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Theorem List for Metamath Proof Explorer - 27201-27300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremstoweidlem53 27201* This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem54 27202* There exists a function as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. Here is used to represent in the paper, because here is used for the subalgebra of functions. is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem55 27203* This lemma proves the existence of a function p as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Here Z is used to represent t0 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem56 27204* This theorem proves Lemma 1 in [BrosowskiDeutsh] p. 90. Here is used to represent t0 in the paper, is used to represent in the paper, and is used to represent ε (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem57 27205* There exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. In this theorem, it is proven the non trivial case (the closed set D is nonempty). Here D is used to represent A in the paper, because the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem58 27206* This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 91. Here D is used to represent the set A of Lemma 2, because here the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem59 27207* This lemma proves that there exists a function as in the proof in [BrosowskiDeutsh] p. 91, after Lemma 2: xj is in the subalgebra, 0 <= xj <= 1, xj < ε / n on Aj (meaning A in the paper), xj > 1 - \epslon / n on Bj. Here is used to represent A in the paper (because A is used for the subalgebra of functions), is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem60 27208* This lemma proves that there exists a function g as in the proof in [BrosowskiDeutsh] p. 91 (this parte of the proof actually spans through pages 91-92): g is in the subalgebra, and for all in , there is a such that (j-4/3)*ε < f(t) <= (j-1/3)*ε and (j-4/3)*ε < g(t) < (j+1/3)*ε. Here is used to represent f in the paper, and is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem61 27209* This lemma proves that there exists a function as in the proof in [BrosowskiDeutsh] p. 92: is in the subalgebra, and for all in , abs( f(t) - g(t) ) < 2*ε. Here is used to represent f in the paper, and is used to represent ε. For this lemma there's the further assumption that the function to be approximated is nonnegative (this assumption is removed in a later theorem). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweidlem62 27210* This theorem proves the Stone Weierstrass theorem for the non trivial case in which T is nonempty. The proof follows [BrosowskiDeutsh] p. 89 (through page 92). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstoweid 27211* This theorem proves the Stone-Weierstrass theorem for real valued functions: let be a compact topology on , and be the set of real continuous functions on . Assume that is a subalgebra of (closed under addition and multiplication of functions) containing constant functions and discriminating points (if and are distinct points in , then there exists a function in such that h(r) is distinct from h(t) ). Then, for any continuous function and for any positive real , there exists a function in the subalgebra , such that approximates up to ( represents the usual ε value). As a classical example, given any a,b reals, the closed interval could be taken, along with the subalgebra of real polynomials on , and then use this theorem to easily prove that real polynomials are dense in the standard metric space of continuous functions on . The proof and lemmas are written following [BrosowskiDeutsh] p. 89 (through page 92). Some effort is put in avoiding the use of the axiom of choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Theoremstowei 27212* This theorem proves the Stone-Weierstrass theorem for real valued functions: let be a compact topology on , and be the set of real continuous functions on . Assume that is a subalgebra of (closed under addition and multiplication of functions) containing constant functions and discriminating points (if and are distinct points in , then there exists a function in such that h(r) is distinct from h(t) ). Then, for any continuous function and for any positive real , there exists a function in the subalgebra , such that approximates up to ( represents the usual ε value). As a classical example, given any a,b reals, the closed interval could be taken, along with the subalgebra of real polynomials on , and then use this theorem to easily prove that real polynomials are dense in the standard metric space of continuous functions on . The proof and lemmas are written following [BrosowskiDeutsh] p. 89 (through page 92). Some effort is put in avoiding the use of the axiom of choice. The deduction version of this theorem is stoweid 27211: often times it will be better to use stoweid 27211 in other proofs (but this version is probably easier to be read and understood). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

18.19.7  Wallis' product for π

Theoremwallispilem1 27213* is monotone: increasing the exponent, the integral decreases. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremwallispilem2 27214* A first set of properties for the sequence that will be used in the proof of the Wallis product formula (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremwallispilem3 27215* I maps to real values (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremwallispilem4 27216* maps to explicit expression for the ratio of two consecutive values of (Contributed by Glauco Siliprandi, 30-Jun-2017.)

