorem	cmbr2 9510	Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23.
|- A e. CH   &   |- B e. CH   =>   |- (A C_H B <-> A = ((A vH B) i^i (A vH (_|_` B))))
Theorem	cmcmi 9511	Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22.
|- A e. CH   &   |- B e. CH   &   |- A C_H B   =>   |- B C_H A
Theorem	cmcm2i 9512	Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39.
|- A e. CH   &   |- B e. CH   &   |- A C_H B   =>   |- A C_H (_|_` B)
Theorem	cmcm3i 9513	Commutation with orthocomplement. Remark in [Kalmbach] p. 23.
|- A e. CH   &   |- B e. CH   &   |- A C_H B   =>   |- (_|_` A) C_H B
Theorem	cmbr3 9514	Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23.
|- A e. CH   &   |- B e. CH   =>   |- (A C_H B <-> (A i^i ((_|_` A) vH B)) = (A i^i B))
Theorem	cmbr4 9515	Alternate definition for the commutes relation.
|- A e. CH   &   |- B e. CH   =>   |- (A C_H B <-> (A i^i ((_|_` A) vH B)) (_ B)
Theorem	lecm 9516	Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40.
|- A e. CH   &   |- B e. CH   =>   |- (A (_ B -> A C_H B)
Theorem	lecmi 9517	Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40.
|- A e. CH   &   |- B e. CH   &   |- A (_ B   =>   |- A C_H B
Theorem	cmj1 9518	A Hilbert lattice element commutes with its join.
|- A e. CH   &   |- B e. CH   =>   |- A C_H (A vH B)
Theorem	cmj2 9519	A Hilbert lattice element commutes with its join.
|- A e. CH   &   |- B e. CH   =>   |- B C_H (A vH B)
Theorem	cmm1 9520	A Hilbert lattice element commutes with its meet.
|- A e. CH   &   |- B e. CH   =>   |- A C_H (A i^i B)
Theorem	cmm2 9521	A Hilbert lattice element commutes with its meet.
|- A e. CH   &   |- B e. CH   =>   |- B C_H (A i^i B)
Theorem	cmbr3t 9522	Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23.
|- ((A e. CH /\ B e. CH) -> (A C_H B <-> (A i^i ((_|_` A) vH B)) = (A i^i B)))
Theorem	cm0t 9523	The zero Hilbert lattice element commutes with every element.
|- (A e. CH -> 0H C_H A)
Theorem	cmid 9524	The commutes relation is reflexive.
|- A e. CH   =>   |- A C_H A
Theorem	pjoml2t 9525	Variation of orthomodular law. Definition in [Kalmbach] p. 22.
|- ((A e. CH /\ B e. CH /\ A (_ B) -> (A vH ((_|_` A) i^i B)) = B)
Theorem	pjoml3t 9526	Variation of orthomodular law.
|- ((A e. CH /\ B e. CH) -> (B (_ A -> (A i^i ((_|_` A) vH B)) = B))
Theorem	pjoml5t 9527	The orthomodular law. Remark in [Kalmbach] p. 22.
|- ((A e. CH /\ B e. CH) -> (A vH ((_|_` A) i^i (A vH B))) = (A vH B))
Theorem	cmcmt 9528	Commutation is symmetric. Theorem 2(v) of [Kalmbach] p. 22.
|- ((A e. CH /\ B e. CH) -> (A C_H B <-> B C_H A))
Theorem	cmcm3t 9529	Commutation with orthocomplement. Remark in [Kalmbach] p. 23.
|- ((A e. CH /\ B e. CH) -> (A C_H B <-> (_|_` A) C_H B))
Theorem	cmcm2t 9530	Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39.
|- ((A e. CH /\ B e. CH) -> (A C_H B <-> A C_H (_|_` B)))
Theorem	lecmt 9531	Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of [Beran] p. 40.
|- ((A e. CH /\ B e. CH /\ A (_ B) -> A C_H B)
Foulis-Holland theorem
Theorem	fh1t 9532	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.
|- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> (A i^i (B vH C)) = ((A i^i B) vH (A i^i C)))
Theorem	fh2t 9533	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.
