mmtheorems73 - Metamath Proof Explorer

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 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17411

Statement List for Metamath Proof Explorer - 7201-7300 - Page 73 of 175

Type	Label	Description
Statement
Theorem	6p3e9 7201	6 + 3 = 9.
|- (6 + 3) = 9
Theorem	6p4e10 7202	6 + 4 = 10.
|- (6 + 4) = 10
Theorem	7p2e9 7203	7 + 2 = 9.
|- (7 + 2) = 9
Theorem	7p3e10 7204	7 + 3 = 10.
|- (7 + 3) = 10
Theorem	8p2e10 7205	8 + 2 = 10.
|- (8 + 2) = 10
Theorem	2t2e4 7206	2 times 2 equals 4.
|- (2 x. 2) = 4
Theorem	3t2e6 7207	3 times 2 equals 6.
|- (3 x. 2) = 6
Theorem	3t3e9 7208	3 times 3 equals 9.
|- (3 x. 3) = 9
Theorem	4t2e8 7209	4 times 2 equals 8.
|- (4 x. 2) = 8
Theorem	5t2e10 7210	5 times 2 equals 10.
|- (5 x. 2) = 10
Theorem	4d2e2 7211	One half of four is two.
|- (4 / 2) = 2
Theorem	1lt2 7212	1 is less than 2.
|- 1 < 2
Theorem	2lt3 7213	2 is less than 3.
|- 2 < 3
Theorem	1lt3 7214	1 is less than 3.
|- 1 < 3
Theorem	halfgt0 7215	One-half is greater than zero.
|- 0 < (1 / 2)
Theorem	halflt1 7216	One-half is less than one.
|- (1 / 2) < 1
Theorem	8th4div3 7217	An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
|- ((1 / 8) x. (4 / 3)) = (1 / 6)
Theorem	halfpm6th 7218	One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)
|- (((1 / 2) - (1 / 6)) = (1 / 3) /\ ((1 / 2) + (1 / 6)) = (2 / 3))
Theorem	halfcl 7219	Closure of half of a number (frequently used special case).
|- (A e. CC -> (A / 2) e. CC)
Theorem	rehalfcl 7220	Real closure of half.
|- (A e. RR -> (A / 2) e. RR)
Theorem	half0 7221	Half of a number is zero iff the number is zero.
|- (A e. CC -> ((A / 2) = 0 <-> A = 0))
Theorem	halfpos 7222	A positive number is greater than its half.
|- (A e. RR -> (0 < A <-> (A / 2) < A))
Theorem	halfpos2 7223	A number is positive iff its half is positive.
|- (A e. RR -> (0 < A <-> 0 < (A / 2)))
Theorem	halfnneg2 7224	A number is nonnegative iff its half is nonnegative.
|- (A e. RR -> (0 <_ A <-> 0 <_ (A / 2)))
Theorem	2halves 7225	Two halves make a whole.
|- (A e. CC -> ((A / 2) + (A / 2)) = A)
Theorem	halfaddsubcl 7226	Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ((A e. CC /\ B e. CC) -> (((A + B) / 2) e. CC /\ ((A - B) / 2) e. CC))
Theorem	halfaddsub 7227	Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ((A e. CC /\ B e. CC) -> ((((A + B) / 2) + ((A - B) / 2)) = A /\ (((A + B) / 2) - ((A - B) / 2)) = B))
Theorem	lt2halves 7228	A sum is less than the whole if each term is less than half.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A < (C / 2) /\ B < (C / 2)) -> (A + B) < C))
Theorem	addltmul 7229	Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)
|- (((A e. RR /\ B e. RR) /\ (2 < A /\ 2 < B)) -> (A + B) < (A x. B))
Theorem	nominpos 7230	There is no smallest positive real number.
|- -. E.x e. RR (0 < x /\ -. E.y e. RR (0 < y /\ y < x))
Theorem	avgle 7231	The average of two numbers is less than or equal to at least one of them.
|- ((A e. RR /\ B e. RR) -> (((A + B) / 2) <_ A \/ ((A + B) / 2) <_ B))
Positive reals (as a subset of complex numbers)
Definition	df-rp 7232	Define the set of positive reals. Definition of positive numbers in [Apostol] p. 20.
|- RR+ = {x e. RR | 0 < x}
Theorem	elrp 7233	Membership in the set of positive reals.
|- (A e. RR+ <-> (A e. RR /\ 0 < A))
Theorem	elrpii 7234	Membership in the set of positive reals.
