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**Statement List for Metamath Proof Explorer -** 9301-9400 - Page 94 of 107
Type	Label	Description
Statement

Theorem	shscom 9301	Commutative law for subspace sum.


Theorem	shsva 9302	Vector sum belongs to subspace sum.
$/\$

Theorem	shsel1 9303	A subspace sum contains a member of one of its subspaces.


Theorem	shsel2 9304	A subspace sum contains a member of one of its subspaces.


Theorem	shsvs 9305	Vector subtraction belongs to subspace sum.
$/\$

Theorem	shunss 9306	Union is smaller than subspace sum.


Theorem	shslej 9307	Subspace sum is smaller than Hilbert lattice join. Remark in [Kalmbach] p. 65.


Theorem	shunssj 9308	Union is smaller than Hilbert lattice join.


Theorem	shjcom 9309	Commutative law for join in .


Theorem	shsub1 9310	Subspace sum is an upper bound of its arguments.


Theorem	shsub2 9311	Subspace sum is an upper bound of its arguments.


Theorem	shub1 9312	Hilbert lattice join is an upper bound of two subspaces.


Theorem	shjcl 9313	Closure of join.


Theorem	shjshcl 9314	closure of join.


Theorem	shlub 9315	Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces.
$/\$

Theorem	shless 9316	Subset implies subset of subspace sum.


Theorem	shlej1 9317	Add disjunct to both sides of Hilbert subspace ordering.


Theorem	shlej2 9318	Add disjunct to both sides of Hilbert subspace ordering.


Theorem	shslejt 9319	Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65.
$/\$

Theorem	shinclt 9320	Closure of intersection of two subspaces.
$/\$

Theorem	shub1t 9321	Hilbert lattice join is an upper bound of two subspaces.
$/\$

Theorem	shub2t 9322	A subspace is a subset of its Hilbert lattice join with another.
$/\$

Theorem	shlubt 9323	Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces.
$/\$ $/\$ $/\$

Theorem	shlej1t 9324	Add disjunct to both sides of Hilbert subspace ordering.
$/\$ $/\$ $/\$

Theorem	shlej2t 9325	Add disjunct to both sides of Hilbert subspace ordering.
$/\$ $/\$ $/\$

Theorem	shsidm 9326	Idempotent law for Hilbert subspace sum.


Theorem	shslub 9327	Least upper bound law for Hilbert subspace sum.
$/\$

Theorem	shlesb1 9328	Hilbert lattice ordering in terms of subspace sum.


Theorem	shsumval2 9329	An alternate way to express subspace sum.


Theorem	shsumval3 9330	An alternate way to express subspace sum.


Theorem	shmods 9331	The modular law holds for subspace sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70.


Theorem	shmod 9332	The modular law is implied by the closure of subspace sum. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70.
$/\$

Hilbert lattice operations

Theorem	sh0let 9333	The zero subspace is the smallest subspace.


Theorem	ch0let 9334	The zero subspace is the smallest member of .


Theorem	shle0t 9335	No subspace is smaller than the zero subspace.


Theorem	chle0t 9336	No Hilbert lattice element is smaller than zero.


Theorem	chnlen0 9337	A Hilbert lattice element that is not a subset of another is nonzero.


Theorem	ch0psst 9338	The zero subspace is a proper subset of non-zero Hilbert lattice elements.


Theorem	orthin 9339	The intersection of orthogonal subspaces is the zero subspace.
$/\$

Theorem	shne0 9340	A non-zero subspace has a non-zero vector.


Theorem	shs0 9341	Hilbert subspace sum with the zero subspace.


Theorem	shs00 9342	Two subspaces are zero iff their join is zero.
$/\$

Theorem	ch0le 9343	The closed subspace zero is the smallest member of .


Theorem	chle0 9344	No Hilbert closed subspace is smaller than zero.


Theorem	chne0 9345	A non-zero closed subspace has a non-zero vector.


Theorem	chocin 9346	Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65.


Theorem	chj0 9347	Join with lattice zero in .


Theorem	chm1 9348	Meet with lattice one in .


Theorem	chjcl 9349	Closure of join.


Theorem	chslej 9350	Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65.


Theorem	chsel 9351	Membership in subspace sum.


Theorem	chincl 9352	Closure of Hilbert lattice intersection.


Theorem	chsscon3 9353	Hilbert lattice contraposition law.


Theorem	chsscon1 9354	Hilbert lattice contraposition law.


Theorem	chsscon2 9355	Hilbert lattice contraposition law.


Theorem	chcon2 9356	Hilbert lattice contraposition law.


Theorem	chcon1 9357	Hilbert lattice contraposition law.


Theorem	chcon3 9358	Hilbert lattice contraposition law.


Theorem	chunssj 9359	Union is smaller than join.


Theorem	chjcom 9360	Commutative law for join in .


Theorem	chub1 9361	join is an upper bound of two elements.


Theorem	chub2 9362	join is an upper bound of two elements.