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Date | Label | Description |
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Theorem | ||
20-Mar-2025 | ccoslid 12686 | Slot property of comp. (Contributed by Jim Kingdon, 20-Mar-2025.) |
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20-Mar-2025 | homslid 12684 |
Slot property of ![]() |
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19-Mar-2025 | ptex 12712 | Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.) |
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18-Mar-2025 | prdsex 12717 | Existence of the structure product. (Contributed by Jim Kingdon, 18-Mar-2025.) |
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13-Mar-2025 | imasex 12725 | Existence of the image structure. (Contributed by Jim Kingdon, 13-Mar-2025.) |
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11-Mar-2025 | imasival 12726 | Value of an image structure. The is a lemma for the theorems imasbas 12727, imasplusg 12728, and imasmulr 12729 and should not be needed once they are proved. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Jim Kingdon, 11-Mar-2025.) (New usage is discouraged.) |
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8-Mar-2025 | subgex 13034 | The class of subgroups of a group is a set. (Contributed by Jim Kingdon, 8-Mar-2025.) |
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28-Feb-2025 | ringressid 13236 | A ring restricted to its base set is a ring. It will usually be the original ring exactly, of course, but to show that needs additional conditions such as those in strressid 12529. (Contributed by Jim Kingdon, 28-Feb-2025.) |
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28-Feb-2025 | grpressid 12930 | A group restricted to its base set is a group. It will usually be the original group exactly, of course, but to show that needs additional conditions such as those in strressid 12529. (Contributed by Jim Kingdon, 28-Feb-2025.) |
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26-Feb-2025 | strext 12563 |
Extending the upper range of a structure. This works because when we
say that a structure has components in ![]() ![]() ![]() |
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23-Feb-2025 | ltlenmkv 14753 |
If ![]() ![]() |
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23-Feb-2025 | neap0mkv 14752 | The analytic Markov principle can be expressed either with two arbitrary real numbers, or one arbitrary number and zero. (Contributed by Jim Kingdon, 23-Feb-2025.) |
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23-Feb-2025 | lringuplu 13335 | If the sum of two elements of a local ring is invertible, then at least one of the summands must be invertible. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.) |
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23-Feb-2025 | lringnz 13334 | A local ring is a nonzero ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.) |
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23-Feb-2025 | lringring 13333 | A local ring is a ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.) |
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23-Feb-2025 | lringnzr 13332 | A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.) |
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23-Feb-2025 | islring 13331 | The predicate "is a local ring". (Contributed by SN, 23-Feb-2025.) |
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23-Feb-2025 | df-lring 13330 | A local ring is a nonzero ring where for any two elements summing to one, at least one is invertible. Any field is a local ring; the ring of integers is an example of a ring which is not a local ring. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.) |
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23-Feb-2025 | 01eq0ring 13328 | If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019.) (Proof shortened by SN, 23-Feb-2025.) |
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23-Feb-2025 | nzrring 13325 | A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof shortened by SN, 23-Feb-2025.) |
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21-Feb-2025 | dftap2 7249 | Tight apartness with the apartness properties from df-pap 7246 expanded. (Contributed by Jim Kingdon, 21-Feb-2025.) |
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20-Feb-2025 | aprap 13342 | The relation given by df-apr 13337 for a local ring is an apartness relation. (Contributed by Jim Kingdon, 20-Feb-2025.) |
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20-Feb-2025 | setscomd 12502 | Different components can be set in any order. (Contributed by Jim Kingdon, 20-Feb-2025.) |
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17-Feb-2025 | aprcotr 13341 | The apartness relation given by df-apr 13337 for a local ring is cotransitive. (Contributed by Jim Kingdon, 17-Feb-2025.) |
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17-Feb-2025 | aprsym 13340 | The apartness relation given by df-apr 13337 for a ring is symmetric. (Contributed by Jim Kingdon, 17-Feb-2025.) |
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17-Feb-2025 | aprval 13338 | Expand Definition df-apr 13337. (Contributed by Jim Kingdon, 17-Feb-2025.) |
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16-Feb-2025 | aprirr 13339 | The apartness relation given by df-apr 13337 for a nonzero ring is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2025.) |
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16-Feb-2025 | aptap 8606 | Complex apartness (as defined at df-ap 8538) is a tight apartness (as defined at df-tap 7248). (Contributed by Jim Kingdon, 16-Feb-2025.) |
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15-Feb-2025 | tapeq2 7251 | Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 15-Feb-2025.) |
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14-Feb-2025 | exmidmotap 7259 | The proposition that every class has at most one tight apartness is equivalent to excluded middle. (Contributed by Jim Kingdon, 14-Feb-2025.) |
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14-Feb-2025 | exmidapne 7258 | Excluded middle implies there is only one tight apartness on any class, namely negated equality. (Contributed by Jim Kingdon, 14-Feb-2025.) |
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14-Feb-2025 | df-pap 7246 |
Apartness predicate. A relation ![]() |
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13-Feb-2025 | df-apr 13337 | The relation between elements whose difference is invertible, which for a local ring is an apartness relation by aprap 13342. (Contributed by Jim Kingdon, 13-Feb-2025.) |
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8-Feb-2025 | 2oneel 7254 |
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8-Feb-2025 | tapeq1 7250 | Equality theorem for tight apartness predicate. (Contributed by Jim Kingdon, 8-Feb-2025.) |
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6-Feb-2025 | 2omotap 7257 |
If there is at most one tight apartness on ![]() |
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6-Feb-2025 | 2omotaplemst 7256 | Lemma for 2omotap 7257. (Contributed by Jim Kingdon, 6-Feb-2025.) |
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6-Feb-2025 | 2omotaplemap 7255 | Lemma for 2omotap 7257. (Contributed by Jim Kingdon, 6-Feb-2025.) |
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6-Feb-2025 | 2onetap 7253 |
Negated equality is a tight apartness on ![]() |
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5-Feb-2025 | netap 7252 | Negated equality on a set with decidable equality is a tight apartness. (Contributed by Jim Kingdon, 5-Feb-2025.) |
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5-Feb-2025 | df-tap 7248 |
Tight apartness predicate. A relation ![]() |
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31-Jan-2025 | 0subg 13057 | The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.) (Proof shortened by SN, 31-Jan-2025.) |
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28-Jan-2025 | dvdsrex 13265 | Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.) |
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24-Jan-2025 | reldvdsrsrg 13259 | The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2025.) |
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18-Jan-2025 | rerecapb 8799 | A real number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 18-Jan-2025.) |
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18-Jan-2025 | recapb 8627 | A complex number has a multiplicative inverse if and only if it is apart from zero. Theorem 11.2.4 of [HoTT], p. (varies), generalized from real to complex numbers. (Contributed by Jim Kingdon, 18-Jan-2025.) |
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17-Jan-2025 | ressval3d 12530 | Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.) |
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17-Jan-2025 | strressid 12529 | Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 17-Jan-2025.) |
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16-Jan-2025 | ressex 12524 | Existence of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.) |
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16-Jan-2025 | ressvalsets 12523 | Value of structure restriction. (Contributed by Jim Kingdon, 16-Jan-2025.) |
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10-Jan-2025 | opprex 13243 |
Existence of the opposite ring. If you know that ![]() |
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10-Jan-2025 | mgpex 13133 |
Existence of the multiplication group. If ![]() |
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5-Jan-2025 | imbibi 252 | The antecedent of one side of a biconditional can be moved out of the biconditional to become the antecedent of the remaining biconditional. (Contributed by BJ, 1-Jan-2025.) (Proof shortened by Wolf Lammen, 5-Jan-2025.) |
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1-Jan-2025 | snss 3727 | The singleton of an element of a class is a subset of the class (inference form of snssg 3726). Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.) (Proof shortened by BJ, 1-Jan-2025.) |
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1-Jan-2025 | snssg 3726 | The singleton formed on a set is included in a class if and only if the set is an element of that class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) (Proof shortened by BJ, 1-Jan-2025.) |
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1-Jan-2025 | snssb 3725 | Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.) |
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9-Dec-2024 | nninfwlpoim 7175 | Decidable equality for ℕ∞ implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.) |
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8-Dec-2024 | nninfwlpoimlemdc 7174 | Lemma for nninfwlpoim 7175. (Contributed by Jim Kingdon, 8-Dec-2024.) |
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8-Dec-2024 | nninfwlpoimlemginf 7173 | Lemma for nninfwlpoim 7175. (Contributed by Jim Kingdon, 8-Dec-2024.) |
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8-Dec-2024 | nninfwlpoimlemg 7172 | Lemma for nninfwlpoim 7175. (Contributed by Jim Kingdon, 8-Dec-2024.) |
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7-Dec-2024 | nninfwlpor 7171 | The Weak Limited Principle of Omniscience (WLPO) implies that equality for ℕ∞ is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.) |
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7-Dec-2024 | nninfwlporlem 7170 | Lemma for nninfwlpor 7171. The result. (Contributed by Jim Kingdon, 7-Dec-2024.) |
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6-Dec-2024 | nninfwlporlemd 7169 | Given two countably infinite sequences of zeroes and ones, they are equal if and only if a sequence formed by pointwise comparing them is all ones. (Contributed by Jim Kingdon, 6-Dec-2024.) |
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3-Dec-2024 | nninfwlpo 7176 | Decidability of equality for ℕ∞ is equivalent to the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 3-Dec-2024.) |
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3-Dec-2024 | nninfdcinf 7168 | The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of ℕ∞ equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.) |
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28-Nov-2024 | basmexd 12521 | A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 28-Nov-2024.) |
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22-Nov-2024 | eliotaeu 5205 | An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.) |
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22-Nov-2024 | eliota 5204 | An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.) |
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18-Nov-2024 | basmex 12520 | A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 18-Nov-2024.) |
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12-Nov-2024 | slotsdifipndx 12632 | The slot for the scalar is not the index of other slots. (Contributed by AV, 12-Nov-2024.) |
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11-Nov-2024 | bj-con1st 14439 | Contraposition when the antecedent is a negated stable proposition. See con1dc 856. (Contributed by BJ, 11-Nov-2024.) |
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11-Nov-2024 | slotsdifdsndx 12675 | The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.) |
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11-Nov-2024 | slotsdifplendx 12664 | The index of the slot for the distance is not the index of other slots. (Contributed by AV, 11-Nov-2024.) |
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11-Nov-2024 | tsetndxnstarvndx 12648 | The slot for the topology is not the slot for the involution in an extensible structure. (Contributed by AV, 11-Nov-2024.) |
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11-Nov-2024 | const 852 | Contraposition when the antecedent is a negated stable proposition. See comment of condc 853. (Contributed by BJ, 18-Nov-2023.) (Proof shortened by BJ, 11-Nov-2024.) |
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10-Nov-2024 | slotsdifunifndx 12682 | The index of the slot for the uniform set is not the index of other slots. (Contributed by AV, 10-Nov-2024.) |
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7-Nov-2024 | ressbasd 12526 | Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.) |
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6-Nov-2024 | oppraddg 13246 | Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.) |
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6-Nov-2024 | opprbasg 13245 | Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.) |
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6-Nov-2024 | opprsllem 13244 | Lemma for opprbasg 13245 and oppraddg 13246. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.) |
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4-Nov-2024 | lgsfvalg 14342 |
Value of the function ![]() |
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1-Nov-2024 | plendxnvscandx 12663 | The slot for the "less than or equal to" ordering is not the slot for the scalar product in an extensible structure. (Contributed by AV, 1-Nov-2024.) |
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1-Nov-2024 | plendxnscandx 12662 | The slot for the "less than or equal to" ordering is not the slot for the scalar in an extensible structure. (Contributed by AV, 1-Nov-2024.) |
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1-Nov-2024 | plendxnmulrndx 12661 | The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 1-Nov-2024.) |
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1-Nov-2024 | qsqeqor 10630 | The squares of two rational numbers are equal iff one number equals the other or its negative. (Contributed by Jim Kingdon, 1-Nov-2024.) |
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31-Oct-2024 | dsndxnmulrndx 12672 | The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.) |
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31-Oct-2024 | tsetndxnmulrndx 12647 | The slot for the topology is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024.) |
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31-Oct-2024 | tsetndxnbasendx 12645 | The slot for the topology is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 31-Oct-2024.) |
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31-Oct-2024 | basendxlttsetndx 12644 | The index of the slot for the base set is less then the index of the slot for the topology in an extensible structure. (Contributed by AV, 31-Oct-2024.) |
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31-Oct-2024 | tsetndxnn 12643 | The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 31-Oct-2024.) |
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30-Oct-2024 | plendxnbasendx 12659 | The slot for the order is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 30-Oct-2024.) |
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30-Oct-2024 | basendxltplendx 12658 |
The index value of the ![]() ![]() |
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30-Oct-2024 | plendxnn 12657 | The index value of the order slot is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 30-Oct-2024.) |
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29-Oct-2024 | dsndxntsetndx 12674 | The slot for the distance function is not the slot for the topology in an extensible structure. (Contributed by AV, 29-Oct-2024.) |
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29-Oct-2024 | slotsdnscsi 12673 |
The slots Scalar, ![]() ![]() ![]() |
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29-Oct-2024 | slotstnscsi 12649 |
The slots Scalar, ![]() ![]() |
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29-Oct-2024 | ipndxnmulrndx 12631 | The slot for the inner product is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.) |
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29-Oct-2024 | ipndxnplusgndx 12630 | The slot for the inner product is not the slot for the group operation in an extensible structure. (Contributed by AV, 29-Oct-2024.) |
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29-Oct-2024 | vscandxnmulrndx 12618 | The slot for the scalar product is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.) |
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29-Oct-2024 | scandxnmulrndx 12613 | The slot for the scalar field is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 29-Oct-2024.) |
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29-Oct-2024 | fiubnn 10809 | A finite set of natural numbers has an upper bound which is a a natural number. (Contributed by Jim Kingdon, 29-Oct-2024.) |
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29-Oct-2024 | fiubz 10808 | A finite set of integers has an upper bound which is an integer. (Contributed by Jim Kingdon, 29-Oct-2024.) |
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29-Oct-2024 | fiubm 10807 | Lemma for fiubz 10808 and fiubnn 10809. A general form of those theorems. (Contributed by Jim Kingdon, 29-Oct-2024.) |
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28-Oct-2024 | unifndxntsetndx 12681 | The slot for the uniform set is not the slot for the topology in an extensible structure. (Contributed by AV, 28-Oct-2024.) |
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28-Oct-2024 | basendxltunifndx 12679 | The index of the slot for the base set is less then the index of the slot for the uniform set in an extensible structure. (Contributed by AV, 28-Oct-2024.) |
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28-Oct-2024 | unifndxnn 12678 | The index of the slot for the uniform set in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.) |
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28-Oct-2024 | dsndxnbasendx 12670 | The slot for the distance is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) (Proof shortened by AV, 28-Oct-2024.) |
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28-Oct-2024 | basendxltdsndx 12669 | The index of the slot for the base set is less then the index of the slot for the distance in an extensible structure. (Contributed by AV, 28-Oct-2024.) |
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28-Oct-2024 | dsndxnn 12668 | The index of the slot for the distance in an extensible structure is a positive integer. (Contributed by AV, 28-Oct-2024.) |
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27-Oct-2024 | bj-nnst 14431 |
Double negation of stability of a formula. Intuitionistic logic refutes
unstability (but does not prove stability) of any formula. This theorem
can also be proved in classical refutability calculus (see
https://us.metamath.org/mpeuni/bj-peircestab.html) but not in minimal
calculus (see https://us.metamath.org/mpeuni/bj-stabpeirce.html). See
nnnotnotr 14678 for the version not using the definition of
stability.