|- A e. RR   &   |- 0 < A   =>   |- A e. RR+
Theorem	1rp 7235	1 is a positive real. (Contributed by Jeffrey Hankins, 23-Nov-2008.)
|- 1 e. RR+
Theorem	rpre 7236	A positive real is a real.
|- (A e. RR+ -> A e. RR)
Theorem	rpcn 7237	A positive real is a complex number.
|- (A e. RR+ -> A e. CC)
Theorem	nnrp 7238	A natural number is a positive real.
|- (A e. NN -> A e. RR+)
Theorem	rpssre 7239	The positive reals are a subset of the reals.
|- RR+ C_ RR
Theorem	rpgt0 7240	A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.)
|- (A e. RR+ -> 0 < A)
Theorem	rpge0 7241	A positive real is greater than or equal to zero.
|- (A e. RR+ -> 0 <_ A)
Theorem	rpregt0 7242	A positive real is a positive real number.
|- (A e. RR+ -> (A e. RR /\ 0 < A))
Theorem	rpne0 7243	A positive real is nonzero.
|- (A e. RR+ -> A =/= 0)
Theorem	rprene0 7244	A positive real is a nonzero real number.
|- (A e. RR+ -> (A e. RR /\ A =/= 0))
Theorem	rpcnne0 7245	A positive real is a nonzero complex number.
|- (A e. RR+ -> (A e. CC /\ A =/= 0))
Theorem	ralrp 7246	Quantification over positive reals.
|- (A.x e. RR+ ph <-> A.x e. RR (0 < x -> ph))
Theorem	rpaddcl 7247	Closure law for addition of positive reals. Part of Axiom 7 of [Apostol] p. 20.
|- ((A e. RR+ /\ B e. RR+) -> (A + B) e. RR+)
Theorem	rpmulcl 7248	Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20.
|- ((A e. RR+ /\ B e. RR+) -> (A x. B) e. RR+)
Theorem	rpdivcl 7249	Closure law for division of positive reals. (Contributed by FL, 27-Dec-2007.)
|- ((A e. RR+ /\ B e. RR+) -> (A / B) e. RR+)
Theorem	rpreccl 7250	Closure law for reciprocation of positive reals. (Contributed by Jeffrey Hankins, 23-Nov-2008.)
|- (A e. RR+ -> (1 / A) e. RR+)
Theorem	rerpdivcl 7251	Closure law for division of a real by a positive real.
|- ((A e. RR /\ B e. RR+) -> (A / B) e. RR)
Theorem	rpneg 7252	Either a nonzero real or its negation is a positive real, but not both. Axiom 8 of [Apostol] p. 20.
|- ((A e. RR /\ A =/= 0) -> (A e. RR+ <-> -. -uA e. RR+))
Theorem	0nrp 7253	Zero is not a positive real. Axiom 9 of [Apostol] p. 20.
|- -. 0 e. RR+
Completeness Axiom and Suprema
Theorem	lbreu 7254	If a set of reals contains a lower bound, it contains a unique lower bound.
|- ((S C_ RR /\ E.x e. S A.y e. S x <_ y) -> E!x e. S A.y e. S x <_ y)
Theorem	lbcl 7255	If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set.
|- ((S C_ RR /\ E.x e. S A.y e. S x <_ y) -> U.{x e. S | A.y e. S x <_ y} e. S)
Theorem	lble 7256	If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set.
|- ((S C_ RR /\ E.x e. S A.y e. S x <_ y /\ A e. S) -> U.{x e. S | A.y e. S x <_ y} <_ A)
Theorem	lbinfm 7257	If a set of reals contains a lower bound, the lower bound is its infimum.
|- ((S C_ RR /\ E.x e. S A.y e. S x <_ y) -> sup(S, RR, `' < ) = U.{x e. S | A.y e. S x <_ y})
Theorem	lbinfmcl 7258	If a set of reals contains a lower bound, it contains its infimum.
|- ((S C_ RR /\ E.x e. S A.y e. S x <_ y) -> sup(S, RR, `' < ) e. S)
Theorem	lbinfmle 7259	If a set of reals contains a lower bound, its infmimum is less than or equal to all members of the set.
|- ((S C_ RR /\ E.x e. S A.y e. S x <_ y /\ A e. S) -> sup(S, RR, `' < ) <_ A)
Theorem	sup2 7260	A non-empty, bounded-above set of reals has a supremum. Stronger version of completeness axiom (it has a slightly weaker antecedent).
|- ((A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A (y < x \/ y = x)) -> E.x e. RR (A.y e. A -. x < y /\ A.y e. RR (y < x -> E.z e. A y < z)))
Theorem	sup3 7261	A version of the completeness axiom for reals.
|- ((A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x) -> E.x e. RR (A.y e. A -. x < y /\ A.y e. RR (y < x -> E.z e. A y < z)))
Theorem	infm3lem 7262	Lemma for infm3 7263.
