Intuitionistic Logic Explorer Most Recent Proofs |
||
Mirrors > Home > ILE Home > Th. List > Recent | MPE Most Recent Other > MM 100 |
See the MPE Most Recent Proofs page for news and some useful links.
Color key: | Intuitionistic Logic Explorer | User Mathboxes |
Date | Label | Description |
---|---|---|
Theorem | ||
13-Apr-2024 | prodmodclem2 11346 | Lemma for prodmodc 11347. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 13-Apr-2024.) |
♯ DECID # | ||
11-Apr-2024 | prodmodclem2a 11345 | Lemma for prodmodc 11347. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.) |
♯ ♯ DECID ♯ | ||
11-Apr-2024 | prodmodclem3 11344 | Lemma for prodmodc 11347. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.) |
♯ ♯ | ||
10-Apr-2024 | jcnd 641 | Deduction joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Apr-2024.) |
4-Apr-2024 | prodrbdclem 11340 | Lemma for prodrbdc 11343. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 4-Apr-2024.) |
DECID | ||
24-Mar-2024 | prodfdivap 11316 | The quotient of two products. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.) |
# | ||
24-Mar-2024 | prodfrecap 11315 | The reciprocal of a finite product. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.) |
# | ||
23-Mar-2024 | prodfap0 11314 | The product of finitely many terms apart from zero is apart from zero. (Contributed by Scott Fenton, 14-Jan-2018.) (Revised by Jim Kingdon, 23-Mar-2024.) |
# # | ||
22-Mar-2024 | prod3fmul 11310 | The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017.) (Revised by Jim Kingdon, 22-Mar-2024.) |
21-Mar-2024 | df-proddc 11320 | Define the product of a series with an index set of integers . This definition takes most of the aspects of df-sumdc 11123 and adapts them for multiplication instead of addition. However, we insist that in the infinite case, there is a nonzero tail of the sequence. This ensures that the convergence criteria match those of infinite sums. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 21-Mar-2024.) |
DECID # | ||
19-Mar-2024 | cos02pilt1 12932 | Cosine is less than one between zero and . (Contributed by Jim Kingdon, 19-Mar-2024.) |
19-Mar-2024 | cosq34lt1 12931 | Cosine is less than one in the third and fourth quadrants. (Contributed by Jim Kingdon, 19-Mar-2024.) |
14-Mar-2024 | coseq0q4123 12915 | Location of the zeroes of cosine in . (Contributed by Jim Kingdon, 14-Mar-2024.) |
14-Mar-2024 | cosq23lt0 12914 | The cosine of a number in the second and third quadrants is negative. (Contributed by Jim Kingdon, 14-Mar-2024.) |
9-Mar-2024 | pilem3 12864 | Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.) |
9-Mar-2024 | exmidonfin 7050 | If a finite ordinal is a natural number, excluded middle follows. That excluded middle implies that a finite ordinal is a natural number is proved in the Metamath Proof Explorer. That a natural number is a finite ordinal is shown at nnfi 6766 and nnon 4523. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
EXMID | ||
9-Mar-2024 | exmidonfinlem 7049 | Lemma for exmidonfin 7050. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
DECID | ||
8-Mar-2024 | sin0pilem2 12863 | Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.) |
8-Mar-2024 | sin0pilem1 12862 | Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.) |
7-Mar-2024 | cosz12 12861 | Cosine has a zero between 1 and 2. (Contributed by Mario Carneiro and Jim Kingdon, 7-Mar-2024.) |
6-Mar-2024 | cos12dec 11474 | Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.) |
25-Feb-2024 | mul2lt0pn 9551 | The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.) |
25-Feb-2024 | mul2lt0np 9550 | The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.) |
25-Feb-2024 | lt0ap0 8410 | A number which is less than zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.) |
# | ||
25-Feb-2024 | negap0d 8393 | The negative of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.) |
# # | ||
24-Feb-2024 | lt0ap0d 8411 | A real number less than zero is apart from zero. Deduction form. (Contributed by Jim Kingdon, 24-Feb-2024.) |
# | ||
20-Feb-2024 | ivthdec 12791 | The intermediate value theorem, decreasing case, for a strictly monotonic function. (Contributed by Jim Kingdon, 20-Feb-2024.) |
20-Feb-2024 | ivthinclemex 12789 | Lemma for ivthinc 12790. Existence of a number between the lower cut and the upper cut. (Contributed by Jim Kingdon, 20-Feb-2024.) |
