mmtheorems95 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 9401-9500 - Page 95 of 108

Type	Label	Description
Statement
Theorem	chub2t 9401	Hilbert lattice join is greater than or equal to its second argument.
|- ((A e. CH /\ B e. CH) -> A (_ (B vH A))
Theorem	chlubt 9402	Hilbert lattice join is the least upper bound of two elements.
|- ((A e. CH /\ B e. CH /\ C e. CH) -> ((A (_ C /\ B (_ C) <-> (A vH B) (_ C))
Theorem	chlej1t 9403	Add join to both sides of Hilbert lattice ordering.
|- (((A e. CH /\ B e. CH /\ C e. CH) /\ A (_ B) -> (A vH C) (_ (B vH C))
Theorem	chlej2t 9404	Add join to both sides of Hilbert lattice ordering.
|- (((A e. CH /\ B e. CH /\ C e. CH) /\ A (_ B) -> (C vH A) (_ (C vH B))
Theorem	chlejb1t 9405	Hilbert lattice ordering in terms of join.
|- ((A e. CH /\ B e. CH) -> (A (_ B <-> (A vH B) = B))
Theorem	chlejb2t 9406	Hilbert lattice ordering in terms of join.
|- ((A e. CH /\ B e. CH) -> (A (_ B <-> (B vH A) = B))
Theorem	chnlet 9407	Equivalent expressions for "not less than" in the Hilbert lattice.
|- ((A e. CH /\ B e. CH) -> (-. B (_ A <-> A (. (A vH B)))
Theorem	chjot 9408	The join of a closed subspace and its orthocomplement is all of Hilbert space.
|- (A e. CH -> (A vH (_|_` A)) = H~)
Theorem	chabs1t 9409	Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14.
|- ((A e. CH /\ B e. CH) -> (A vH (A i^i B)) = A)
Theorem	chabs2t 9410	Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14.
|- ((A e. CH /\ B e. CH) -> (A i^i (A vH B)) = A)
Theorem	chabs1 9411	Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14.
|- A e. CH   &   |- B e. CH   =>   |- (A vH (A i^i B)) = A
Theorem	chabs2 9412	Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14.
|- A e. CH   &   |- B e. CH   =>   |- (A i^i (A vH B)) = A
Theorem	chjidmt 9413	Idempotent law for Hilbert lattice join.
|- (A e. CH -> (A vH A) = A)
Theorem	chjidm 9414	Idempotent law for Hilbert lattice join.
|- A e. CH   =>   |- (A vH A) = A
Theorem	chj12 9415	A rearrangement of Hilbert lattice join.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   =>   |- (A vH (B vH C)) = (B vH (A vH C))
Theorem	chj4 9416	Rearrangement of the join of 4 Hilbert lattice elements.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   &   |- D e. CH   =>   |- ((A vH B) vH (C vH D)) = ((A vH C) vH (B vH D))
Theorem	chjjdir 9417	Hilbert lattice join distributes over itself.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   =>   |- ((A vH B) vH C) = ((A vH C) vH (B vH C))
Theorem	chdmm1t 9418	DeMorgan's law for meet in a Hilbert lattice.
|- ((A e. CH /\ B e. CH) -> (_|_` (A i^i B)) = ((_|_` A) vH (_|_` B)))
Theorem	chdmm2t 9419	DeMorgan's law for meet in a Hilbert lattice.
|- ((A e. CH /\ B e. CH) -> (_|_` ((_|_` A) i^i B)) = (A vH (_|_` B)))
Theorem	chdmm3t 9420	DeMorgan's law for meet in a Hilbert lattice.
|- ((A e. CH /\ B e. CH) -> (_|_` (A i^i (_|_` B))) = ((_|_` A) vH B))
Theorem	chdmm4t 9421	DeMorgan's law for meet in a Hilbert lattice.
|- ((A e. CH /\ B e. CH) -> (_|_` ((_|_` A) i^i (_|_` B))) = (A vH B))
Theorem	chdmj1t 9422	DeMorgan's law for join in a Hilbert lattice.
|- ((A e. CH /\ B e. CH) -> (_|_` (A vH B)) = ((_|_` A) i^i (_|_` B)))
Theorem	chdmj2t 9423	DeMorgan's law for join in a Hilbert lattice.
|- ((A e. CH /\ B e. CH) -> (_|_` ((_|_` A) vH B)) = (A i^i (_|_` B)))
Theorem	chdmj3t 9424	DeMorgan's law for join in a Hilbert lattice.
|- ((A e. CH /\ B e. CH) -> (_|_` (A vH (_|_` B))) = ((_|_` A) i^i B))
Theorem	chdmj4t 9425	DeMorgan's law for join in a Hilbert lattice.
|- ((A e. CH /\ B e. CH) -> (_|_` ((_|_` A) vH (_|_` B))) = (A i^i B))
Theorem	chjasst 9426	Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14.
|- ((A e. CH /\ B e. CH /\ C e. CH) -> ((A vH B) vH C) = (A vH (B vH C)))
Theorem	chj12t 9427	A rearrangement of Hilbert lattice join.
|- ((A e. CH /\ B e. CH /\ C e. CH) -> (A vH (B vH C)) = (B vH (A vH C)))
Theorem	chj4t 9428	Rearrangement of the join of 4 Hilbert lattice elements.
