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Theorem List for Metamath Proof Explorer - 27701-27800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremstirlinglem12 27701* The sequence is bounded below. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirlinglem13 27702* 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 27703* 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 27704* The Stirling's formula is proven using a number of local definitions. The main theorem stirling 27705 will use this final lemma, but it will not expose the local definitions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Theoremstirling 27705 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 27706 Stirling's approximation formula for factorial: here convergence is expressed with respect to the standard topology on the reals. The main theorem stirling 27705 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 27707* Define the signed area by treating complex numbers as vectors with two components. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarim 27708* Signed area takes value in reals. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarac 27709* Signed area is anticommutative. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigaraf 27710* Signed area is additive by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarmf 27711* Signed area is additive (with respect to subtraction) by the first argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigaras 27712* Signed area is additive by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarms 27713* Signed area is additive (with respect to subtraction) by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarls 27714* Signed area is linear by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.)

Theoremsigarid 27715* Signed area of a flat parallelogram is zero. (Contributed by Saveliy Skresanov, 20-Sep-2017.)

Theoremsigarexp 27716* Expand the signed area formula by linearity. (Contributed by Saveliy Skresanov, 20-Sep-2017.)

Theoremsigarperm 27717* 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 27718* If signed area between vectors and is zero, then those vectors lie on the same line. (Contributed by Saveliy Skresanov, 22-Sep-2017.)

Theoremsigarimcd 27719* Signed area takes value in complex numbers. Deduction version. (Contributed by Saveliy Skresanov, 23-Sep-2017.)

Theoremsigariz 27720* 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 27721* 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 27722* 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 27723* Subtracting (double) area of from yields the (double) area of . (Contributed by Saveliy Skresanov, 23-Sep-2017.)

Theoremcevathlem1 27724 Ceva's theorem first lemma. Multiplies three identities and divides by the common factors. (Contributed by Saveliy Skresanov, 24-Sep-2017.)

Theoremcevathlem2 27725* Ceva's theorem second lemma. Relate (doubled) areas of triangles and with of segments and . (Contributed by Saveliy Skresanov, 24-Sep-2017.)

Theoremcevath 27726* 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 27725 three times and then using cevathlem1 27724 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 27721. (Contributed by Saveliy Skresanov, 24-Sep-2017.)

19.21  Mathbox for Jarvin Udandy

TheoremhirstL-ax3 27727 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 27728 Recovery of ax-3 7 from hirstL-ax3 27727. (Contributed by Jarvin Udandy, 3-Jul-2015.)

Theoremaibandbiaiffaiffb 27729 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 27730 A closed form showing (a implies b and b implies a) implies (a same-as b) (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremnotatnand 27731 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 27732 Given a is equivalent to T., there exists a proof for a. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theoremaisfina 27733 Given a is equivalent to F., there exists a proof for not a. (Contributed by Jarvin Udandy, 30-Aug-2016.)

Theorembothtbothsame 27734 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 27735 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 27736 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 27737 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 27738 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 27739 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 27740 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 27741 Given a implies b, not b, there exists a proof for not a. (Contributed by Jarvin Udandy, 1-Sep-2016.)

Theoremaibnbaif 27742 Given a implies b, not b, there exists a proof for a is F. (Contributed by Jarvin Udandy, 1-Sep-2016.)

Theoremaiffbtbat 27743 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 27744 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 27745 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 27746 Given a is equivalent to b, there exists a proof for (not (a xor b)). (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theoremiatbtatnnb 27747 Given a implies b, there exists a proof for a implies not not b. (Contributed by Jarvin Udandy, 2-Sep-2016.)

Theorematbiffatnnb 27748 If a implies b, then a implies not not b (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theorembisaiaisb 27749 Application of bicom1 with a, b swapped. (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theorematbiffatnnbalt 27750 If a implies b, then a implies not not b (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremabnotbtaxb 27751 Assuming a, not b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremabnotataxb 27752 Assuming not a, b, there exists a proof a-xor-b.) (Contributed by Jarvin Udandy, 31-Aug-2016.)

Theoremconimpf 27753 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 28-Aug-2016.)

Theoremconimpfalt 27754 Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 29-Aug-2016.)

Theoremaistbisfiaxb 27755 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 27756 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 27757 Given (((a and b) and c) and d), there exists a proof for a (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremabcdtb 27758 Given (((a and b) and c) and d), there exists a proof for b (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremabcdtc 27759 Given (((a and b) and c) and d), there exists a proof for c (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremabcdtd 27760 Given (((a and b) and c) and d), there exists a proof for d (Contributed by Jarvin Udandy, 3-Sep-2016.)

Theoremmdandyv0 27761 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 27762 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 27763 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 27764 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 27765 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 27766 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 27767 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 27768 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 27769 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 27770 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 27771 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 27772 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 27773 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 27774 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 27775 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 27776 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 27777 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 27778 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 27779 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 27780 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 27781 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 27782 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 27783 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 27784 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 27785 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 27786 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 27787 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 27788 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.)

Theoremmdandyvr12 27789 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.)

Theoremmdandyvr13 27790 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.)

Theoremmdandyvr14 27791 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.)

Theoremmdandyvr15 27792 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.)

Theoremmdandyvrx0 27793 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvrx1 27794 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvrx2 27795 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvrx3 27796 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvrx4 27797 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvrx5 27798 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvrx6 27799 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

Theoremmdandyvrx7 27800 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)

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