(Contributed by BJ, 9-Oct-2019.) Prove it in ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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27-Oct-2024 | bj-imnimnn 14426 | If a formula is implied by both a formula and its negation, then it is not refutable. There is another proof using the inference associated with bj-nnclavius 14425 as its last step. (Contributed by BJ, 27-Oct-2024.) |
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25-Oct-2024 | nnwosdc 12039 | Well-ordering principle: any inhabited decidable set of positive integers has a least element (schema form). (Contributed by NM, 17-Aug-2001.) (Revised by Jim Kingdon, 25-Oct-2024.) |
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23-Oct-2024 | nnwodc 12036 | Well-ordering principle: any inhabited decidable set of positive integers has a least element. Theorem I.37 (well-ordering principle) of [Apostol] p. 34. (Contributed by NM, 17-Aug-2001.) (Revised by Jim Kingdon, 23-Oct-2024.) |
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22-Oct-2024 | uzwodc 12037 | Well-ordering principle: any inhabited decidable subset of an upper set of integers has a least element. (Contributed by NM, 8-Oct-2005.) (Revised by Jim Kingdon, 22-Oct-2024.) |
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21-Oct-2024 | nnnotnotr 14678 | Double negation of double negation elimination. Suggested by an online post by Martin Escardo. Although this statement resembles nnexmid 850, it can be proved with reference only to implication and negation (that is, without use of disjunction). (Contributed by Jim Kingdon, 21-Oct-2024.) |
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21-Oct-2024 | unifndxnbasendx 12680 | The slot for the uniform set is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) |
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21-Oct-2024 | ipndxnbasendx 12629 | The slot for the inner product is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) |
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21-Oct-2024 | scandxnbasendx 12611 | The slot for the scalar is not the slot for the base set in an extensible structure. (Contributed by AV, 21-Oct-2024.) |
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20-Oct-2024 | isprm5lem 12140 |
Lemma for isprm5 12141. The interesting direction (showing that
one only
needs to check prime divisors up to the square root of ![]() |
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19-Oct-2024 | resseqnbasd 12531 | The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.) |
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18-Oct-2024 | mgpress 13139 | Subgroup commutes with the multiplicative group operator. (Contributed by Mario Carneiro, 10-Jan-2015.) (Proof shortened by AV, 18-Oct-2024.) |
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18-Oct-2024 | dsndxnplusgndx 12671 | The slot for the distance function is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.) |
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18-Oct-2024 | plendxnplusgndx 12660 | The slot for the "less than or equal to" ordering is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.) |
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18-Oct-2024 | tsetndxnplusgndx 12646 | The slot for the topology is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.) |
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18-Oct-2024 | vscandxnscandx 12619 | The slot for the scalar product is not the slot for the scalar field in an extensible structure. (Contributed by AV, 18-Oct-2024.) |
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18-Oct-2024 | vscandxnplusgndx 12617 | The slot for the scalar product is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.) |
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18-Oct-2024 | vscandxnbasendx 12616 | The slot for the scalar product is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.) |
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18-Oct-2024 | scandxnplusgndx 12612 | The slot for the scalar field is not the slot for the group operation in an extensible structure. (Contributed by AV, 18-Oct-2024.) |
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18-Oct-2024 | starvndxnmulrndx 12601 | The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.) |
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18-Oct-2024 | starvndxnplusgndx 12600 | The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.) |
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18-Oct-2024 | starvndxnbasendx 12599 | The slot for the involution function is not the slot for the base set in an extensible structure. (Contributed by AV, 18-Oct-2024.) |
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17-Oct-2024 | basendxltplusgndx 12571 | The index of the slot for the base set is less then the index of the slot for the group operation in an extensible structure. (Contributed by AV, 17-Oct-2024.) |
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17-Oct-2024 | plusgndxnn 12569 | The index of the slot for the group operation in an extensible structure is a positive integer. (Contributed by AV, 17-Oct-2024.) |
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17-Oct-2024 | elnndc 9611 |
Membership of an integer in ![]() |
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14-Oct-2024 | 2zinfmin 11250 | Two ways to express the minimum of two integers. Because order of integers is decidable, we have more flexibility than for real numbers. (Contributed by Jim Kingdon, 14-Oct-2024.) |
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14-Oct-2024 | mingeb 11249 |
Equivalence of ![]() |
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13-Oct-2024 | pcxnn0cl 12309 | Extended nonnegative integer closure of the general prime count function. (Contributed by Jim Kingdon, 13-Oct-2024.) |
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13-Oct-2024 | xnn0letri 9802 | Dichotomy for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.) |
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13-Oct-2024 | xnn0dcle 9801 |
Decidability of ![]() |
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9-Oct-2024 | nn0leexp2 10689 | Ordering law for exponentiation. (Contributed by Jim Kingdon, 9-Oct-2024.) |
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8-Oct-2024 | pclemdc 12287 | Lemma for the prime power pre-function's properties. (Contributed by Jim Kingdon, 8-Oct-2024.) |
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8-Oct-2024 | elnn0dc 9610 |
Membership of an integer in ![]() |
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7-Oct-2024 | pclemub 12286 | Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Jim Kingdon, 7-Oct-2024.) |
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7-Oct-2024 | pclem0 12285 | Lemma for the prime power pre-function's properties. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Jim Kingdon, 7-Oct-2024.) |
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7-Oct-2024 | nn0ltexp2 10688 | Special case of ltexp2 14296 which we use here because we haven't yet defined df-rpcxp 14216 which is used in the current proof of ltexp2 14296. (Contributed by Jim Kingdon, 7-Oct-2024.) |
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6-Oct-2024 | suprzcl2dc 11955 | The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 7931.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 6-Oct-2024.) |
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5-Oct-2024 | zsupssdc 11954 | An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 7931.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.) |
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5-Oct-2024 | suprzubdc 11952 | The supremum of a bounded-above decidable set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.) |
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1-Oct-2024 | infex2g 7032 | Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.) |
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30-Sep-2024 | unbendc 12454 | An unbounded decidable set of positive integers is infinite. (Contributed by NM, 5-May-2005.) (Revised by Jim Kingdon, 30-Sep-2024.) |
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30-Sep-2024 | prmdc 12129 | Primality is decidable. (Contributed by Jim Kingdon, 30-Sep-2024.) |
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30-Sep-2024 | dcfi 6979 | Decidability of a family of propositions indexed by a finite set. (Contributed by Jim Kingdon, 30-Sep-2024.) |
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29-Sep-2024 | ssnnct 12447 |
A decidable subset of ![]() |
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29-Sep-2024 | ssnnctlemct 12446 | Lemma for ssnnct 12447. The result. (Contributed by Jim Kingdon, 29-Sep-2024.) |
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28-Sep-2024 | nninfdcex 11953 | A decidable set of natural numbers has an infimum. (Contributed by Jim Kingdon, 28-Sep-2024.) |
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27-Sep-2024 | infregelbex 9597 | Any lower bound of a set of real numbers with an infimum is less than or equal to the infimum. (Contributed by Jim Kingdon, 27-Sep-2024.) |
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26-Sep-2024 | nninfdclemp1 12450 |
Lemma for nninfdc 12453. Each element of the sequence ![]() |
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26-Sep-2024 | nnminle 12035 | The infimum of a decidable subset of the natural numbers is less than an element of the set. The infimum is also a minimum as shown at nnmindc 12034. (Contributed by Jim Kingdon, 26-Sep-2024.) |
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25-Sep-2024 | nninfdclemcl 12448 | Lemma for nninfdc 12453. (Contributed by Jim Kingdon, 25-Sep-2024.) |
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24-Sep-2024 | nninfdclemlt 12451 | Lemma for nninfdc 12453. The function from nninfdclemf 12449 is strictly monotonic. (Contributed by Jim Kingdon, 24-Sep-2024.) |
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23-Sep-2024 | nninfdc 12453 | An unbounded decidable set of positive integers is infinite. (Contributed by Jim Kingdon, 23-Sep-2024.) |
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23-Sep-2024 | nninfdclemf1 12452 | Lemma for nninfdc 12453. The function from nninfdclemf 12449 is one-to-one. (Contributed by Jim Kingdon, 23-Sep-2024.) |
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23-Sep-2024 | nninfdclemf 12449 |
Lemma for nninfdc 12453. A function from the natural numbers into
![]() |
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23-Sep-2024 | nnmindc 12034 | An inhabited decidable subset of the natural numbers has a minimum. (Contributed by Jim Kingdon, 23-Sep-2024.) |
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19-Sep-2024 | ssomct 12445 |
A decidable subset of ![]() |
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14-Sep-2024 | nnpredlt 4623 | The predecessor (see nnpredcl 4622) of a nonzero natural number is less than (see df-iord 4366) that number. (Contributed by Jim Kingdon, 14-Sep-2024.) |
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13-Sep-2024 | nninfisollemeq 7129 |
Lemma for nninfisol 7130. The case where ![]() ![]() ![]() |
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13-Sep-2024 | nninfisollemne 7128 |
Lemma for nninfisol 7130. A case where ![]() ![]() ![]() |
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13-Sep-2024 | nninfisollem0 7127 |
Lemma for nninfisol 7130. The case where ![]() |
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12-Sep-2024 | nninfisol 7130 |
Finite elements of ℕ∞ are isolated. That is, given a
natural
number and any element of ℕ∞, it is decidable
whether the
natural number (when converted to an element of
ℕ∞) is equal to
the given element of ℕ∞. Stated in an online
post by Martin
Escardo. One way to understand this theorem is that you do not need to
look at an unbounded number of elements of the sequence ![]() ![]() ![]() |
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7-Sep-2024 | eulerthlemfi 12227 |
Lemma for eulerth 12232. The set ![]() |
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7-Sep-2024 | modqexp 10646 | Exponentiation property of the modulo operation, see theorem 5.2(c) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.) |
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5-Sep-2024 | eulerthlemh 12230 |
Lemma for eulerth 12232. A permutation of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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2-Sep-2024 | eulerthlemth 12231 | Lemma for eulerth 12232. The result. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.) |