|- (((A e. CH /\ B e. CH) /\ (C e. CH /\ D e. CH)) -> ((A vH B) vH (C vH D)) = ((A vH C) vH (B vH D)))
Theorem	ledi 9429	An ortholattice is distributive in one ordering direction.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   =>   |- ((A i^i B) vH (A i^i C)) (_ (A i^i (B vH C))
Theorem	ledir 9430	An ortholattice is distributive in one ordering direction.
|- A e. CH   &   |- B e. CH   &   |- C e. CH   =>   |- ((A i^i C) vH (B i^i C)) (_ ((A vH B) i^i C)
Theorem	lejdi 9431	An ortholattice is distributive in one ordering direction (join version).
|- A e. CH   &   |- B e. CH   &   |- C e. CH   =>   |- (A vH (B i^i C)) (_ ((A vH B) i^i (A vH C))
Theorem	lejdir 9432	An ortholattice is distributive in one ordering direction (join version).
|- A e. CH   &   |- B e. CH   &   |- C e. CH   =>   |- ((A i^i B) vH C) (_ ((A vH C) i^i (B vH C))
Theorem	ledit 9433	An ortholattice is distributive in one ordering direction.
|- ((A e. CH /\ B e. CH /\ C e. CH) -> ((A i^i B) vH (A i^i C)) (_ (A i^i (B vH C)))
Span (cont.) and one-dimensional subspaces
Theorem	spansn0 9434	The span of the singleton of the zero vector is the zero subspace.
|- (span` {0h}) = 0H
Theorem	span0 9435	The span of the empty set is the zero subspace. Remark 11.6.e of [Schechter] p. 276.
|- (span` (/)) = 0H
Theorem	elspan 9436	Membership in the span of a subset of Hilbert space.
|- B e. V   =>   |- (A (_ H~ -> (B e. (span` A) <-> A.x e. SH (A (_ x -> B e. x)))
Theorem	spanun 9437	The span of a union is the subspace sum of spans.
|- A (_ H~   &   |- B (_ H~   =>   |- (span` (A u. B)) = ((span` A) +H (span` B))
Theorem	spanunt 9438	The span of a union is the subspace sum of spans.
|- ((A (_ H~ /\ B (_ H~) -> (span` (A u. B)) = ((span` A) +H (span` B)))
Theorem	sshhococ 9439	The join of two Hilbert space subsets (not necessarily closed subspaces) equals the join of their closures (double orthocomplements).
|- A (_ H~   &   |- B (_ H~   =>   |- (A vH B) = ((_|_` (_|_` A)) vH (_|_` (_|_` B)))
Theorem	hne0 9440	Hilbert space has a nonzero vector iff it is not trivial.
|- (H~ =/= 0H <-> E.x e. H~ x =/= 0h)
Theorem	chsup0 9441	The supremum of the empty set.
|- ( \/H ` (/)) = 0H
Theorem	h1deot 9442	Membership in orthocomplement of 1-dimensional subspace.
|- B e. H~   =>   |- (A e. (_|_` {B}) <-> (A e. H~ /\ (A .ih B) = 0))
Theorem	h1det 9443	Membership in 1-dimensional subspace.
|- B e. H~   =>   |- (A e. (_|_` (_|_` {B})) <-> (A e. H~ /\ A.x e. H~ ((B .ih x) = 0 -> (A .ih x) = 0)))
Theorem	h1did 9444	A generating vector belongs to the 1-dimensional subspace it generates.
|- (A e. H~ -> A e. (_|_` (_|_` {A})))
Theorem	h1dn0 9445	A non-zero vector generates a (non-zero) 1-dimensional subspace.
|- ((A e. H~ /\ A =/= 0h) -> (_|_` (_|_` {A})) =/= 0H)
Theorem	h1de2 9446	Membership in 1-dimensional subspace. All members are collinear with the generating vector.
|- A e. H~   &   |- B e. H~   =>   |- (A e. (_|_` (_|_` {B})) -> ((B .ih B) .h A) = ((A .ih B) .h B))
Theorem	h1de2b 9447	Membership in 1-dimensional subspace. All members are collinear with the generating vector.
|- A e. H~   &   |- B e. H~   =>   |- (B =/= 0h -> (A e. (_|_` (_|_` {B})) <-> A = (((A .ih B) / (B .ih B)) .h B)))
Theorem	h1de2ctlem 9448	Lemma for h1de2ct 9449.
Theorem	h1de2ct 9449	Membership in 1-dimensional subspace. All members are collinear with the generating vector.
|- B e. H~   =>   |- (A e. (_|_` (_|_` {B})) <-> E.x e. CC A = (x .h B))
Theorem	spansn 9450	The span of a singleton in Hilbert space equals its closure.
|- A e. H~   =>   |- (span` {A}) = (_|_` (_|_` {A}))
Theorem	elspansn 9451	Membership in the span of a singleton.
|- A e. H~   =>   |- (B e. (span` {A}) <-> E.x e. CC B = (x .h A))
Theorem	spansnt 9452	The span of a singleton in Hilbert space equals its closure.
|- (A e.