mmtheorems70 - 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 10401-10487

Color key:

Metamath Proof Explorer (1-8967)

Hilbert Space Explorer (8968-10487)

**Statement List for Metamath Proof Explorer -** 6901-7000 - Page 70 of 105
Type	Label	Description
Statement

Theorem	fsump1slem 6901	Lemma for fsump1s 6902.

Theorem	fsump1s 6902	The addition of the next term in a finite sum of is the previous term plus .
$/\$

Theorem	fsumcllem 6903	- Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.)
$/\$ $/\$

Theorem	fsumclt 6904	Closure of a finite sum of complex numbers .
$/\$

Theorem	fsum0clt 6905	Closure of a finite sum of complex numbers , starting at index zero.
$/\$

Theorem	fsumreclt 6906	Closure of a finite sum of reals.
$/\$

Theorem	fsum1ps 6907	Separate out the first term in a finite sum.
$/\$ $/\$

Theorem	fsum1p 6908	Separate out the first term in a finite sum.
$/\$ $/\$

Theorem	fsumsplit 6909	Split a finite sum into two parts. Warning: The HTML proof page is 0.6 megabyte in size.
$/\$ $/\$

Theorem	fsum0split 6910	Split a finite sum into two parts.
$/\$ $/\$

Theorem	fsumadd 6911	The sum of two finite sums.
$/\$ $/\$

Theorem	fsum2 6912	The sum of two terms.


Theorem	fsum3 6913	The sum of three terms.


Theorem	fsum4 6914	The sum of four terms.


Theorem	csbfsumlem 6915	Lemma for csbfsum 6916.

Theorem	csbfsum 6916	Distribute substitution for classes over a finite sum.
$/\$ $/\$

Theorem	fsumcom 6917	Interchange order of summation. Warning: The HTML proof page is 0.6MB in size.
$/\$ $/\$

Theorem	fsumrev 6918	Reversal of a finite sum. Warning: The HTML proof page is 0.6 MB in size.
$/\$ $/\$

Theorem	fsumrev2 6919	Reversal of a finite sum.
$/\$

Theorem	fsumshft 6920	Index shift of a finite sum.
$/\$ $/\$

Theorem	fsumshftm 6921	Negative index shift of a finite sum.
$/\$ $/\$

Theorem	fsummulc1 6922	A finite sum multiplied by a constant.
$/\$ $/\$

Theorem	fsummulc2 6923	A finite sum multiplied by a constant.
$/\$ $/\$

Theorem	fsumdivc 6924	A finite sum divided by a constant.
$/\$ $/\$ $/\$

Theorem	fsumdivcALT 6925	A finite sum divided by a constant. (An experiment: this version of fsumdivc 6924 adds 5 bytes and 233 bytes to the compressed and uncompressed proofs, but saves 540 bytes on the HTML page.)
$/\$ $/\$ $/\$

Theorem	fsum2mul 6926	Separate the nested sum of the product .
$/\$ $/\$ $/\$

Theorem	fsumconst 6927	The sum of constant terms ( is not free in ).
$/\$

Theorem	fsum0 6928	If all of the terms of a finite sum are zero, so is the sum.


Theorem	fsumcmp 6929	If all of the terms of finite sums compare, so do the sums.
$/\$ $/\$ $/\$

Theorem	fsumcmp0 6930	If all of the terms of a finite sum are nonnegative, so is the sum.
$/\$ $/\$

Theorem	fsumcmpndx2 6931	A shorter sum of nonnegative terms is smaller than a longer one.
$/\$ $/\$ $/\$ $/\$

Theorem	fsumabs 6932	Generalized triangle inequality: the absolute value of a finite sum is less than or equal to the sum of absolute values.
$/\$

Theorem	fsumabs2mul 6933	The sum of absolute values of the product is less than or equal to the product of the two sums of absolute values.