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2-Sep-2024 | eulerthlema 12229 | Lemma for eulerth 12232. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 2-Sep-2024.) |
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2-Sep-2024 | eulerthlemrprm 12228 |
Lemma for eulerth 12232. ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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30-Aug-2024 | fprodap0f 11643 | A finite product of terms apart from zero is apart from zero. A version of fprodap0 11628 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by Jim Kingdon, 30-Aug-2024.) |
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28-Aug-2024 | fprodrec 11636 | The finite product of reciprocals is the reciprocal of the product. (Contributed by Jim Kingdon, 28-Aug-2024.) |
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26-Aug-2024 | exmidontri2or 7241 | Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.) |
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26-Aug-2024 | exmidontri 7237 | Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.) |
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26-Aug-2024 | ontri2orexmidim 4571 | Ordinal trichotomy implies excluded middle. Closed form of ordtri2or2exmid 4570. (Contributed by Jim Kingdon, 26-Aug-2024.) |
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26-Aug-2024 | ontriexmidim 4521 | Ordinal trichotomy implies excluded middle. Closed form of ordtriexmid 4520. (Contributed by Jim Kingdon, 26-Aug-2024.) |
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25-Aug-2024 | onntri2or 7244 | Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.) |
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25-Aug-2024 | onntri3or 7243 | Double negated ordinal trichotomy. (Contributed by Jim Kingdon, 25-Aug-2024.) |
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25-Aug-2024 | csbcow 3068 | Composition law for chained substitutions into a class. Version of csbco 3067 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 10-Nov-2005.) (Revised by Gino Giotto, 25-Aug-2024.) |
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25-Aug-2024 | cbvreuvw 2709 | Version of cbvreuv 2705 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.) |
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25-Aug-2024 | cbvrexvw 2708 | Version of cbvrexv 2704 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.) |
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25-Aug-2024 | cbvralvw 2707 | Version of cbvralv 2703 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.) |
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25-Aug-2024 | cbvabw 2300 | Version of cbvab 2301 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.) |
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25-Aug-2024 | nfsbv 1947 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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25-Aug-2024 | cbvexvw 1920 | Change bound variable. See cbvexv 1918 for a version with fewer disjoint variable conditions. (Contributed by NM, 19-Apr-2017.) Avoid ax-7 1448. (Revised by Gino Giotto, 25-Aug-2024.) |
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25-Aug-2024 | cbvalvw 1919 | Change bound variable. See cbvalv 1917 for a version with fewer disjoint variable conditions. (Contributed by NM, 9-Apr-2017.) Avoid ax-7 1448. (Revised by Gino Giotto, 25-Aug-2024.) |
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25-Aug-2024 | nfal 1576 |
If ![]() ![]() ![]() ![]() ![]() |
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24-Aug-2024 | gcdcomd 11974 |
The ![]() |
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21-Aug-2024 | dvds2addd 11835 | Deduction form of dvds2add 11831. (Contributed by SN, 21-Aug-2024.) |
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17-Aug-2024 | fprodcl2lem 11612 | Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.) (Revised by Jim Kingdon, 17-Aug-2024.) |
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16-Aug-2024 | if0ab 14493 |
Expression of a conditional class as a class abstraction when the False
alternative is the empty class: in that case, the conditional class is
the extension, in the True alternative, of the condition.
Remark: a consequence which could be formalized is the inclusion
|
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16-Aug-2024 | fprodunsn 11611 |
Multiply in an additional term in a finite product. See also
fprodsplitsn 11640 which is the same but with a ![]() ![]() ![]() ![]() ![]() |
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15-Aug-2024 | bj-charfundcALT 14497 | Alternate proof of bj-charfundc 14496. It was expected to be much shorter since it uses bj-charfun 14495 for the main part of the proof and the rest is basic computations, but these turn out to be lengthy, maybe because of the limited library of available lemmas. (Contributed by BJ, 15-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
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15-Aug-2024 | bj-charfun 14495 |
Properties of the characteristic function on the class ![]() ![]() |
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15-Aug-2024 | fmelpw1o 14494 |
With a formula ![]() ![]() ![]() ![]() ![]() ![]() ![]()
As proved in if0ab 14493, the associated element of |
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15-Aug-2024 | cnstab 8601 |
Equality of complex numbers is stable. Stability here means
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15-Aug-2024 | subap0d 8600 | Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.) (Proof shortened by BJ, 15-Aug-2024.) |
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15-Aug-2024 | ifexd 4484 | Existence of a conditional class (deduction form). (Contributed by BJ, 15-Aug-2024.) |
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15-Aug-2024 | ifelpwun 4483 | Existence of a conditional class, quantitative version (inference form). (Contributed by BJ, 15-Aug-2024.) |
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15-Aug-2024 | ifelpwund 4482 | Existence of a conditional class, quantitative version (deduction form). (Contributed by BJ, 15-Aug-2024.) |
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15-Aug-2024 | ifelpwung 4481 | Existence of a conditional class, quantitative version (closed form). (Contributed by BJ, 15-Aug-2024.) |
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15-Aug-2024 | ifidss 3549 | A conditional class whose two alternatives are equal is included in that alternative. With excluded middle, we can prove it is equal to it. (Contributed by BJ, 15-Aug-2024.) |
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15-Aug-2024 | ifssun 3548 | A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.) |
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12-Aug-2024 | exmidontriimlem2 7220 | Lemma for exmidontriim 7223. (Contributed by Jim Kingdon, 12-Aug-2024.) |
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12-Aug-2024 | exmidontriimlem1 7219 | Lemma for exmidontriim 7223. A variation of r19.30dc 2624. (Contributed by Jim Kingdon, 12-Aug-2024.) |
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11-Aug-2024 | nndc 851 |
Double negation of decidability of a formula. Intuitionistic logic
refutes the negation of decidability (but does not prove decidability) of
any formula.
This should not trick the reader into thinking that
Actually, |
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10-Aug-2024 | exmidontriim 7223 | Excluded middle implies ordinal trichotomy. Lemma 10.4.1 of [HoTT], p. (varies). The proof follows the proof from the HoTT book fairly closely. (Contributed by Jim Kingdon, 10-Aug-2024.) |
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10-Aug-2024 | exmidontriimlem4 7222 |
Lemma for exmidontriim 7223. The induction step for the induction on
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10-Aug-2024 | exmidontriimlem3 7221 |
Lemma for exmidontriim 7223. What we get to do based on induction on
both
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10-Aug-2024 | nnnninf2 7124 |
Canonical embedding of ![]() ![]() |
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10-Aug-2024 | infnninf 7121 |
The point at infinity in ℕ∞ is the constant sequence
equal to
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9-Aug-2024 | ss1o0el1o 6911 |
Reformulation of ss1o0el1 4197 using ![]() ![]() ![]() ![]() |
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9-Aug-2024 | pw1dc0el 6910 | Another equivalent of excluded middle, which is a mere reformulation of the definition. (Contributed by BJ, 9-Aug-2024.) |
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9-Aug-2024 | ss1o0el1 4197 |
A subclass of ![]() ![]() ![]() ![]() ![]() ![]() |
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8-Aug-2024 | pw1dc1 6912 | If, in the set of truth values (the powerset of 1o), equality to 1o is decidable, then excluded middle holds (and conversely). (Contributed by BJ and Jim Kingdon, 8-Aug-2024.) |
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7-Aug-2024 | pw1fin 6909 |
Excluded middle is equivalent to the power set of ![]() |
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7-Aug-2024 | elomssom 4604 | A natural number ordinal is, as a set, included in the set of natural number ordinals. (Contributed by NM, 21-Jun-1998.) Extract this result from the previous proof of elnn 4605. (Revised by BJ, 7-Aug-2024.) |
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6-Aug-2024 | bj-charfunbi 14499 |
In an ambient set ![]() ![]() ![]()
This characterization can be applied to singletons when the set |
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6-Aug-2024 | bj-charfunr 14498 |
If a class ![]() ![]() ![]() ![]() ![]()
The hypothesis imposes that
The theorem would still hold if the codomain of |
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6-Aug-2024 | bj-charfundc 14496 |
Properties of the characteristic function on the class ![]() ![]() ![]() ![]() |
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6-Aug-2024 | prodssdc 11596 | Change the index set to a subset in an upper integer product. (Contributed by Scott Fenton, 11-Dec-2017.) (Revised by Jim Kingdon, 6-Aug-2024.) |
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5-Aug-2024 | fnmptd 14492 | The maps-to notation defines a function with domain (deduction form). (Contributed by BJ, 5-Aug-2024.) |
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5-Aug-2024 | funmptd 14491 |
The maps-to notation defines a function (deduction form).
Note: one should similarly prove a deduction form of funopab4 5253, then prove funmptd 14491 from it, and then prove funmpt 5254 from that: this would reduce global proof length. (Contributed by BJ, 5-Aug-2024.) |
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5-Aug-2024 | bj-dcfal 14443 | The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.) |
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5-Aug-2024 | bj-dctru 14441 | The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.) |
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5-Aug-2024 | bj-stfal 14430 | The false truth value is stable. (Contributed by BJ, 5-Aug-2024.) |
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5-Aug-2024 | bj-sttru 14428 | The true truth value is stable. (Contributed by BJ, 5-Aug-2024.) |
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5-Aug-2024 | prod1dc 11593 | Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.) (Revised by Jim Kingdon, 5-Aug-2024.) |
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5-Aug-2024 | 2ssom 6524 | The ordinal 2 is included in the set of natural number ordinals. (Contributed by BJ, 5-Aug-2024.) |
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2-Aug-2024 | onntri52 7242 | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
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2-Aug-2024 | onntri24 7240 | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
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2-Aug-2024 | onntri45 7239 | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
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2-Aug-2024 | onntri51 7238 | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
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2-Aug-2024 | onntri13 7236 | Double negated ordinal trichotomy. (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
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2-Aug-2024 | onntri35 7235 |
Double negated ordinal trichotomy.