Theorem	infm3 7263	The completeness axiom for reals in terms of infimum: a non-empty, bounded-below set of reals has a infimum. (This theorem is the dual of sup3 7261.)
|- ((A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A x <_ y) -> E.x e. RR (A.y e. A -. y < x /\ A.y e. RR (x < y -> E.z e. A z < y)))
Theorem	suprcl 7264	Closure of supremum of a non-empty bounded set of reals.
|- ((A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x) -> sup(A, RR, < ) e. RR)
Theorem	suprub 7265	A member of a non-empty bounded set of reals is less than or equal to the set's upper bound.
|- (((A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x) /\ B e. A) -> B <_ sup(A, RR, < ))
Theorem	suprlub 7266	The supremum of a non-empty bounded set of reals is the least upper bound.
|- (((A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x) /\ (B e. RR /\ B < sup(A, RR, < ))) -> E.z e. A B < z)
Theorem	suprnub 7267	An upper bound is not less than the supremum of a non-empty bounded set of reals.
|- (((A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x) /\ (B e. RR /\ A.z e. A -. B < z)) -> -. B < sup(A, RR, < ))
Theorem	suprleub 7268	The supremum of a non-empty bounded set of reals is less than or equal to an upper bound.
|- (((A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x) /\ (B e. RR /\ A.z e. A z <_ B)) -> sup(A, RR, < ) <_ B)
Theorem	sup3ii 7269	A version of the completeness axiom for reals.
|- (A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x)   =>   |- E.x e. RR (A.y e. A -. x < y /\ A.y e. RR (y < x -> E.z e. A y < z))
Theorem	suprclii 7270	Closure of supremum of a non-empty bounded set of reals.
|- (A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x)   =>   |- sup(A, RR, < ) e. RR
Theorem	suprubii 7271	A member of a non-empty bounded set of reals is less than or equal to the set's upper bound.
|- (A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x)   =>   |- (B e. A -> B <_ sup(A, RR, < ))
Theorem	suprlubii 7272	The supremum of a non-empty bounded set of reals is the least upper bound.
|- (A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x)   =>   |- ((B e. RR /\ B < sup(A, RR, < )) -> E.z e. A B < z)
Theorem	suprnubii 7273	An upper bound is not less than the supremum of a non-empty bounded set of reals.
|- (A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x)   =>   |- ((B e. RR /\ A.z e. A -. B < z) -> -. B < sup(A, RR, < ))
Theorem	suprleubii 7274	The supremum of a non-empty bounded set of reals is less than or equal to an upper bound.
|- (A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x)   =>   |- ((B e. RR /\ A.z e. A z <_ B) -> sup(A, RR, < ) <_ B)
Theorem	reuunineg 7275	The negative of the unique real such that ph.
|- (x = -uy -> (ph <-> ps))   =>   |- (E!x e. RR ph -> U.{x e. RR | ph} = -uU.{y e. RR | ps})
Theorem	dfinfmr 7276	The infimum (expressed as supremum with converse 'less-than') of a set of reals A.
|- (A C_ RR -> sup(A, RR, `' < ) = U.{x e. RR | (A.y e. A x <_ y /\ A.y e. RR (x < y -> E.z e. A z < y))})
Theorem	infmsup 7277	The infimum (expressed as supremum with converse 'less-than') of a set of reals A is the negative of the supremum of the negatives of its elements. The antecedent ensures that A is nonempty and has a lower bound.
|- ((A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A x <_ y) -> sup(A, RR, `' < ) = -usup({z e. RR | -uz e. A}, RR, < ))
Theorem	infmrcl 7278	Closure of infimum of a non-empty bounded set of reals.
|- ((A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A x <_ y) -> sup(A, RR, `' < ) e. RR)
Theorem	nnunb 7279	The set of natural numbers is unbounded above. Theorem I.28 of [Apostol] p. 26.
|- -. E.x e. RR A.y e. NN (y < x \/ y = x)
Theorem	arch 7280	Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26.
|- (A e. RR -> E.n e. NN A < n)
Theorem	nnrecl 7281	There exists a natural number whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28.
|- ((A e. RR /\ 0 < A) -> E.n e. NN (1 / n) < A)
Theorem	bndndx 7282	A bounded real sequence A(k) is less than or equal to at least one of its indices.
