mmtheorems94 - Metamath Proof Explorer

Home	Metamath Proof Explorer	< Previous Next >
		Browser slow? Try the Unicode version.

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10691

Color key:

Metamath Proof Explorer (1-8743)

Hilbert Space Explorer (8744-10691)

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

Theorem	shjshcl 9301	closure of join.


Theorem	shlub 9302	Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces.
$/\$

Theorem	shless 9303	Subset implies subset of subspace sum.


Theorem	shlej1 9304	Add disjunct to both sides of Hilbert subspace ordering.


Theorem	shlej2 9305	Add disjunct to both sides of Hilbert subspace ordering.


Theorem	shslejt 9306	Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65.
$/\$

Theorem	shinclt 9307	Closure of intersection of two subspaces.
$/\$

Theorem	shub1t 9308	Hilbert lattice join is an upper bound of two subspaces.
$/\$

Theorem	shub2t 9309	A subspace is a subset of its Hilbert lattice join with another.
$/\$

Theorem	shlubt 9310	Hilbert lattice join is the least upper bound (among Hilbert lattice elements) of two subspaces.
$/\$ $/\$ $/\$

Theorem	shlej1t 9311	Add disjunct to both sides of Hilbert subspace ordering.
$/\$ $/\$ $/\$

Theorem	shlej2t 9312	Add disjunct to both sides of Hilbert subspace ordering.
$/\$ $/\$ $/\$

Theorem	shsidm 9313	Idempotent law for Hilbert subspace sum.


Theorem	shslub 9314	Least upper bound law for Hilbert subspace sum.
$/\$

Theorem	shlesb1 9315	Hilbert lattice ordering in terms of subspace sum.


Theorem	shsumval2 9316	An alternate way to express subspace sum.


Theorem	shsumval3 9317	An alternate way to express subspace sum.


Theorem	shmods 9318	The modular law holds for subspace sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70.


Theorem	shmod 9319	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 9320	The zero subspace is the smallest subspace.


Theorem	ch0let 9321	The zero subspace is the smallest member of .


Theorem	shle0t 9322	No subspace is smaller than the zero subspace.


Theorem	chle0t 9323	No Hilbert lattice element is smaller than zero.


Theorem	chnlen0 9324	A Hilbert lattice element that is not a subset of another is nonzero.


Theorem	ch0psst 9325	The zero subspace is a proper subset of non-zero Hilbert lattice elements.


Theorem	orthin 9326	The intersection of orthogonal subspaces is the zero subspace.
$/\$

Theorem	shne0 9327	A non-zero subspace has a non-zero vector.


Theorem	shs0 9328	Hilbert subspace sum with the zero subspace.


Theorem	shs00 9329	Two subspaces are zero iff their join is zero.
$/\$

Theorem	ch0le 9330	The closed subspace zero is the smallest member of .


Theorem	chle0 9331	No Hilbert closed subspace is smaller than zero.


Theorem	chne0 9332	A non-zero closed subspace has a non-zero vector.


Theorem	chocin 9333	Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65.


Theorem	chj0 9334	Join with lattice zero in .


Theorem	chm1 9335	Meet with lattice one in .


Theorem	chjcl 9336	Closure of join.


Theorem	chslej 9337	Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65.


Theorem	chsel 9338	Membership in subspace sum.


Theorem	chincl 9339	Closure of Hilbert lattice intersection.


Theorem	chsscon3 9340	Hilbert lattice contraposition law.


Theorem	chsscon1 9341	Hilbert lattice contraposition law.


Theorem	chsscon2 9342	Hilbert lattice contraposition law.


Theorem	chcon2 9343	Hilbert lattice contraposition law.


Theorem	chcon1 9344	Hilbert lattice contraposition law.


Theorem	chcon3 9345	Hilbert lattice contraposition law.


Theorem	chunssj 9346	Union is smaller than join.


Theorem	chjcom 9347	Commutative law for join in .


Theorem	chub1 9348	join is an upper bound of two elements.


Theorem	chub2 9349	join is an upper bound of two elements.


Theorem	chlub 9350	Hilbert lattice join is the least upper bound of two elements.
$/\$

Theorem	chlubi 9351	Hilbert lattice join is the least upper bound of two elements (one direction of chlub 9350).
$/\$

Theorem	chlej1 9352	Add join to both sides of a Hilbert lattice ordering.


Theorem	chlej2 9353	Add join to both sides of a Hilbert lattice ordering.


Theorem	chlej12 9354	Add join to both sides of a Hilbert lattice ordering.
$/\$

Theorem	chlejb1 9355	Hilbert lattice ordering in terms of join.


Theorem	chdmm1 9356	DeMorgan's law for meet in a Hilbert lattice.


Theorem	chdmm2 9357	DeMorgan's law for meet in a Hilbert lattice.


Theorem	chdmm3 9358	DeMorgan's law for meet in a Hilbert lattice.


Theorem	chdmm4 9359	DeMorgan's law for meet in a Hilbert lattice.