mmtheorems93 - 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 10601-10700 108 10701-10766

Color key:

Metamath Proof Explorer
(1-8795)

Hilbert Space Explorer
(8796-10377)

User Sandboxes
(10378-10766)

**Statement List for Metamath Proof Explorer -** 9201-9300 - Page 93 of 108
Type	Label	Description
Statement

Theorem	projlem8 9201	Part of Lemma 3.6 of [Beran] p. 100. The set is a non-empty set of reals with an upper bound. Part of Lemma 3.6 of [Beran] p. 100. Used by projlem9 9202 projlem12 9205 projlem13 9206 projlem15 9208. Note we use 'supremum'; its negative is the infimum.
$/\$ $/\$

Theorem	projlem9 9202	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Real closure of the infimum of norms. Used by projlem11 9204 projlem12 9205 projlem13 9206 projlem15 9208.


Theorem	projlem10 9203	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). A member of the infimum set. Used by projlem12 9205.


Theorem	projlem11 9204	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). is the infimum of the set of norms. Show it is real. Used by projlem12 9205 projlem13 9206 projlem14 9207 projlem15 9208 projlem18 9211 projlem19 9212 projlem26 9219 projlem28 9221 projlem31 9224.


Theorem	projlem12 9205	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). The infimum is less than any norm in the set of norms. Used by projlem14 9207 projlem18 9211 projlem31 9224.


Theorem	projlem13 9206	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). The infimum of the set of norms is nonnegative. Used by projlem18 9211 projlem19 9212 projlem28 9221.


Theorem	projlem14 9207	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Used by projlem16 9209.


Theorem	projlem15 9208	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Used by projlem16 9209.


Theorem	projlem16 9209	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). Existence of a vector sequence member, used to show (via Axiom of Choice) the vector sequence existence. Used by projlem17 9210.
$/\$

Theorem	projlem17 9210	Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). This uses the Axiom of Choice to show the existence of a vector sequence satisfying the assumption of Lemma 3.6 of [Beran] p. 100: "Let {y_n } be a sequence of W such that i₀ - 1/n < \|\|x₀ - y_n \|\| < i₀ + 1/n." Here, corresponds to "W"; to "{y_n }"; to "n"; to "i₀ "; and to "\|\|x₀ - y_n \|\|". Used by projlem 9225.
$/\$ $/\$

Theorem	projlem18 9211	Part of Lemma 3.6 of [Beran] p. 101, top. Used by projlem19 9212.


Theorem	projlem19 9212	Part of Lemma 3.6 of [Beran] p. 101. Used by projlem20 9213.
$/\$ $/\$

Theorem	projlem20 9213	Part of Lemma 3.6 of [Beran] p. 101. Used by projlem27 9220.
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	projlem21 9214	Part of Lemma 3.6 of [Beran] p. 100. The hypothesis lets us work with our postulated vector sequence (whose existence was shown by projlem17 9210). Here we just show the sequence value belongs to the closed subspace . Used by projlem27 9220 projlem28 9221.
$/\$ $/\$

Theorem	projlem22 9215	Part of Lemma 3.6 of [Beran] p. 100. Here we show a member of the vector sequence is bounded. Used by projlem27 9220.
$/\$ $/\$

Theorem	projlem23 9216	Part of Lemma 3.6 of [Beran] p. 101. The hypothesis lets us work with the sequence which corresponds to Beran's "{\|\|yn-x0\|\|}". Used by projlem25 9218 projlem26 9219.
$/\$

Theorem	projlem24 9217	Part of Lemma 3.6 of [Beran] p. 101. Here we show our vector sequence implies the real numbers sequence corresponding to Beran's "{\|\|yn-x0\|\|}". Used by projlem25 9218 projlem26 9219.
$/\$

Theorem	projlem25 9218	Part of Lemma 3.6 of [Beran] p. 101. "The sequence {\|\|yn-x0\|\|} of real numbers converges to the number \|\|y0-x0\|\|" (here, "y0" is and "x0" is ). Used by projlem31 9224.
$/\$

Theorem	projlem26 9219	Part of Lemma 3.6 of [Beran] p. 101. The real sequence (Beran's "{\|\|yn-x0\|\|}") converges to the infimum of norms. Used by projlem31 9224.
$/\$ $/\$ $/\$