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Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremwallispilem4 27501* maps to explicit expression for the ratio of two consecutive values of (Contributed by Glauco Siliprandi, 30-Jun-2017.)

Theoremwallispilem5 27502* The sequence converges to 1. (Contributed by Glauco Siliprandi, 30-Jun-2017.)

Theoremwallispi 27503* Wallis' formula for π : Wallis' product converges to π / 2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremwallispi2lem1 27504 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 27505 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 27506 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.)

19.19.8  Stirling's approximation formula for ` n ` factorial

Theoremstirlinglem1 27507 A simple limit of fractions is computed. (Contributed by Glauco Siliprandi, 30-Jun-2017.)

Theoremstirlinglem2 27508 maps to positive reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem3 27509 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 27510* 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 27511* 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 27512* A series that converges to log (N+1)/N (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem7 27513* Algebraic manipulation of the formula for J(n) (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem8 27514 If converges to , then converges to C^2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem9 27515* 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 27516* 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 27517* is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem12 27518* The sequence is bounded below. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem13 27519* is decreasing and has a lower bound, then it converges. Since is , in another theorem it is proven that converges as well. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem14 27520* 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 27521* The Stirling's formula is proven using a number of local definitions. The main theorem stirling 27522 will use this final lemma, but it will not expose the local definitions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirling 27522 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 27523 Stirling's approximation formula for factorial: here convergence is expressed with respect to the standard topology on the reals. The main theorem stirling 27522 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.)

19.20  Mathbox for Saveliy Skresanov

19.20.1  Ceva's theorem

Theoremsigarval 27524* Define the signed area by treating complex numbers as vectors with two components. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarim 27525* Signed area takes value in reals. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarac 27526* Signed area is anticommutative. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigaraf 27527* Signed area is additive by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarmf 27528* Signed area is additive (with respect to subtraction) by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigaras 27529* Signed area is additive by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarms 27530* Signed area is additive (with respect to subtraction) by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarls 27531* Signed area is linear by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarid 27532* Signed area of a flat parallelogram is zero. (Contributed by Saveliy Skresanov, 20-Sep-2017.)

Theoremsigarexp 27533* Expand the signed area formula by linearity. (Contributed by Saveliy Skresanov, 20-Sep-2017.)

Theoremsigarperm 27534* Signed area acts as a double area of a triangle . Here we prove that cyclically permuting the vertices doesn't change the area. (Contributed by Saveliy Skresanov, 20-Sep-2017.)

Theoremsigardiv 27535* If signed area between vectors and is zero, then those vectors lie on the same line. (Contributed by Saveliy Skresanov, 22-Sep-2017.)

Theoremsigarimcd 27536* Signed area takes value in complex numbers. Deduction version. (Contributed by Saveliy Skresanov, 23-Sep-2017.)

Theoremsigariz 27537* If signed area is zero, the signed area with swapped arguments is also zero. Deduction version. ( Contributed by Saveliy Skresanov, 23-Sep-2017.) (Contributed by Saveliy Skresanov, 24-Sep-2017.)

Theoremsigarcol 27538* Given three points , and such that , the point lies on the line going through and iff the corresponding signed area is zero. That justifies the usage of signed area as a collinearity indicator. (Contributed by Saveliy Skresanov, 22-Sep-2017.)

Theoremsharhght 27539* Let be a triangle, and let lie on the line . Then (doubled) areas of triangles and relate as lengths of corresponding bases and . (Contributed by Saveliy Skresanov, 23-Sep-2017.)

Theoremsigaradd 27540* Subtracting (double) area of from yields the (double) area of . (Contributed by Saveliy Skresanov, 23-Sep-2017.)

Theoremcevathlem1 27541 Ceva's theorem first lemma. Multiplies three identities and divides by the common factors. (Contributed by Saveliy Skresanov, 24-Sep-2017.)

Theoremcevathlem2 27542* Ceva's theorem second lemma. Relate (doubled) areas of triangles and with of segments and . (Contributed by Saveliy Skresanov, 24-Sep-2017.)

Theoremcevath 27543* Ceva's theorem. Let be a triangle and let points , and lie on sides , , correspondingly. Suppose that cevians , and intersect at one point . Then triangle's sides are partitioned into segments and their lengths satisfy a certain identity. Here we obtain a bit stronger version by using complex numbers themselves instead of their absolute values.

The proof goes by applying cevathlem2 27542 three times and then using cevathlem1 27541 to multiply obtained identities and prove the theorem.

In the theorem statement we are using function as a collinearity indicator. For justification of that use, see sigarcol 27538. (Contributed by Saveliy Skresanov, 24-Sep-2017.)

19.21  Mathbox for Jarvin Udandy

TheoremhirstL-ax3 27544 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 27545 Recovery of ax-3 7 from hirstL-ax3 27544. (Contributed by Jarvin Udandy, 3-Jul-2015.)

Theoremaibandbiaiffaiffb 27546 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 27547 A closed form showing (a implies b and b implies a) implies (a same-as b) (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremnotatnand 27548 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 27549 Given a is equivalent to T., there exists a proof for a. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theoremaisfina 27550 Given a is equivalent to F., there exists a proof for not a. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theorembothtbothsame 27551 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 27552 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 27553 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 27554 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 27555 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 27556 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 27557 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 27558 Given a implies b, not b, there exists a proof for not a. (Contributed by Jarvin Udandy, 1-Sep-2016.)

Theoremaibnbaif 27559 Given a implies b, not b, there exists a proof for a is F. (Contributed by Jarvin Udandy, 1-Sep-2016.)

Theoremaiffbtbat 27560 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 27561 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 27562 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 27563 Given a is equivalent to b, there exists a proof for (not (a xor b)). (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theoremiatbtatnnb 27564 Given a implies b, there exists a proof for a implies not not b. (Contributed by Jarvin Udandy, 2-Sep-2016.)

Theorematbiffatnnb 27565 If a implies b, then a implies not not b (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theorembisaiaisb 27566 Application of bicom1 with a, b swapped. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theorematbiffatnnbalt 27567 If a implies b, then a implies not not b (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremabnotbtaxb 27568 Assuming a, not b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremabnotataxb 27569 Assuming not a, b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremconimpf 27570 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theoremconimpfalt 27571 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremaistbisfiaxb 27572 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 27573 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 27574 Given (((a and b) and c) and d), there exists a proof for a (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremabcdtb 27575 Given (((a and b) and c) and d), there exists a proof for b (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremabcdtc 27576 Given (((a and b) and c) and d), there exists a proof for c (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremabcdtd 27577 Given (((a and b) and c) and d), there exists a proof for d (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremmdandyv0 27578 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 27579 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 27580 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 27581 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 27582 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 27583 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 27584 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 27585 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 27586 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 27587 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 27588 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 27589 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 27590 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 27591 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 27592 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 27593 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 27594 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 27595 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 27596 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 27597 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 27598 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 27599 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 27600 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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