Theoremwallispilem5 27217* The sequence converges to 1. (Contributed by Glauco Siliprandi, 30-Jun-2017.)

Theoremwallispi 27218* Wallis' formula for π : Wallis' product converges to π / 2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremwallispi2lem1 27219 An intermediate step between the first version of the Wallis' formula for π and the second version of Wallis' formula. This second version will then be used to prove Stirling's approximation formula for the factorial. (Contributed by Glauco Siliprandi, 30-Jun-2017.)

Theoremwallispi2lem2 27220 Two expressions are proven to be equal, and this is used to complete the proof of the second version of Wallis' formula for π . (Contributed by Glauco Siliprandi, 30-Jun-2017.)

Theoremwallispi2 27221 An alternative version of Wallis' formula for π ; this second formula uses factorials and it is later used to proof Stirling's approximation formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

18.19.8  Stirling's approximation formula for ` n ` factorial

Theoremstirlinglem1 27222 A simple limit of fractions is computed. (Contributed by Glauco Siliprandi, 30-Jun-2017.)

Theoremstirlinglem2 27223 maps to positive reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem3 27224 Long but simple algebraic transformations are applied to show that , the Wallis formula for π , can be expressed in terms of , the Stirling's approximation formula for the factorial, up to a constant factor. This will allow (in a later theorem) to determine the right constant factor to be put into the , in order to get the exact Stirling's formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem4 27225* Algebraic manipulation of n ) - ( B . It will be used in other theorems to show that is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem5 27226* If is between and , then a series (without alternating negative and positive terms) is given that converges to log (1+T)/(1-T) . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem6 27227* A series that converges to log (N+1)/N (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem7 27228* Algebraic manipulation of the formula for J(n) (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem8 27229 If converges to , then converges to C^2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem9 27230* is expressed as a limit of a series. This result will be used both to prove that is decreasing and to prove that is bounded (below). It will follow that converges in the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem10 27231* A bound for any B(N)-B(N + 1) that will allow to find a lower bound for the whole sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem11 27232* is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem12 27233* The sequence is bounded below. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem13 27234* is decreasing and has a lower bound, then it converges. Since is , in another theorem it is proven that converges also. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem14 27235* The sequence converges to a positive real. This proves that the Stirling's formula converges to the factorial, up to a constant. In another theorem, using Wallis' formula for π& , such constant is exactly determined, thus proving the Stirling's formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem15 27236* The Stirling's formula is proven using a number of local definitions. The main theorem stirling 27237 will use this final lemma, but it will not expose the local definitions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirling 27237 Stirling's approximation formula for factorial. The proof follows two major steps: first it is proven that and factorial are asymptotically equivalent, up to an unknown constant. Then, using Wallis' formula for π it is proven that the unknown constant is the square root of π and then the exact Stirling's formula is established. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlingr 27238 Stirling's approximation formula for factorial: here convergence is expressed with respect to the standard topology on the reals. The main theorem stirling 27237 is proven for convergence in the topology of complex numbers. The variable is used to denote convergence with respect to the standard topology on the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

18.20  Mathbox for Jarvin Udandy

TheoremhirstL-ax3 27239 The third axiom of a system called "L" but proven to be a theorem since set.mm uses a different third axiom. This is named hirst after Holly P. Hirst and Jeffry L. Hirst. Axiom A3 of [Mendelson] p. 35. (Contributed by Jarvin Udandy, 7-Feb-2015.) (Proof modification is discouraged.)

Theoremax3h 27240 Recovery of ax-3 9 from hirstL-ax3 27239. (Contributed by Jarvin Udandy, 3-Jul-2015.)