There are five equivalent statements: (1)
Another way of stating this is that EXMID is equivalent
to
trichotomy, either the (Contributed by James E. Hanson and Jim Kingdon, 2-Aug-2024.) |
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1-Aug-2024 | nnral 2467 | The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1649. (Contributed by Jim Kingdon, 1-Aug-2024.) |
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31-Jul-2024 | 3nsssucpw1 7234 |
Negated excluded middle implies that ![]() ![]() |
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31-Jul-2024 | sucpw1nss3 7233 |
Negated excluded middle implies that the successor of the power set of
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30-Jul-2024 | 3nelsucpw1 7232 |
Three is not an element of the successor of the power set of ![]() |
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30-Jul-2024 | sucpw1nel3 7231 |
The successor of the power set of ![]() ![]() |
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30-Jul-2024 | sucpw1ne3 7230 |
Negated excluded middle implies that the successor of the power set of
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30-Jul-2024 | pw1nel3 7229 |
Negated excluded middle implies that the power set of ![]() ![]() |
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30-Jul-2024 | pw1ne3 7228 |
The power set of ![]() |
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30-Jul-2024 | pw1ne1 7227 |
The power set of ![]() |
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30-Jul-2024 | pw1ne0 7226 |
The power set of ![]() |
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29-Jul-2024 | grpcld 12889 | Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.) |
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29-Jul-2024 | pw1on 7224 |
The power set of ![]() |
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28-Jul-2024 | exmidpweq 6908 |
Excluded middle is equivalent to the power set of ![]() ![]() |
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27-Jul-2024 | dcapnconstALT 14745 | Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 14744 by means of dceqnconst 14743. (Contributed by Jim Kingdon, 27-Jul-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
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27-Jul-2024 | reap0 14742 | Real number trichotomy is equivalent to decidability of apartness from zero. (Contributed by Jim Kingdon, 27-Jul-2024.) |
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26-Jul-2024 | nconstwlpolemgt0 14747 | Lemma for nconstwlpo 14749. If one of the terms of series is positive, so is the sum. (Contributed by Jim Kingdon, 26-Jul-2024.) |
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26-Jul-2024 | nconstwlpolem0 14746 | Lemma for nconstwlpo 14749. If all the terms of the series are zero, so is their sum. (Contributed by Jim Kingdon, 26-Jul-2024.) |
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24-Jul-2024 | tridceq 14740 | Real trichotomy implies decidability of real number equality. Or in other words, analytic LPO implies analytic WLPO (see trilpo 14727 and redcwlpo 14739). Thus, this is an analytic analogue to lpowlpo 7165. (Contributed by Jim Kingdon, 24-Jul-2024.) |
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24-Jul-2024 | iswomni0 14735 |
Weak omniscience stated in terms of equality with ![]() |
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24-Jul-2024 | lpowlpo 7165 | LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7164. There is an analogue in terms of analytic omniscience principles at tridceq 14740. (Contributed by Jim Kingdon, 24-Jul-2024.) |
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23-Jul-2024 | nconstwlpolem 14748 | Lemma for nconstwlpo 14749. (Contributed by Jim Kingdon, 23-Jul-2024.) |
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23-Jul-2024 | dceqnconst 14743 | Decidability of real number equality implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See redcwlpo 14739 for more discussion of decidability of real number equality. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) (Revised by Jim Kingdon, 23-Jul-2024.) |
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23-Jul-2024 | redc0 14741 | Two ways to express decidability of real number equality. (Contributed by Jim Kingdon, 23-Jul-2024.) |
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23-Jul-2024 | canth 5828 |
No set ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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22-Jul-2024 | nconstwlpo 14749 |
Existence of a certain non-constant function from reals to integers
implies ![]() ![]() |
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15-Jul-2024 | fprodseq 11590 | The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.) (Revised by Jim Kingdon, 15-Jul-2024.) |
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14-Jul-2024 | rexbid2 2482 | Formula-building rule for restricted existential quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.) |
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14-Jul-2024 | ralbid2 2481 | Formula-building rule for restricted universal quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.) |
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12-Jul-2024 | 2irrexpqap 14332 |
There exist real numbers ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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12-Jul-2024 | 2logb9irrap 14331 | Example for logbgcd1irrap 14324. The logarithm of nine to base two is irrational (in the sense of being apart from any rational number). (Contributed by Jim Kingdon, 12-Jul-2024.) |
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12-Jul-2024 | erlecpbl 12750 | Translate the relation compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.) |
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12-Jul-2024 | ercpbl 12749 | Translate the function compatibility relation to a quotient set. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.) |
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12-Jul-2024 | ercpbllemg 12748 | Lemma for ercpbl 12749. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by AV, 12-Jul-2024.) |
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12-Jul-2024 | divsfvalg 12747 | Value of the function in qusval 12743. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.) |
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11-Jul-2024 | logbgcd1irraplemexp 14322 |
Lemma for logbgcd1irrap 14324. Apartness of ![]() ![]() ![]() ![]() ![]() ![]() |
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11-Jul-2024 | reapef 14135 | Apartness and the exponential function for reals. (Contributed by Jim Kingdon, 11-Jul-2024.) |
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10-Jul-2024 | apcxp2 14294 | Apartness and real exponentiation. (Contributed by Jim Kingdon, 10-Jul-2024.) |
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9-Jul-2024 | logbgcd1irraplemap 14323 | Lemma for logbgcd1irrap 14324. The result, with the rational number expressed as numerator and denominator. (Contributed by Jim Kingdon, 9-Jul-2024.) |
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9-Jul-2024 | apexp1 10697 | Exponentiation and apartness. (Contributed by Jim Kingdon, 9-Jul-2024.) |
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5-Jul-2024 | logrpap0 14234 | The logarithm is apart from 0 if its argument is apart from 1. (Contributed by Jim Kingdon, 5-Jul-2024.) |
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3-Jul-2024 | rplogbval 14299 | Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by Jim Kingdon, 3-Jul-2024.) |
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3-Jul-2024 | logrpap0d 14235 | Deduction form of logrpap0 14234. (Contributed by Jim Kingdon, 3-Jul-2024.) |
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3-Jul-2024 | logrpap0b 14233 | The logarithm is apart from 0 if and only if its argument is apart from 1. (Contributed by Jim Kingdon, 3-Jul-2024.) |
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28-Jun-2024 | 2o01f 14682 |
Mapping zero and one between ![]() ![]() |
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28-Jun-2024 | 012of 14681 |
Mapping zero and one between ![]() ![]() |
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27-Jun-2024 | iooreen 14719 | An open interval is equinumerous to the real numbers. (Contributed by Jim Kingdon, 27-Jun-2024.) |
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27-Jun-2024 | iooref1o 14718 | A one-to-one mapping from the real numbers onto the open unit interval. (Contributed by Jim Kingdon, 27-Jun-2024.) |
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25-Jun-2024 | neapmkvlem 14750 | Lemma for neapmkv 14751. The result, with a few hypotheses broken out for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.) |
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25-Jun-2024 | ismkvnn 14737 | The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 25-Jun-2024.) |
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25-Jun-2024 | ismkvnnlem 14736 | Lemma for ismkvnn 14737. The result, with a hypothesis to give a name to an expression for convenience. (Contributed by Jim Kingdon, 25-Jun-2024.) |
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25-Jun-2024 | enmkvlem 7158 | Lemma for enmkv 7159. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.) |
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24-Jun-2024 | neapmkv 14751 | If negated equality for real numbers implies apartness, Markov's Principle follows. Exercise 11.10 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 24-Jun-2024.) |
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24-Jun-2024 | dcapnconst 14744 |
Decidability of real number apartness implies the existence of a certain
non-constant function from real numbers to integers. Variation of
Exercise 11.6(i) of [HoTT], p. (varies).
See trilpo 14727 for more
discussion of decidability of real number apartness.
This is a weaker form of dceqnconst 14743 and in fact this theorem can be proved using dceqnconst 14743 as shown at dcapnconstALT 14745. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) |
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24-Jun-2024 | enmkv 7159 |
Being Markov is invariant with respect to equinumerosity. For example,
this means that we can express the Markov's Principle as either
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21-Jun-2024 | redcwlpolemeq1 14738 | Lemma for redcwlpo 14739. A biconditionalized version of trilpolemeq1 14724. (Contributed by Jim Kingdon, 21-Jun-2024.) |
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20-Jun-2024 | redcwlpo 14739 |
Decidability of real number equality implies the Weak Limited Principle
of Omniscience (WLPO). We expect that we'd need some form of countable
choice to prove the converse.
Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence is all ones or it is not. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. This real number will equal one if and only if the sequence is all ones (redcwlpolemeq1 14738). Therefore decidability of real number equality would imply decidability of whether the sequence is all ones. Because of this theorem, decidability of real number equality is sometimes called "analytic WLPO". WLPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qdceq 10246 for real numbers. (Contributed by Jim Kingdon, 20-Jun-2024.) |
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20-Jun-2024 | iswomninn 14734 |
Weak omniscience stated in terms of natural numbers. Similar to
iswomnimap 7163 but it will sometimes be more convenient to
use ![]() ![]() ![]() ![]() |
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20-Jun-2024 | iswomninnlem 14733 | Lemma for iswomnimap 7163. The result, with a hypothesis for convenience. (Contributed by Jim Kingdon, 20-Jun-2024.) |
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20-Jun-2024 | enwomni 7167 |
Weak omniscience is invariant with respect to equinumerosity. For
example, this means that we can express the Weak Limited Principle of
Omniscience as either ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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20-Jun-2024 | enwomnilem 7166 | Lemma for enwomni 7167. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.) |
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19-Jun-2024 | rpabscxpbnd 14295 | Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Jim Kingdon, 19-Jun-2024.) |
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16-Jun-2024 | rpcxpsqrt 14278 |
The exponential function with exponent ![]() ![]() ![]() ![]() |
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13-Jun-2024 | rpcxpadd 14262 | Sum of exponents law for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 13-Jun-2024.) |
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12-Jun-2024 | cxpap0 14261 | Complex exponentiation is apart from zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
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12-Jun-2024 | rpcncxpcl 14259 | Closure of the complex power function. (Contributed by Jim Kingdon, 12-Jun-2024.) |
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12-Jun-2024 | rpcxp0 14255 | Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
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12-Jun-2024 | cxpexpnn 14253 | Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
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12-Jun-2024 | cxpexprp 14252 | Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
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12-Jun-2024 | rpcxpef 14251 | Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.) |
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12-Jun-2024 | df-rpcxp 14216 | Define the power function on complex numbers. Because df-relog 14215 is only defined on positive reals, this definition only allows for a base which is a positive real. (Contributed by Jim Kingdon, 12-Jun-2024.) |
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10-Jun-2024 | trirec0xor 14729 |
Version of trirec0 14728 with exclusive-or.
The definition of a discrete field is sometimes stated in terms of exclusive-or but as proved here, this is equivalent to inclusive-or because the two disjuncts cannot be simultaneously true. (Contributed by Jim Kingdon, 10-Jun-2024.) |
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10-Jun-2024 | trirec0 14728 |
Every real number having a reciprocal or equaling zero is equivalent to
real number trichotomy.
This is the key part of the definition of what is known as a discrete field, so "the real numbers are a discrete field" can be taken as an equivalent way to state real trichotomy (see further discussion at trilpo 14727). (Contributed by Jim Kingdon, 10-Jun-2024.) |
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9-Jun-2024 | omniwomnimkv 7164 |
A set is omniscient if and only if it is weakly omniscient and Markov.