19-Feb-2024 | ivthinclemuopn 12785 | Lemma for ivthinc 12790. The upper cut is open. (Contributed by Jim Kingdon, 19-Feb-2024.) |
19-Feb-2024 | dedekindicc 12780 | A Dedekind cut identifies a unique real number. Similar to df-inp 7274 except that the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 19-Feb-2024.) |
18-Feb-2024 | ivthinclemloc 12788 | Lemma for ivthinc 12790. Locatedness. (Contributed by Jim Kingdon, 18-Feb-2024.) |
18-Feb-2024 | ivthinclemdisj 12787 | Lemma for ivthinc 12790. The lower and upper cuts are disjoint. (Contributed by Jim Kingdon, 18-Feb-2024.) |
18-Feb-2024 | ivthinclemur 12786 | Lemma for ivthinc 12790. The upper cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.) |
18-Feb-2024 | ivthinclemlr 12784 | Lemma for ivthinc 12790. The lower cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.) |
18-Feb-2024 | ivthinclemum 12782 | Lemma for ivthinc 12790. The upper cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.) |
18-Feb-2024 | ivthinclemlm 12781 | Lemma for ivthinc 12790. The lower cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.) |
15-Feb-2024 | dedekindicclemeu 12778 | Lemma for dedekindicc 12780. Part of proving uniqueness. (Contributed by Jim Kingdon, 15-Feb-2024.) |
15-Feb-2024 | dedekindicclemlu 12777 | Lemma for dedekindicc 12780. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 15-Feb-2024.) |
15-Feb-2024 | dedekindicclemlub 12776 | Lemma for dedekindicc 12780. The set L has a least upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.) |
15-Feb-2024 | dedekindicclemloc 12775 | Lemma for dedekindicc 12780. The set L is located. (Contributed by Jim Kingdon, 15-Feb-2024.) |
15-Feb-2024 | dedekindicclemub 12774 | Lemma for dedekindicc 12780. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.) |
15-Feb-2024 | dedekindicclemuub 12773 | Lemma for dedekindicc 12780. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 15-Feb-2024.) |
14-Feb-2024 | suplociccex 12772 | An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 7837 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.) |
14-Feb-2024 | suplociccreex 12771 | An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 7837 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.) |
6-Feb-2024 | ivthinclemlopn 12783 | Lemma for ivthinc 12790. The lower cut is open. (Contributed by Jim Kingdon, 6-Feb-2024.) |
5-Feb-2024 | ivthinc 12790 | The intermediate value theorem, increasing case, for a strictly monotonic function. Theorem 5.5 of [Bauer], p. 494. This is Metamath 100 proof #79. (Contributed by Jim Kingdon, 5-Feb-2024.) |
2-Feb-2024 | dedekindeulemuub 12764 | Lemma for dedekindeu 12770. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-2024.) |
31-Jan-2024 | dedekindeulemeu 12769 | Lemma for dedekindeu 12770. Part of proving uniqueness. (Contributed by Jim Kingdon, 31-Jan-2024.) |
31-Jan-2024 | dedekindeulemlu 12768 | Lemma for dedekindeu 12770. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 31-Jan-2024.) |
31-Jan-2024 | dedekindeulemlub 12767 | Lemma for dedekindeu 12770. The set L has a least upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.) |
31-Jan-2024 | dedekindeulemloc 12766 | Lemma for dedekindeu 12770. The set L is located. (Contributed by Jim Kingdon, 31-Jan-2024.) |
31-Jan-2024 | dedekindeulemub 12765 | Lemma for dedekindeu 12770. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.) |
30-Jan-2024 | axsuploc 7837 | An inhabited, bounded-above, located set of reals has a supremum. Axiom for real and complex numbers, derived from ZF set theory. (This restates ax-pre-suploc 7741 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.) |
24-Jan-2024 | axpre-suploclemres 7709 | Lemma for axpre-suploc 7710. The result. The proof just needs to define as basically the same set as (but expressed as a subset of rather than a subset of ), and apply suplocsr 7617. (Contributed by Jim Kingdon, 24-Jan-2024.) |
23-Jan-2024 | ax-pre-suploc 7741 |
An inhabited, bounded-above, located set of reals has a supremum.
Locatedness here means that given , either there is an element of the set greater than , or is an upper bound. Although this and ax-caucvg 7740 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 7740. (Contributed by Jim Kingdon, 23-Jan-2024.) |
23-Jan-2024 | axpre-suploc 7710 |
An inhabited, bounded-above, located set of reals has a supremum.