Theoremaibandbiaiffaiffb 27241 A closed form showing (a implies b and b implies a) same-as (a same-as b) (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremaibandbiaiaiffb 27242 A closed form showing (a implies b and b implies a) implies (a same-as b) (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremnotatnand 27243 Do not use. Use intnanr instead. Given not a, there exists a proof for not (a and b). (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremaistia 27244 Given a is equivalent to T., there exists a proof for a. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theoremaisfina 27245 Given a is equivalent to F., there exists a proof for not a. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theorembothtbothsame 27246 Given both a,b are equivalent to T., there exists a proof for a is the same as b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theorembothfbothsame 27247 Given both a,b are equivalent to F., there exists a proof for a is the same as b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremaiffbbtat 27248 Given a is equivalent to b, b is equivalent to T. there exists a proof for a is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremaisbbisfaisf 27249 Given a is equivalent to b, b is equivalent to F. there exists a proof for a is equivalent to F. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theoremaxorbtnotaiffb 27250 Given a is exclusive to b, there exists a proof for (not (a if-and-only-if b)) df-xor is a closed form of this. (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremaiffnbandciffatnotciffb 27251 Given a is equivalent to NOT b, c is equivalent to a. there exists a proof for ( not ( c iff b ) ). (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremaxorbciffatcxorb 27252 Given a is equivalent to NOT b, c is equivalent to a. there exists a proof for ( c xor b ) . (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremaibnbna 27253 Given a implies b, not b, there exists a proof for not a. (Contributed by Jarvin Udandy, 1-Sep-2016.)

Theoremaibnbaif 27254 Given a implies b, not b, there exists a proof for a is F. (Contributed by Jarvin Udandy, 1-Sep-2016.)

Theoremaiffbtbat 27255 Given a is equivalent to b, T. is equivalent to b. there exists a proof for a is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremastbstanbst 27256 Given a is equivalent to T., also given that b is equivalent to T, there exists a proof for a and b is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremaistbistaandb 27257 Given a is equivalent to T., also given that b is equivalent to T, there exists a proof for (a and b). (Contributed by Jarvin Udandy, 9-Sep-2016.)

Theoremaisbnaxb 27258 Given a is equivalent to b, there exists a proof for (not (a xor b)). (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theoremiatbtatnnb 27259 Given a implies b, there exists a proof for a implies not not b. (Contributed by Jarvin Udandy, 2-Sep-2016.)

Theorematbiffatnnb 27260 If a implies b, is is implied a implies not not b (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theorembisaiaisb 27261 Application of bicom1 with a, b swapped. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theorematbiffatnnbalt 27262 If a implies b, it is implied a implies not not b (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremabnotbtaxb 27263 Assuming a, not b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremabnotataxb 27264 Assuming not a, b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremconimpf 27265 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theoremconimpfalt 27266 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremaistbisfiaxb 27267 Given a is equivalent to T., Given b is equivalent to F. there exists a proof for a-xor-b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremaisfbistiaxb 27268 Given a is equivalent to F., Given b is equivalent to T., there exists a proof for a-xor-b. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremabcdta 27269 Given (((a and b) and c) and d), there exists a proof for a (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremabcdtb 27270 Given (((a and b) and c) and d), there exists a proof for b (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremabcdtc 27271 Given (((a and b) and c) and d), there exists a proof for c (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremabcdtd 27272 Given (((a and b) and c) and d), there exists a proof for d (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremmdandyv0 27273 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyv1 27274 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyv2 27275 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyv3 27276 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyv4 27277 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyv5 27278 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyv6 27279 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyv7 27280 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyv8 27281 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyv9 27282 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyv10 27283 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyv11 27284 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyv12 27285 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyv13 27286 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyv14 27287 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyv15 27288 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly (Contributed by Jarvin Udandy, 6-Sep-2016.)

Theoremmdandyvr0 27289 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvr1 27290 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvr2 27291 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvr3 27292 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvr4 27293 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvr5 27294 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvr6 27295 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvr7 27296 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvr8 27297 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvr9 27298 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvr10 27299 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvr11 27300 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

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