|- (E.x e. RR A.k e. NN (A e. RR /\ A <_ x) -> E.k e. NN A <_ k)
Supremum on the extended reals
Theorem	xrsupexmnf 7283	Adding minus infinity to a set does not affect the existence of its supremum.
|- (E.x e. RR* (A.y e. A -. x < y /\ A.y e. RR* (y < x -> E.z e. A y < z)) -> E.x e. RR* (A.y e. (A u. { -oo}) -. x < y /\ A.y e. RR* (y < x -> E.z e. (A u. { -oo})y < z)))
Theorem	xrinfmexpnf 7284	Adding plus infinity to a set does not affect the existence of its infimum.
|- (E.x e. RR* (A.y e. A -. y < x /\ A.y e. RR* (x < y -> E.z e. A z < y)) -> E.x e. RR* (A.y e. (A u. { +oo}) -. y < x /\ A.y e. RR* (x < y -> E.z e. (A u. { +oo})z < y)))
Theorem	xrsupsslem 7285	Lemma for xrsupss 7287.
Theorem	xrinfmsslem 7286	Lemma for xrinfmss 7288.
Theorem	xrsupss 7287	Any subset of extended reals has a supremum.
|- (A C_ RR* -> E.x e. RR* (A.y e. A -. x < y /\ A.y e. RR* (y < x -> E.z e. A y < z)))
Theorem	xrinfmss 7288	Any subset of extended reals has an infimum.
|- (A C_ RR* -> E.x e. RR* (A.y e. A -. y < x /\ A.y e. RR* (x < y -> E.z e. A z < y)))
Theorem	xrub 7289	By quantifying only over reals, we can specify any extended real upper bound for any set of extended reals.
|- ((A C_ RR* /\ B e. RR*) -> (A.x e. RR (x < B -> E.y e. A x < y) <-> A.x e. RR* (x < B -> E.y e. A x < y)))
Theorem	supxr 7290	The supremum of a set of extended reals.
|- (((A C_ RR* /\ B e. RR*) /\ (A.x e. A -. B < x /\ A.x e. RR (x < B -> E.y e. A x < y))) -> sup(A, RR*, < ) = B)
Theorem	supxr2 7291	The supremum of a set of extended reals.
|- (((A C_ RR* /\ B e. RR*) /\ (A.x e. A x <_ B /\ A.x e. RR (x < B -> E.y e. A x < y))) -> sup(A, RR*, < ) = B)
Theorem	supxrre 7292	The real and extended real suprema match when the real supremum exists.
|- ((A C_ RR /\ A =/= (/) /\ E.x e. RR A.y e. A y <_ x) -> sup(A, RR*, < ) = sup(A, RR, < ))
Theorem	supxrcl 7293	The supremum of an arbitrary set of extended reals is an extended real.
|- (A C_ RR* -> sup(A, RR*, < ) e. RR*)
Theorem	supxrun 7294	The supremum of the union of two sets of extended reals equals the largest of their suprema.
|- ((A C_ RR* /\ B C_ RR* /\ sup(A, RR*, < ) <_ sup(B, RR*, < )) -> sup((A u. B), RR*, < ) = sup(B, RR*, < ))
Theorem	infmxrcl 7295	The infimum of an arbitrary set of extended reals is an extended real.
|- (A C_ RR* -> sup(A, RR*, `' < ) e. RR*)
Theorem	supxrmnf 7296	Adding minus infinity to a set does not affect its supremum.
|- (A C_ RR* -> sup((A u. { -oo}), RR*, < ) = sup(A, RR*, < ))
Theorem	supxrpnf 7297	The supremum of a set of extended reals containing plus infnity is plus infinity.
|- ((A C_ RR* /\ +oo e. A) -> sup(A, RR*, < ) = +oo)
Theorem	supxrunb1 7298	The supremum of an unbounded-above set of extended reals is plus infinity.
|- (A C_ RR* -> (A.x e. RR E.y e. A x <_ y <-> sup(A, RR*, < ) = +oo))
Theorem	supxrunb2 7299	The supremum of an unbounded-above set of extended reals is plus infinity.
|- (A C_ RR* -> (A.x e. RR E.y e. A x < y <-> sup(A, RR*, < ) = +oo))
Theorem	supxrbnd 7300	The supremum of a bounded-above nonempty set of reals is real.
|- ((A C_ RR /\ A =/= (/) /\ sup(A, RR*, < ) < +oo) -> sup(A, RR*, < ) e. RR)