Locatedness here means that given , either there is an element of the set greater than , or is an upper bound. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-suploc 7741. (Contributed by Jim Kingdon, 23-Jan-2024.) (New usage is discouraged.) |
22-Jan-2024 | suplocsr 7617 | An inhabited, bounded, located set of signed reals has a supremum. (Contributed by Jim Kingdon, 22-Jan-2024.) |
21-Jan-2024 | bj-el2oss1o 12981 | Shorter proof of el2oss1o 13188 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
21-Jan-2024 | ltm1sr 7585 | Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.) |
19-Jan-2024 | suplocsrlempr 7615 | Lemma for suplocsr 7617. The set has a least upper bound. (Contributed by Jim Kingdon, 19-Jan-2024.) |
18-Jan-2024 | suplocsrlemb 7614 | Lemma for suplocsr 7617. The set is located. (Contributed by Jim Kingdon, 18-Jan-2024.) |
16-Jan-2024 | suplocsrlem 7616 | Lemma for suplocsr 7617. The set has a least upper bound. (Contributed by Jim Kingdon, 16-Jan-2024.) |
14-Jan-2024 | suplocexprlemlub 7532 | Lemma for suplocexpr 7533. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.) |
14-Jan-2024 | suplocexprlemub 7531 | Lemma for suplocexpr 7533. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.) |
10-Jan-2024 | cbvcsbw 3007 | Version of cbvcsb 3008 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
10-Jan-2024 | cbvsbcw 2936 | Version of cbvsbc 2937 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
10-Jan-2024 | cbvreuvw 2660 | Version of cbvreuv 2656 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
10-Jan-2024 | cbvrexvw 2659 | Version of cbvrexv 2655 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
10-Jan-2024 | cbvralvw 2658 | Version of cbvralv 2654 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
10-Jan-2024 | cbvabw 2262 | Version of cbvab 2263 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
9-Jan-2024 | suplocexprlemloc 7529 | Lemma for suplocexpr 7533. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.) |
9-Jan-2024 | suplocexprlemdisj 7528 | Lemma for suplocexpr 7533. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.) |
9-Jan-2024 | suplocexprlemru 7527 | Lemma for suplocexpr 7533. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.) |
9-Jan-2024 | suplocexprlemrl 7525 | Lemma for suplocexpr 7533. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.) |
9-Jan-2024 | suplocexprlem2b 7522 | Lemma for suplocexpr 7533. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.) |
9-Jan-2024 | suplocexprlemell 7521 | Lemma for suplocexpr 7533. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.) |
7-Jan-2024 | suplocexpr 7533 | An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.) |
7-Jan-2024 | suplocexprlemex 7530 | Lemma for suplocexpr 7533. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.) |
7-Jan-2024 | suplocexprlemmu 7526 | Lemma for suplocexpr 7533. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.) |
7-Jan-2024 | suplocexprlemml 7524 | Lemma for suplocexpr 7533. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.) |
7-Jan-2024 | suplocexprlemss 7523 | Lemma for suplocexpr 7533. is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.) |
5-Jan-2024 | dedekindicclemicc 12779 | Lemma for dedekindicc 12780. Same as dedekindicc 12780, except that we merely show to be an element of . Later we will strengthen that to . (Contributed by Jim Kingdon, 5-Jan-2024.) |
5-Jan-2024 | dedekindeu 12770 | A Dedekind cut identifies a unique real number. Similar to df-inp 7274 except that the the Dedekind cut is formed by sets of reals (rather than positive rationals). But in both cases the defining property of a Dedekind cut is that it is inhabited (bounded), rounded, disjoint, and located. (Contributed by Jim Kingdon, 5-Jan-2024.) |
31-Dec-2023 | dvmptsubcn 12854 | Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.) |
31-Dec-2023 | dvmptnegcn 12853 | Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.) |
31-Dec-2023 | dvmptcmulcn 12852 | Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.) |
31-Dec-2023 | brm 3978 | If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.) |
30-Dec-2023 | dvmptccn 12848 | Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.) |
30-Dec-2023 | dvmptidcn 12847 | Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.) |
25-Dec-2023 | ctfoex 7003 | A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.) |
⊔ | ||
23-Dec-2023 | enct 11946 | Countability is invariant relative to equinumerosity. (Contributed by Jim Kingdon, 23-Dec-2023.) |
⊔ ⊔ | ||
23-Dec-2023 | enctlem 11945 | Lemma for enct 11946. One direction of the biconditional. (Contributed by Jim Kingdon, 23-Dec-2023.) |
⊔ ⊔ | ||
23-Dec-2023 | omct 7002 | is countable. (Contributed by Jim Kingdon, 23-Dec-2023.) |
⊔ | ||
21-Dec-2023 | dvcoapbr 12840 | The chain rule for derivatives at a point. The # # hypothesis constrains what functions work for . (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 21-Dec-2023.) |
# # | ||
19-Dec-2023 | apsscn 8409 | The points apart from a given point are complex numbers. (Contributed by Jim Kingdon, 19-Dec-2023.) |
# | ||
19-Dec-2023 | aprcl 8408 | Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.) |
# | ||
18-Dec-2023 | limccoap 12816 | Composition of two limits. This theorem is only usable in the case where # implies R(x) # so it is less general than might appear at first. (Contributed by Mario Carneiro, 29-Dec-2016.) (Revised by Jim Kingdon, 18-Dec-2023.) |
# # # # lim # lim # lim | ||
16-Dec-2023 | cnreim 10750 | Complex apartness in terms of real and imaginary parts. See also apreim 8365 which is similar but with different notation. (Contributed by Jim Kingdon, 16-Dec-2023.) |
# # # | ||
14-Dec-2023 | cnopnap 12763 | The complex numbers apart from a given complex number form an open set. (Contributed by Jim Kingdon, 14-Dec-2023.) |
# | ||
14-Dec-2023 | cnovex 12365 | The class of all continuous functions from a topology to another is a set. (Contributed by Jim Kingdon, 14-Dec-2023.) |
Copyright terms: Public domain | W3C HTML validation [external] |