The case ![]() ![]() ![]() ![]() ![]() |
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9-Jun-2024 | iswomnimap 7163 | The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.) |
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9-Jun-2024 | iswomni 7162 | The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.) |
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9-Jun-2024 | df-womni 7161 |
A weakly omniscient set is one where we can decide whether a predicate
(here represented by a function ![]() ![]()
In particular, The term WLPO is common in the literature; there appears to be no widespread term for what we are calling a weakly omniscient set. (Contributed by Jim Kingdon, 9-Jun-2024.) |
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1-Jun-2024 | cmnmndd 13109 | A commutative monoid is a monoid. (Contributed by SN, 1-Jun-2024.) |
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1-Jun-2024 | grpmndd 12888 | A group is a monoid. (Contributed by SN, 1-Jun-2024.) |
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29-May-2024 | pw1nct 14688 | A condition which ensures that the powerset of a singleton is not countable. The antecedent here can be referred to as the uniformity principle. Based on Mastodon posts by Andrej Bauer and Rahul Chhabra. (Contributed by Jim Kingdon, 29-May-2024.) |
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28-May-2024 | sssneq 14687 | Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.) |
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26-May-2024 | elpwi2 4158 | Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof shortened by Wolf Lammen, 26-May-2024.) |
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24-May-2024 | dvmptcjx 14122 | Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 24-May-2024.) |
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23-May-2024 | cbvralfw 2694 | Rule used to change bound variables, using implicit substitution. Version of cbvralf 2696 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1507 and ax-bndl 1509 in the proof. (Contributed by NM, 7-Mar-2004.) (Revised by Gino Giotto, 23-May-2024.) |
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22-May-2024 | efltlemlt 14131 | Lemma for eflt 14132. The converse of efltim 11705 plus the epsilon-delta setup. (Contributed by Jim Kingdon, 22-May-2024.) |
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21-May-2024 | eflt 14132 | The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 21-May-2024.) |
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19-May-2024 | apdifflemr 14731 | Lemma for apdiff 14732. (Contributed by Jim Kingdon, 19-May-2024.) |
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18-May-2024 | apdifflemf 14730 |
Lemma for apdiff 14732. Being apart from the point halfway between
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17-May-2024 | apdiff 14732 | The irrationals (reals apart from any rational) are exactly those reals that are a different distance from every rational. (Contributed by Jim Kingdon, 17-May-2024.) |
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16-May-2024 | crnggrpd 13191 | A commutative ring is a group. (Contributed by SN, 16-May-2024.) |
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16-May-2024 | crngringd 13190 | A commutative ring is a ring. (Contributed by SN, 16-May-2024.) |
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16-May-2024 | ringgrpd 13186 | A ring is a group. (Contributed by SN, 16-May-2024.) |
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15-May-2024 | reeff1oleme 14129 | Lemma for reeff1o 14130. (Contributed by Jim Kingdon, 15-May-2024.) |
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14-May-2024 | df-relog 14215 | Define the natural logarithm function. Defining the logarithm on complex numbers is similar to square root - there are ways to define it but they tend to make use of excluded middle. Therefore, we merely define logarithms on positive reals. See http://en.wikipedia.org/wiki/Natural_logarithm and https://en.wikipedia.org/wiki/Complex_logarithm. (Contributed by Jim Kingdon, 14-May-2024.) |
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12-May-2024 | dvdstrd 11836 | The divides relation is transitive, a deduction version of dvdstr 11834. (Contributed by metakunt, 12-May-2024.) |
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7-May-2024 | ioocosf1o 14211 | The cosine function is a bijection when restricted to its principal domain. (Contributed by Mario Carneiro, 12-May-2014.) (Revised by Jim Kingdon, 7-May-2024.) |
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7-May-2024 | cos0pilt1 14209 |
Cosine is between minus one and one on the open interval between zero and
![]() |
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6-May-2024 | cos11 14210 |
Cosine is one-to-one over the closed interval from ![]() ![]() |
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5-May-2024 | omiunct 12444 | The union of a countably infinite collection of countable sets is countable. Theorem 8.1.28 of [AczelRathjen], p. 78. Compare with ctiunct 12440 which has a stronger hypothesis but does not require countable choice. (Contributed by Jim Kingdon, 5-May-2024.) |
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5-May-2024 | ctiunctal 12441 |
Variation of ctiunct 12440 which allows ![]() ![]() |
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3-May-2024 | cc4n 7269 |
Countable choice with a simpler restriction on how every set in the
countable collection needs to be inhabited. That is, compared with
cc4 7268, the hypotheses only require an A(n) for each
value of ![]() ![]() ![]() ![]() ![]() |
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3-May-2024 | cc4f 7267 |
Countable choice by showing the existence of a function ![]() ![]() ![]() |
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1-May-2024 | cc4 7268 |
Countable choice by showing the existence of a function ![]() ![]() ![]() |
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29-Apr-2024 | cc3 7266 | Countable choice using a sequence F(n) . (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Jim Kingdon, 29-Apr-2024.) |
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27-Apr-2024 | cc2 7265 | Countable choice using sequences instead of countable sets. (Contributed by Jim Kingdon, 27-Apr-2024.) |
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27-Apr-2024 | cc2lem 7264 | Lemma for cc2 7265. (Contributed by Jim Kingdon, 27-Apr-2024.) |
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27-Apr-2024 | cc1 7263 | Countable choice in terms of a choice function on a countably infinite set of inhabited sets. (Contributed by Jim Kingdon, 27-Apr-2024.) |
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19-Apr-2024 | omctfn 12443 | Using countable choice to find a sequence of enumerations for a collection of countable sets. Lemma 8.1.27 of [AczelRathjen], p. 77. (Contributed by Jim Kingdon, 19-Apr-2024.) |
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13-Apr-2024 | prodmodclem2 11584 | Lemma for prodmodc 11585. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 13-Apr-2024.) |
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11-Apr-2024 | prodmodclem2a 11583 | Lemma for prodmodc 11585. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.) |
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11-Apr-2024 | prodmodclem3 11582 | Lemma for prodmodc 11585. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.) |
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10-Apr-2024 | jcnd 652 | Deduction joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Apr-2024.) |
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4-Apr-2024 | prodrbdclem 11578 | Lemma for prodrbdc 11581. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 4-Apr-2024.) |
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24-Mar-2024 | prodfdivap 11554 | The quotient of two products. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.) |
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24-Mar-2024 | prodfrecap 11553 | The reciprocal of a finite product. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.) |
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23-Mar-2024 | prodfap0 11552 | 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.) |
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22-Mar-2024 | prod3fmul 11548 | The product of two infinite products. (Contributed by Scott Fenton, 18-Dec-2017.) (Revised by Jim Kingdon, 22-Mar-2024.) |
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21-Mar-2024 | df-proddc 11558 |
Define the product of a series with an index set of integers ![]() |
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19-Mar-2024 | cos02pilt1 14208 |
Cosine is less than one between zero and ![]() ![]() ![]() |
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19-Mar-2024 | cosq34lt1 14207 | Cosine is less than one in the third and fourth quadrants. (Contributed by Jim Kingdon, 19-Mar-2024.) |
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14-Mar-2024 | coseq0q4123 14191 |
Location of the zeroes of cosine in
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14-Mar-2024 | cosq23lt0 14190 | The cosine of a number in the second and third quadrants is negative. (Contributed by Jim Kingdon, 14-Mar-2024.) |
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9-Mar-2024 | pilem3 14140 | Lemma for pi related theorems. (Contributed by Jim Kingdon, 9-Mar-2024.) |
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9-Mar-2024 | exmidonfin 7192 | 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 6871 and nnon 4609. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
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9-Mar-2024 | exmidonfinlem 7191 | Lemma for exmidonfin 7192. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
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8-Mar-2024 | sin0pilem2 14139 | Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.) |
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8-Mar-2024 | sin0pilem1 14138 | Lemma for pi related theorems. (Contributed by Mario Carneiro and Jim Kingdon, 8-Mar-2024.) |
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7-Mar-2024 | cosz12 14137 | Cosine has a zero between 1 and 2. (Contributed by Mario Carneiro and Jim Kingdon, 7-Mar-2024.) |
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6-Mar-2024 | cos12dec 11774 | Cosine is decreasing from one to two. (Contributed by Mario Carneiro and Jim Kingdon, 6-Mar-2024.) |
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2-Mar-2024 | dvrfvald 13300 | Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.) |
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2-Mar-2024 | plusffvalg 12780 | The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.) |
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25-Feb-2024 | insubm 12871 | The intersection of two submonoids is a submonoid. (Contributed by AV, 25-Feb-2024.) |
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25-Feb-2024 | mul2lt0pn 9763 | The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.) |
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25-Feb-2024 | mul2lt0np 9762 | The product of multiplicands of different signs is negative. (Contributed by Jim Kingdon, 25-Feb-2024.) |
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25-Feb-2024 | lt0ap0 8604 | A number which is less than zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.) |
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25-Feb-2024 | negap0d 8587 | The negative of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.) |
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24-Feb-2024 | lt0ap0d 8605 | A real number less than zero is apart from zero. Deduction form. (Contributed by Jim Kingdon, 24-Feb-2024.) |
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20-Feb-2024 | ivthdec 14058 | The intermediate value theorem, decreasing case, for a strictly monotonic function. (Contributed by Jim Kingdon, 20-Feb-2024.) |
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20-Feb-2024 | ivthinclemex 14056 | Lemma for ivthinc 14057. Existence of a number between the lower cut and the upper cut. (Contributed by Jim Kingdon, 20-Feb-2024.) |
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19-Feb-2024 | ivthinclemuopn 14052 | Lemma for ivthinc 14057. The upper cut is open. (Contributed by Jim Kingdon, 19-Feb-2024.) |
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19-Feb-2024 | dedekindicc 14047 | A Dedekind cut identifies a unique real number. Similar to df-inp 7464 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.) |
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19-Feb-2024 | grpsubfvalg 12917 | Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.) |
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18-Feb-2024 | ivthinclemloc 14055 | Lemma for ivthinc 14057. Locatedness. (Contributed by Jim Kingdon, 18-Feb-2024.) |
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18-Feb-2024 | ivthinclemdisj 14054 | Lemma for ivthinc 14057. The lower and upper cuts are disjoint. (Contributed by Jim Kingdon, 18-Feb-2024.) |
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18-Feb-2024 | ivthinclemur 14053 | Lemma for ivthinc 14057. The upper cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.) |
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18-Feb-2024 | ivthinclemlr 14051 | Lemma for ivthinc 14057. The lower cut is rounded. (Contributed by Jim Kingdon, 18-Feb-2024.) |
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18-Feb-2024 | ivthinclemum 14049 | Lemma for ivthinc 14057. The upper cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.) |
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18-Feb-2024 | ivthinclemlm 14048 | Lemma for ivthinc 14057. The lower cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.) |
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17-Feb-2024 | 0subm 12870 | The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.) |
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17-Feb-2024 | mndissubm 12865 | If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the the other monoid is contained in the base set of the monoid, then the (base set of the) monoid is a submonoid of the other monoid. (Contributed by AV, 17-Feb-2024.) |
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17-Feb-2024 | mgmsscl 12779 | If the base set of a magma is contained in the base set of another magma, and the group operation of the magma is the restriction of the group operation of the other magma to its base set, then the base set of the magma is closed under the group operation of the other magma. (Contributed by AV, 17-Feb-2024.) |
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15-Feb-2024 | dedekindicclemeu 14045 | Lemma for dedekindicc 14047. Part of proving uniqueness. (Contributed by Jim Kingdon, 15-Feb-2024.) |
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15-Feb-2024 | dedekindicclemlu 14044 | Lemma for dedekindicc 14047. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 15-Feb-2024.) |
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15-Feb-2024 | dedekindicclemlub 14043 | Lemma for dedekindicc 14047. The set L has a least upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.) |
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15-Feb-2024 | dedekindicclemloc 14042 | Lemma for dedekindicc 14047. The set L is located. (Contributed by Jim Kingdon, 15-Feb-2024.) |
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15-Feb-2024 | dedekindicclemub 14041 | Lemma for dedekindicc 14047. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.) |
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15-Feb-2024 | dedekindicclemuub 14040 | Lemma for dedekindicc 14047. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 15-Feb-2024.) |
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14-Feb-2024 | suplociccex 14039 | An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8029 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.) |
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14-Feb-2024 | suplociccreex 14038 | An inhabited, bounded-above, located set of reals in a closed interval has a supremum. A similar theorem is axsuploc 8029 but that one is for the entire real line rather than a closed interval. (Contributed by Jim Kingdon, 14-Feb-2024.) |
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6-Feb-2024 | ivthinclemlopn 14050 | Lemma for ivthinc 14057. The lower cut is open. (Contributed by Jim Kingdon, 6-Feb-2024.) |
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5-Feb-2024 | ivthinc 14057 | 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.) |
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2-Feb-2024 | dedekindeulemuub 14031 | Lemma for dedekindeu 14037. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-2024.) |
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31-Jan-2024 | dedekindeulemeu 14036 | Lemma for dedekindeu 14037. Part of proving uniqueness. (Contributed by Jim Kingdon, 31-Jan-2024.) |
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31-Jan-2024 | dedekindeulemlu 14035 | Lemma for dedekindeu 14037. There is a number which separates the lower and upper cuts. (Contributed by Jim Kingdon, 31-Jan-2024.) |
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31-Jan-2024 | dedekindeulemlub 14034 | Lemma for dedekindeu 14037. The set L has a least upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.) |
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31-Jan-2024 | dedekindeulemloc 14033 | Lemma for dedekindeu 14037. The set L is located. (Contributed by Jim Kingdon, 31-Jan-2024.) |
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31-Jan-2024 | dedekindeulemub 14032 | Lemma for dedekindeu 14037. The lower cut has an upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.) |
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30-Jan-2024 | axsuploc 8029 | 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 7931 with ordering on the extended reals.) (Contributed by Jim Kingdon, 30-Jan-2024.) |
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30-Jan-2024 | iotam 5208 |
Representation of "the unique element such that ![]() ![]() ![]() |
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29-Jan-2024 | sgrpidmndm 12820 | A semigroup with an identity element which is inhabited is a monoid. Of course there could be monoids with the empty set as identity element, but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.) |
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24-Jan-2024 | axpre-suploclemres 7899 |
Lemma for axpre-suploc 7900. The result. The proof just needs to define
![]() ![]() ![]() ![]() |
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23-Jan-2024 | ax-pre-suploc 7931 |
An inhabited, bounded-above, located set of reals has a supremum.