|- (((A e. CH /\ B e. CH /\ C e. CH) /\ (B C_H A /\ B C_H C)) -> (A i^i (B vH C)) = ((A i^i B) vH (A i^i C)))
Theorem	cm2jt 9534	A lattice element that commutes with two others also commutes with their join. Theorem 4.2 of [Beran] p. 49.
|- (((A e. CH /\ B e. CH /\ C e. CH) /\ (A C_H B /\ A C_H C)) -> A C_H (B vH C))
Theorem	fh1 9535	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.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- A C_H B   &   |- A C_H C   =>   |- (A i^i (B vH C)) = ((A i^i B) vH (A i^i C))
Theorem	fh2 9536	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.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- A C_H B   &   |- A C_H C   =>   |- (B i^i (A vH C)) = ((B i^i A) vH (B i^i C))
Theorem	fh3 9537	Variation of the Foulis-Holland Theorem.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- A C_H B   &   |- A C_H C   =>   |- (A vH (B i^i C)) = ((A vH B) i^i (A vH C))
Theorem	fh4 9538	Variation of the Foulis-Holland Theorem.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- A C_H B   &   |- A C_H C   =>   |- (B vH (A i^i C)) = ((B vH A) i^i (B vH C))
Quantum Logic Explorer axioms
Theorem	qlax1 9539	One of the equations showing CH is an ortholattice. (This corresponds to axiom "ax-1" in the Quantum Logic Explorer.)
|- A e. CH   =>   |- A = (_|_` (_|_` A))
Theorem	qlax2 9540	One of the equations showing CH is an ortholattice. (This corresponds to axiom "ax-2" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   =>   |- (A vH B) = (B vH A)
Theorem	qlax3 9541	One of the equations showing CH is an ortholattice. (This corresponds to axiom "ax-3" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   &   |- C e. CH   =>   |- ((A vH B) vH C) = (A vH (B vH C))
Theorem	qlax4 9542	One of the equations showing CH is an ortholattice. (This corresponds to axiom "ax-4" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   =>   |- (A vH (B vH (_|_` B))) = (B vH (_|_` B))
Theorem	qlax5 9543	One of the equations showing CH is an ortholattice. (This corresponds to axiom "ax-5" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   =>   |- (A vH (_|_` ((_|_` A) vH B))) = A
Theorem	qlaxr1 9544	One of the conditions showing CH is an ortholattice. (This corresponds to axiom "ax-r1" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   &   |- A = B   =>   |- B = A
Theorem	qlaxr2 9545	One of the conditions showing CH is an ortholattice. (This corresponds to axiom "ax-r2" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- A = B   &   |- B = C   =>   |- A = C
Theorem	qlaxr4 9546	One of the conditions showing CH is an ortholattice. (This corresponds to axiom "ax-r4" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   &   |- A = B   =>   |- (_|_` A) = (_|_` B)
Theorem	qlaxr5 9547	One of the conditions showing CH is an ortholattice. (This corresponds to axiom "ax-r5" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- A = B   =>   |- (A vH C) = (B vH C)
Theorem	qlaxr3 9548	A variation of the orthomodular law, showing CH is an orthomodular lattice. (This corresponds to axiom "ax-r3" in the Quantum Logic Explorer.)
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- (C vH (_|_` C)) = ((_|_` ((_|_` A) vH (_|_` B))) vH (_|_` (A vH B)))   =>   |- A = B
Orthogonal subspaces
Theorem	osumlem1 9549	Lemma for osum 9557.
Theorem	osumlem2 9550	Lemma for osum 9557.
Theorem	osumlem3 9551	Lemma for osum 9557.
Theorem	osumlem4 9552	Lemma for osum 9557.
Theorem	osumlem5 9553	Lemma for osum 9557.
Theorem	osumlem6 9554	Lemma for osum 9557.
Theorem	osumlem7 9555	Lemma for osum 9557.
Theorem	osumlem8 9556	Lemma for osum 9557.
Theorem	osum 9557	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 9553 and via pjpjtht 9227 in osumlem7 9555).
|- A e.