Locatedness here means that given Although this and ax-caucvg 7930 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 7930. (Contributed by Jim Kingdon, 23-Jan-2024.) |
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23-Jan-2024 | axpre-suploc 7900 |
An inhabited, bounded-above, located set of reals has a supremum.
Locatedness here means that given This construction-dependent theorem should not be referenced directly; instead, use ax-pre-suploc 7931. (Contributed by Jim Kingdon, 23-Jan-2024.) (New usage is discouraged.) |
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22-Jan-2024 | suplocsr 7807 | An inhabited, bounded, located set of signed reals has a supremum. (Contributed by Jim Kingdon, 22-Jan-2024.) |
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21-Jan-2024 | bj-el2oss1o 14462 | Shorter proof of el2oss1o 6443 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
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21-Jan-2024 | ltm1sr 7775 | Adding minus one to a signed real yields a smaller signed real. (Contributed by Jim Kingdon, 21-Jan-2024.) |
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20-Jan-2024 | mndinvmod 12845 | Uniqueness of an inverse element in a monoid, if it exists. (Contributed by AV, 20-Jan-2024.) |
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19-Jan-2024 | suplocsrlempr 7805 |
Lemma for suplocsr 7807. The set ![]() |
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18-Jan-2024 | suplocsrlemb 7804 |
Lemma for suplocsr 7807. The set ![]() |
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16-Jan-2024 | suplocsrlem 7806 |
Lemma for suplocsr 7807. The set ![]() |
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14-Jan-2024 | suplocexprlemlub 7722 | Lemma for suplocexpr 7723. The putative supremum is a least upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.) |
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14-Jan-2024 | suplocexprlemub 7721 | Lemma for suplocexpr 7723. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.) |
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10-Jan-2024 | nfcsbw 3093 | Bound-variable hypothesis builder for substitution into a class. Version of nfcsb 3094 with a disjoint variable condition. (Contributed by Mario Carneiro, 12-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.) |
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10-Jan-2024 | nfsbcdw 3091 | Version of nfsbcd 2982 with a disjoint variable condition. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.) |
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10-Jan-2024 | cbvcsbw 3061 | Version of cbvcsb 3062 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
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10-Jan-2024 | cbvsbcw 2990 | Version of cbvsbc 2991 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
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10-Jan-2024 | cbvrex2vw 2715 | Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvrex2v 2717 with a disjoint variable condition, which does not require ax-13 2150. (Contributed by FL, 2-Jul-2012.) (Revised by Gino Giotto, 10-Jan-2024.) |
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10-Jan-2024 | cbvral2vw 2714 | Change bound variables of double restricted universal quantification, using implicit substitution. Version of cbvral2v 2716 with a disjoint variable condition, which does not require ax-13 2150. (Contributed by NM, 10-Aug-2004.) (Revised by Gino Giotto, 10-Jan-2024.) |
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10-Jan-2024 | cbvralw 2698 | Rule used to change bound variables, using implicit substitution. Version of cbvral 2699 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1507 and ax-bndl 1509 in the proof. (Contributed by NM, 31-Jul-2003.) (Revised by Gino Giotto, 10-Jan-2024.) |
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10-Jan-2024 | cbvrexfw 2695 | Rule used to change bound variables, using implicit substitution. Version of cbvrexf 2697 with a disjoint variable condition, which does not require ax-13 2150. (Contributed by FL, 27-Apr-2008.) (Revised by Gino Giotto, 10-Jan-2024.) |
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10-Jan-2024 | nfralw 2514 |
Bound-variable hypothesis builder for restricted quantification. See
nfralya 2517 for a version with ![]() ![]() ![]() ![]() |
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10-Jan-2024 | nfraldw 2509 |
Not-free for restricted universal quantification where ![]() ![]() ![]() ![]() |
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10-Jan-2024 | nfabdw 2338 | Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2339 with a disjoint variable condition. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.) |
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10-Jan-2024 | cbv2w 1750 | Rule used to change bound variables, using implicit substitution. Version of cbv2 1749 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by Gino Giotto, 10-Jan-2024.) |
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9-Jan-2024 | suplocexprlemloc 7719 | Lemma for suplocexpr 7723. The putative supremum is located. (Contributed by Jim Kingdon, 9-Jan-2024.) |
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9-Jan-2024 | suplocexprlemdisj 7718 | Lemma for suplocexpr 7723. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.) |
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9-Jan-2024 | suplocexprlemru 7717 | Lemma for suplocexpr 7723. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.) |
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9-Jan-2024 | suplocexprlemrl 7715 | Lemma for suplocexpr 7723. The lower cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-Jan-2024.) |
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9-Jan-2024 | suplocexprlem2b 7712 | Lemma for suplocexpr 7723. Expression for the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.) |
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9-Jan-2024 | suplocexprlemell 7711 | Lemma for suplocexpr 7723. Membership in the lower cut of the putative supremum. (Contributed by Jim Kingdon, 9-Jan-2024.) |
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7-Jan-2024 | suplocexpr 7723 | An inhabited, bounded-above, located set of positive reals has a supremum. (Contributed by Jim Kingdon, 7-Jan-2024.) |
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7-Jan-2024 | suplocexprlemex 7720 | Lemma for suplocexpr 7723. The putative supremum is a positive real. (Contributed by Jim Kingdon, 7-Jan-2024.) |
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7-Jan-2024 | suplocexprlemmu 7716 | Lemma for suplocexpr 7723. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.) |
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7-Jan-2024 | suplocexprlemml 7714 | Lemma for suplocexpr 7723. The lower cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-Jan-2024.) |
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7-Jan-2024 | suplocexprlemss 7713 |
Lemma for suplocexpr 7723. ![]() |
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5-Jan-2024 | dedekindicclemicc 14046 |
Lemma for dedekindicc 14047. Same as dedekindicc 14047, except that we
merely show ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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5-Jan-2024 | dedekindeu 14037 | A Dedekind cut identifies a unique real number. Similar to df-inp 7464 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.) |
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31-Dec-2023 | dvmptsubcn 14121 | Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.) |
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31-Dec-2023 | dvmptnegcn 14120 | Function-builder for derivative, product rule for negatives. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.) |
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31-Dec-2023 | dvmptcmulcn 14119 | Function-builder for derivative, product rule for constant multiplier. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.) |
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31-Dec-2023 | rinvmod 13110 | Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovimo 6067. (Contributed by AV, 31-Dec-2023.) |
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31-Dec-2023 | brm 4053 | If two sets are in a binary relation, the relation is inhabited. (Contributed by Jim Kingdon, 31-Dec-2023.) |
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30-Dec-2023 | dvmptccn 14115 | Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.) |
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30-Dec-2023 | dvmptidcn 14114 | Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 30-Dec-2023.) |
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29-Dec-2023 | mndbn0 12831 | The base set of a monoid is not empty. (It is also inhabited, as seen at mndidcl 12830). Statement in [Lang] p. 3. (Contributed by AV, 29-Dec-2023.) |
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26-Dec-2023 | lidrididd 12800 |
If there is a left and right identity element for any binary operation
(group operation) ![]() |
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26-Dec-2023 | lidrideqd 12799 |
If there is a left and right identity element for any binary operation
(group operation) ![]() |
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25-Dec-2023 | ctfoex 7116 | A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.) |
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23-Dec-2023 | enct 12433 | Countability is invariant relative to equinumerosity. (Contributed by Jim Kingdon, 23-Dec-2023.) |
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23-Dec-2023 | enctlem 12432 | Lemma for enct 12433. One direction of the biconditional. (Contributed by Jim Kingdon, 23-Dec-2023.) |
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23-Dec-2023 | omct 7115 |
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21-Dec-2023 | dvcoapbr 14107 |
The chain rule for derivatives at a point. The
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19-Dec-2023 | apsscn 8603 | The points apart from a given point are complex numbers. (Contributed by Jim Kingdon, 19-Dec-2023.) |
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19-Dec-2023 | aprcl 8602 | Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.) |
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18-Dec-2023 | limccoap 14083 |
Composition of two limits. This theorem is only usable in the case
where ![]() ![]() ![]() |
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16-Dec-2023 | cnreim 10986 | Complex apartness in terms of real and imaginary parts. See also apreim 8559 which is similar but with different notation. (Contributed by Jim Kingdon, 16-Dec-2023.) |
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14-Dec-2023 | cnopnap 14030 | The complex numbers apart from a given complex number form an open set. (Contributed by Jim Kingdon, 14-Dec-2023.) |
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14-Dec-2023 | cnovex 13632 | The class of all continuous functions from a topology to another is a set. (Contributed by Jim Kingdon, 14-Dec-2023.) |
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13-Dec-2023 | reopnap 13974 | The real numbers apart from a given real number form an open set. (Contributed by Jim Kingdon, 13-Dec-2023.) |
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12-Dec-2023 | cnopncntop 13973 | The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.) |
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12-Dec-2023 | unicntopcntop 13972 | The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by Jim Kingdon, 12-Dec-2023.) |
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4-Dec-2023 | bj-pm2.18st 14438 | Clavius law for stable formulas. See pm2.18dc 855. (Contributed by BJ, 4-Dec-2023.) |
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4-Dec-2023 | bj-nnclavius 14425 | Clavius law with doubly negated consequent. (Contributed by BJ, 4-Dec-2023.) |
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2-Dec-2023 | dvmulxx 14104 | The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 14102. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 2-Dec-2023.) |
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1-Dec-2023 | dvmulxxbr 14102 | The product rule for derivatives at a point. For the (simpler but more limited) function version, see dvmulxx 14104. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 1-Dec-2023.) |
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29-Nov-2023 | subctctexmid 14686 | If every subcountable set is countable and Markov's principle holds, excluded middle follows. Proposition 2.6 of [BauerSwan], p. 14:4. The proof is taken from that paper. (Contributed by Jim Kingdon, 29-Nov-2023.) |
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29-Nov-2023 | ismkvnex 7152 |
The predicate of being Markov stated in terms of double negation and
comparison with ![]() |
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28-Nov-2023 | ccfunen 7262 | Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.) |
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28-Nov-2023 | exmid1stab 4208 |
If every proposition is stable, excluded middle follows. We are
thinking of ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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27-Nov-2023 | df-cc 7261 | The expression CCHOICE will be used as a readable shorthand for any form of countable choice, analogous to df-ac 7204 for full choice. (Contributed by Jim Kingdon, 27-Nov-2023.) |
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26-Nov-2023 | offeq 6095 | Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.) |
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25-Nov-2023 | dvaddxx 14103 | The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 14101. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.) |
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25-Nov-2023 | dvaddxxbr 14101 |
The sum rule for derivatives at a point. That is, if the derivative
of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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25-Nov-2023 | dcnn 848 | Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 842. The relation between dcn 842 and dcnn 848 is analogous to that between notnot 629 and notnotnot 634 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 848 means that a proposition is testable if and only if its negation is testable, and dcn 842 means that decidability implies testability. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof shortened by BJ, 25-Nov-2023.) |
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24-Nov-2023 | bj-dcst 14449 | Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.) |
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24-Nov-2023 | bj-nnbidc 14445 | If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 14432. (Contributed by BJ, 24-Nov-2023.) |
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24-Nov-2023 | bj-dcstab 14444 | A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.) |
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24-Nov-2023 | bj-fadc 14442 | A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.) |
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24-Nov-2023 | bj-trdc 14440 | A provable formula is decidable. (Contributed by BJ, 24-Nov-2023.) |
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24-Nov-2023 | bj-stal 14437 | The universal quantification of a stable formula is stable. See bj-stim 14434 for implication, stabnot 833 for negation, and bj-stan 14435 for conjunction. (Contributed by BJ, 24-Nov-2023.) |
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24-Nov-2023 | bj-stand 14436 | The conjunction of two stable formulas is stable. Deduction form of bj-stan 14435. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 14435 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.) |
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24-Nov-2023 | bj-stan 14435 | The conjunction of two stable formulas is stable. See bj-stim 14434 for implication, stabnot 833 for negation, and bj-stal 14437 for universal quantification. (Contributed by BJ, 24-Nov-2023.) |
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24-Nov-2023 | bj-stim 14434 | A conjunction with a stable consequent is stable. See stabnot 833 for negation , bj-stan 14435 for conjunction , and bj-stal 14437 for universal quantification. (Contributed by BJ, 24-Nov-2023.) |
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24-Nov-2023 | bj-nnbist 14432 |
If a formula is not refutable, then it is stable if and only if it is
provable. By double-negation translation, if ![]() ![]() ![]() ![]() ![]() ![]() |
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24-Nov-2023 | bj-fast 14429 | A refutable formula is stable. (Contributed by BJ, 24-Nov-2023.) |
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24-Nov-2023 | bj-trst 14427 | A provable formula is stable. (Contributed by BJ, 24-Nov-2023.) |
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24-Nov-2023 | bj-nnan 14424 | The double negation of a conjunction implies the conjunction of the double negations. (Contributed by BJ, 24-Nov-2023.) |
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24-Nov-2023 | bj-nnim 14423 | The double negation of an implication implies the implication with the consequent doubly negated. (Contributed by BJ, 24-Nov-2023.) |
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24-Nov-2023 | bj-nnsn 14421 | As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.) |
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24-Nov-2023 | nnal 1649 | The double negation of a universal quantification implies the universal quantification of the double negation. (Contributed by BJ, 24-Nov-2023.) |
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22-Nov-2023 | ofvalg 6091 | Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.) |
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21-Nov-2023 | exmidac 7207 | The axiom of choice implies excluded middle. See acexmid 5873 for more discussion of this theorem and a way of stating it without using CHOICE or EXMID. (Contributed by Jim Kingdon, 21-Nov-2023.) |
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21-Nov-2023 | exmidaclem 7206 | Lemma for exmidac 7207. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.) |
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21-Nov-2023 | exmid1dc 4200 |
A convenience theorem for proving that something implies EXMID.
Think of this as an alternative to using a proposition, as in proofs
like undifexmid 4193 or ordtriexmid 4520. In this context ![]() ![]() ![]() ![]() ![]() |
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20-Nov-2023 | acfun 7205 | A convenient form of choice. The goal here is to state choice as the existence of a choice function on a set of inhabited sets, while making full use of our notation around functions and function values. (Contributed by Jim Kingdon, 20-Nov-2023.) |
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18-Nov-2023 | condc 853 |
Contraposition of a decidable proposition.
This theorem swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky". This theorem (without the decidability condition, of course) is called Transp or "the principle of transposition" in Principia Mathematica (Theorem *2.17 of [WhiteheadRussell] p. 103) and is Axiom A3 of [Margaris] p. 49. We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning. (Contributed by Jim Kingdon, 13-Mar-2018.) (Proof shortened by BJ, 18-Nov-2023.) |
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18-Nov-2023 | stdcn 847 | A formula is stable if and only if the decidability of its negation implies its decidability. Note that the right-hand side of this biconditional is the converse of dcn 842. (Contributed by BJ, 18-Nov-2023.) |
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17-Nov-2023 | cnplimclemr 14074 | Lemma for cnplimccntop 14075. The reverse direction. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.) |
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17-Nov-2023 | cnplimclemle 14073 | Lemma for cnplimccntop 14075. Satisfying the epsilon condition for continuity. (Contributed by Mario Carneiro and Jim Kingdon, 17-Nov-2023.) |
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14-Nov-2023 | limccnp2cntop 14082 | The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 14-Nov-2023.) |
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10-Nov-2023 | rpmaxcl 11231 | The maximum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 10-Nov-2023.) |
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9-Nov-2023 | limccnp2lem 14081 | Lemma for limccnp2cntop 14082. This is most of the result, expressed in epsilon-delta form, with a large number of hypotheses so that lengthy expressions do not need to be repeated. (Contributed by Jim Kingdon, 9-Nov-2023.) |
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4-Nov-2023 | ellimc3apf 14065 | Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 4-Nov-2023.) |
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3-Nov-2023 | limcmpted 14068 | Express the limit operator for a function defined by a mapping, via epsilon-delta. (Contributed by Jim Kingdon, 3-Nov-2023.) |
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1-Nov-2023 | unct 12442 | The union of two countable sets is countable. Corollary 8.1.20 of [AczelRathjen], p. 75. (Contributed by Jim Kingdon, 1-Nov-2023.) |
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31-Oct-2023 | ctiunct 12440 |
A sequence of enumerations gives an enumeration of the union. We refer
to "sequence of enumerations" rather than "countably many
countable
sets" because the hypothesis provides more than countability for
each
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For "countably many countable sets" the key hypothesis would
be
Compare with the case of two sets instead of countably many, as seen at unct 12442, which says that the union of two countable sets is countable .
The proof proceeds by mapping a natural number to a pair of natural
numbers (by xpomen 12395) and using the first number to map to an
element
(Contributed by Jim Kingdon, 31-Oct-2023.) |
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30-Oct-2023 | ctssdccl 7109 |
A mapping from a decidable subset of the natural numbers onto a
countable set. This is similar to one direction of ctssdc 7111 but
expressed in terms of classes rather than ![]() |
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28-Oct-2023 | ctiunctlemfo 12439 | Lemma for ctiunct 12440. (Contributed by Jim Kingdon, 28-Oct-2023.) |
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28-Oct-2023 | ctiunctlemf 12438 | Lemma for ctiunct 12440. (Contributed by Jim Kingdon, 28-Oct-2023.) |
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28-Oct-2023 | ctiunctlemudc 12437 | Lemma for ctiunct 12440. (Contributed by Jim Kingdon, 28-Oct-2023.) |
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28-Oct-2023 | ctiunctlemuom 12436 | Lemma for ctiunct 12440. (Contributed by Jim Kingdon, 28-Oct-2023.) |
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28-Oct-2023 | ctiunctlemu2nd 12435 | Lemma for ctiunct 12440. (Contributed by Jim Kingdon, 28-Oct-2023.) |
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28-Oct-2023 | ctiunctlemu1st 12434 | Lemma for ctiunct 12440. (Contributed by Jim Kingdon, 28-Oct-2023.) |
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28-Oct-2023 | pm2.521gdc 868 | A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107, under a decidability condition. (Contributed by BJ, 28-Oct-2023.) |
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28-Oct-2023 | stdcndc 845 | A formula is decidable if and only if its negation is decidable and it is stable (that is, it is testable and stable). (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof shortened by BJ, 28-Oct-2023.) |
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28-Oct-2023 | conax1k 654 | Weakening of conax1 653. General instance of pm2.51 655 and of pm2.52 656. (Contributed by BJ, 28-Oct-2023.) |
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28-Oct-2023 | conax1 653 | Contrapositive of ax-1 6. (Contributed by BJ, 28-Oct-2023.) |
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25-Oct-2023 | divcnap 13991 | Complex number division is a continuous function, when the second argument is apart from zero. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Jim Kingdon, 25-Oct-2023.) |
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23-Oct-2023 | cnm 7830 | A complex number is an inhabited set. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by Jim Kingdon, 23-Oct-2023.) (New usage is discouraged.) |
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23-Oct-2023 | oprssdmm 6171 | Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.) |
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22-Oct-2023 | addcncntoplem 13987 | Lemma for addcncntop 13988, subcncntop 13989, and mulcncntop 13990. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 22-Oct-2023.) |
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22-Oct-2023 | txmetcnp 13954 | Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by Jim Kingdon, 22-Oct-2023.) |
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22-Oct-2023 | xmetxpbl 13944 |
The maximum metric (Chebyshev distance) on the product of two sets,
expressed in terms of balls centered on a point ![]() ![]() |
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15-Oct-2023 | xmettxlem 13945 | Lemma for xmettx 13946. (Contributed by Jim Kingdon, 15-Oct-2023.) |
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11-Oct-2023 | xmettx 13946 | The maximum metric (Chebyshev distance) on the product of two sets, expressed as a binary topological product. (Contributed by Jim Kingdon, 11-Oct-2023.) |
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11-Oct-2023 | xmetxp 13943 | The maximum metric (Chebyshev distance) on the product of two sets. (Contributed by Jim Kingdon, 11-Oct-2023.) |
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7-Oct-2023 | df-iress 12469 |
Define a multifunction restriction operator for extensible structures,
which can be used to turn statements about rings into statements about
subrings, modules into submodules, etc. This definition knows nothing
about individual structures and merely truncates the ![]() (Credit for this operator, as well as the 2023 modification for iset.mm, goes to Mario Carneiro.) (Contributed by Stefan O'Rear, 29-Nov-2014.) (Revised by Jim Kingdon, 7-Oct-2023.) |
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29-Sep-2023 | syl2anc2 412 | Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023.) |
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27-Sep-2023 | fnpr2ob 12758 | Biconditional version of fnpr2o 12757. (Contributed by Jim Kingdon, 27-Sep-2023.) |
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25-Sep-2023 | xpsval 12770 | Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.) |
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25-Sep-2023 | fvpr1o 12760 | The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.) |
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25-Sep-2023 | fvpr0o 12759 | The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.) |
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25-Sep-2023 | fnpr2o 12757 |
Function with a domain of ![]() |
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25-Sep-2023 | df-xps 12724 | Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.) |
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12-Sep-2023 | pwntru 4199 | A slight strengthening of pwtrufal 14683. (Contributed by Mario Carneiro and Jim Kingdon, 12-Sep-2023.) |
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11-Sep-2023 | pwtrufal 14683 |
A subset of the singleton ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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9-Sep-2023 | mathbox 14420 |
(This theorem is a dummy placeholder for these guidelines. The label
of this theorem, "mathbox", is hard-coded into the Metamath
program to
identify the start of the mathbox section for web page generation.)
A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of iset.mm. For contributors: By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of iset.mm. Guidelines: Mathboxes in iset.mm follow the same practices as in set.mm, so refer to the mathbox guidelines there for more details. (Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (New usage is discouraged.) |
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6-Sep-2023 | djuexb 7042 | The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.) |
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3-Sep-2023 | pwf1oexmid 14685 |
An exercise related to ![]() |
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3-Sep-2023 | pwle2 14684 |
An exercise related to ![]() |
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30-Aug-2023 | isomninn 14715 |
Omniscience stated in terms of natural numbers. Similar to isomnimap 7134
but it will sometimes be more convenient to use ![]() ![]() ![]() ![]() |
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30-Aug-2023 | isomninnlem 14714 | Lemma for isomninn 14715. The result, with a hypothesis to provide a convenient notation. (Contributed by Jim Kingdon, 30-Aug-2023.) |
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28-Aug-2023 | trilpolemisumle 14722 | Lemma for trilpo 14727. An upper bound for the sum of the digits beyond a certain point. (Contributed by Jim Kingdon, 28-Aug-2023.) |
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25-Aug-2023 | cvgcmp2n 14717 | A comparison test for convergence of a real infinite series. (Contributed by Jim Kingdon, 25-Aug-2023.) |
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25-Aug-2023 | cvgcmp2nlemabs 14716 |
Lemma for cvgcmp2n 14717. The partial sums get closer to each other
as
we go further out. The proof proceeds by rewriting
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24-Aug-2023 | trilpolemclim 14720 | Lemma for trilpo 14727. Convergence of the series. (Contributed by Jim Kingdon, 24-Aug-2023.) |
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23-Aug-2023 | trilpo 14727 |
Real number trichotomy implies the Limited Principle of Omniscience
(LPO). We expect that we'd need some form of countable choice to prove
the converse.
Here's the outline of the proof. Given an infinite sequence F of zeroes and ones, we need to show the sequence contains a zero or it is all ones. Construct a real number A whose representation in base two consists of a zero, a decimal point, and then the numbers of the sequence. Compare it with one using trichotomy. The three cases from trichotomy are trilpolemlt1 14725 (which means the sequence contains a zero), trilpolemeq1 14724 (which means the sequence is all ones), and trilpolemgt1 14723 (which is not possible). Equivalent ways to state real number trichotomy (sometimes called "analytic LPO") include decidability of real number apartness (see triap 14713) or that the real numbers are a discrete field (see trirec0 14728). LPO is known to not be provable in IZF (and most constructive foundations), so this theorem establishes that we will be unable to prove an analogue to qtri3or 10242 for real numbers. (Contributed by Jim Kingdon, 23-Aug-2023.) |
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23-Aug-2023 | trilpolemres 14726 | Lemma for trilpo 14727. The result. (Contributed by Jim Kingdon, 23-Aug-2023.) |
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23-Aug-2023 | trilpolemlt1 14725 |
Lemma for trilpo 14727. The ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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23-Aug-2023 | trilpolemeq1 14724 |
Lemma for trilpo 14727. The ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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23-Aug-2023 | trilpolemgt1 14723 |
Lemma for trilpo 14727. The ![]() ![]() ![]() |
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23-Aug-2023 | trilpolemcl 14721 | Lemma for trilpo 14727. The sum exists. (Contributed by Jim Kingdon, 23-Aug-2023.) |
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23-Aug-2023 | triap 14713 | Two ways of stating real number trichotomy. (Contributed by Jim Kingdon, 23-Aug-2023.) |
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19-Aug-2023 | djuenun 7210 | Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.) |
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16-Aug-2023 | ctssdclemr 7110 | Lemma for ctssdc 7111. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.) |
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16-Aug-2023 | ctssdclemn0 7108 |
Lemma for ctssdc 7111. The ![]() ![]() ![]() ![]() |
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15-Aug-2023 | ctssexmid 7147 | The decidability condition in ctssdc 7111 is needed. More specifically, ctssdc 7111 minus that condition, plus the Limited Principle of Omniscience (LPO), implies excluded middle. (Contributed by Jim Kingdon, 15-Aug-2023.) |
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15-Aug-2023 | ctssdc 7111 | A set is countable iff there is a surjection from a decidable subset of the natural numbers onto it. The decidability condition is needed as shown at ctssexmid 7147. (Contributed by Jim Kingdon, 15-Aug-2023.) |
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14-Aug-2023 | mpoexw 6213 | Weak version of mpoex 6214 that holds without ax-coll 4118. If the domain and codomain of an operation given by maps-to notation are sets, the operation is a set. (Contributed by Rohan Ridenour, 14-Aug-2023.) |
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13-Aug-2023 | grpinvfvalg 12914 | The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) (Revised by Rohan Ridenour, 13-Aug-2023.) |
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13-Aug-2023 | ltntri 8084 |
Negative trichotomy property for real numbers. It is well known that we
cannot prove real number trichotomy, ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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13-Aug-2023 | mptexw 6113 | Weak version of mptex 5742 that holds without ax-coll 4118. If the domain and codomain of a function given by maps-to notation are sets, the function is a set. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
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13-Aug-2023 | funexw 6112 | Weak version of funex 5739 that holds without ax-coll 4118. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.) |
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11-Aug-2023 | qnnen 12431 | The rational numbers are countably infinite. Corollary 8.1.23 of [AczelRathjen], p. 75. This is Metamath 100 proof #3. (Contributed by Jim Kingdon, 11-Aug-2023.) |
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10-Aug-2023 | ctinfomlemom 12427 |
Lemma for ctinfom 12428. Converting between ![]() ![]() |
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9-Aug-2023 | difinfsnlem 7097 |
Lemma for difinfsn 7098. The case where we need to swap ![]() ![]() ![]() ![]() ![]() ![]() |
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8-Aug-2023 | difinfinf 7099 | An infinite set minus a finite subset is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.) |
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8-Aug-2023 | difinfsn 7098 | An infinite set minus one element is infinite. We require that the set has decidable equality. (Contributed by Jim Kingdon, 8-Aug-2023.) |
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7-Aug-2023 | ctinf 12430 | A set is countably infinite if and only if it has decidable equality, is countable, and is infinite. (Contributed by Jim Kingdon, 7-Aug-2023.) |
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7-Aug-2023 | inffinp1 12429 | An infinite set contains an element not contained in a given finite subset. (Contributed by Jim Kingdon, 7-Aug-2023.) |
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7-Aug-2023 | ctinfom 12428 |
A condition for a set being countably infinite. Restates ennnfone 12425 in
terms of ![]() ![]() |
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6-Aug-2023 | rerestcntop 13986 | The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.) |
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6-Aug-2023 | tgioo2cntop 13985 | The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Mario Carneiro, 13-Aug-2014.) (Revised by Jim Kingdon, 6-Aug-2023.) |
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4-Aug-2023 | nninffeq 14705 |
Equality of two functions on ℕ∞ which agree at every
integer and
at the point at infinity. From an online post by Martin Escardo.
Remark: the last two hypotheses can be grouped into one,
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3-Aug-2023 | txvalex 13690 |
Existence of the binary topological product. If ![]() ![]() |
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3-Aug-2023 | ablgrpd 13092 | An Abelian group is a group, deduction form of ablgrp 13091. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | 1nsgtrivd 13077 | A group with exactly one normal subgroup is trivial. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | triv1nsgd 13076 | A trivial group has exactly one normal subgroup. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | trivnsgd 13075 | The only normal subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | 0idnsgd 13074 | The whole group and the zero subgroup are normal subgroups of a group. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | trivsubgsnd 13059 | The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | trivsubgd 13058 | The only subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | mulgcld 13003 | Deduction associated with mulgcl 12999. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | hashfingrpnn 12908 | A finite group has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | hashfinmndnn 12832 | A finite monoid has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | dvdsgcdidd 11994 | The greatest common divisor of a positive integer and another integer it divides is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | gcdmultipled 11993 |
The greatest common divisor of a nonnegative integer ![]() ![]() |
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3-Aug-2023 | fihashelne0d 10776 | A finite set with an element has nonzero size. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | phpeqd 6931 | Corollary of the Pigeonhole Principle using equality. Strengthening of phpm 6864 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | enpr2d 6816 | A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | elrnmpt2d 4882 | Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | elrnmptdv 4881 | Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | rspcime 2848 | Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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3-Aug-2023 | neqcomd 2182 | Commute an inequality. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
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2-Aug-2023 | dvid 14098 | Derivative of the identity function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.) |
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2-Aug-2023 | dvconst 14097 | Derivative of a constant function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.) |
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2-Aug-2023 | dvidlemap 14096 | Lemma for dvid 14098 and dvconst 14097. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 2-Aug-2023.) |
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2-Aug-2023 | diveqap1bd 8792 | If two complex numbers are equal, their quotient is one. One-way deduction form of diveqap1 8661. Converse of diveqap1d 8754. (Contributed by David Moews, 28-Feb-2017.) (Revised by Jim Kingdon, 2-Aug-2023.) |
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31-Jul-2023 | mul0inf 11248 | Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11070 and mulap0bd 8613 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.) |
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31-Jul-2023 | mul0eqap 8626 | If two numbers are apart from each other and their product is zero, one of them must be zero. (Contributed by Jim Kingdon, 31-Jul-2023.) |
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31-Jul-2023 | apcon4bid 8580 | Contrapositive law deduction for apartness. (Contributed by Jim Kingdon, 31-Jul-2023.) |
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30-Jul-2023 | uzennn 10435 | An upper integer set is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 30-Jul-2023.) |
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30-Jul-2023 | djuen 7209 | Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.) |
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30-Jul-2023 | endjudisj 7208 | Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by Jim Kingdon, 30-Jul-2023.) |
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30-Jul-2023 | eninr 7096 | Equinumerosity of a set and its image under right injection. (Contributed by Jim Kingdon, 30-Jul-2023.) |
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30-Jul-2023 | eninl 7095 | Equinumerosity of a set and its image under left injection. (Contributed by Jim Kingdon, 30-Jul-2023.) |
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29-Jul-2023 | exmidunben 12426 |
If any unbounded set of positive integers is equinumerous to ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
29-Jul-2023 | exmidsssnc 4203 |
Excluded middle in terms of subsets of a singleton. This is similar to
exmid01 4198 but lets you choose any set as the element of
the singleton
rather than just ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
28-Jul-2023 | dvfcnpm 14095 | The derivative is a function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 28-Jul-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
28-Jul-2023 | dvfpm 14094 | The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 28-Jul-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
23-Jul-2023 | ennnfonelemhdmp1 12409 | Lemma for ennnfone 12425. Domain at a successor where we need to add an element to the sequence. (Contributed by Jim Kingdon, 23-Jul-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
23-Jul-2023 | ennnfonelemp1 12406 |
Lemma for ennnfone 12425. Value of ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
22-Jul-2023 | nntr2 6503 | Transitive law for natural numbers. (Contributed by Jim Kingdon, 22-Jul-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
22-Jul-2023 | nnsssuc 6502 | A natural number is a subset of another natural number if and only if it belongs to its successor. (Contributed by Jim Kingdon, 22-Jul-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
21-Jul-2023 | ennnfoneleminc 12411 |
Lemma for ennnfone 12425. We only add elements to ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
20-Jul-2023 | ennnfonelemg 12403 |
Lemma for ennnfone 12425. Closure for ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
20-Jul-2023 | ennnfonelemjn 12402 |
Lemma for ennnfone 12425. Non-initial state for ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
20-Jul-2023 | ennnfonelemj0 12401 |
Lemma for ennnfone 12425. Initial state for ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
20-Jul-2023 | seqp1cd 10465 |
Value of the sequence builder function at a successor. A version of
seq3p1 10461 which provides two classes ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
20-Jul-2023 | seqovcd 10462 | A closure law for the recursive sequence builder. This is a lemma for theorems such as seqf2 10463 and seq1cd 10464 and is unlikely to be needed once such theorems are proved. (Contributed by Jim Kingdon, 20-Jul-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
19-Jul-2023 | ennnfonelemhom 12415 |
Lemma for ennnfone 12425. The sequences in ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
19-Jul-2023 | ennnfonelemex 12414 |
Lemma for ennnfone 12425. Extending the sequence ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
19-Jul-2023 | ennnfonelemkh 12412 | Lemma for ennnfone 12425. Because we add zero or one entries for each new index, the length of each sequence is no greater than its index. (Contributed by Jim Kingdon, 19-Jul-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
19-Jul-2023 | ennnfonelemom 12408 |
Lemma for ennnfone 12425. ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
19-Jul-2023 | ennnfonelem1 12407 | Lemma for ennnfone 12425. Second value. (Contributed by Jim Kingdon, 19-Jul-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
19-Jul-2023 | seq1cd 10464 |
Initial value of the recursive sequence builder. A version of seq3-1 10459
which provides two classes ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
17-Jul-2023 | ennnfonelemhf1o 12413 |
Lemma for ennnfone 12425. Each of the functions in ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
16-Jul-2023 | ennnfonelemen 12421 | Lemma for ennnfone 12425. The result. (Contributed by Jim Kingdon, 16-Jul-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
16-Jul-2023 | ennnfonelemdm 12420 |
Lemma for ennnfone 12425. The function ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
16-Jul-2023 | ennnfonelemrn 12419 |
Lemma for ennnfone 12425. ![]() ![]() |
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16-Jul-2023 | ennnfonelemf1 12418 |
Lemma for ennnfone 12425. ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
16-Jul-2023 | ennnfonelemfun 12417 |
Lemma for ennnfone 12425. ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
16-Jul-2023 | ennnfonelemrnh 12416 | Lemma for ennnfone 12425. A consequence of ennnfonelemss 12410. (Contributed by Jim Kingdon, 16-Jul-2023.) |
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15-Jul-2023 | ennnfonelemss 12410 |
Lemma for ennnfone 12425. We only add elements to ![]() |
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15-Jul-2023 | ennnfonelem0 12405 | Lemma for ennnfone 12425. Initial value. (Contributed by Jim Kingdon, 15-Jul-2023.) |
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15-Jul-2023 | ennnfonelemk 12400 | Lemma for ennnfone 12425. (Contributed by Jim Kingdon, 15-Jul-2023.) |
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15-Jul-2023 | ennnfonelemdc 12399 | Lemma for ennnfone 12425. A direct consequence of fidcenumlemrk 6952. (Contributed by Jim Kingdon, 15-Jul-2023.) |
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14-Jul-2023 | djur 7067 | A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) Upgrade implication to biconditional and shorten proof. (Revised by BJ, 14-Jul-2023.) |
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13-Jul-2023 | sbthomlem 14709 | Lemma for sbthom 14710. (Contributed by Mario Carneiro and Jim Kingdon, 13-Jul-2023.) |
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12-Jul-2023 | caseinr 7090 | Applying the "case" construction to a right injection. (Contributed by Jim Kingdon, 12-Jul-2023.) |
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12-Jul-2023 | inl11 7063 | Left injection is one-to-one. (Contributed by Jim Kingdon, 12-Jul-2023.) |
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11-Jul-2023 | djudomr 7218 | A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.) |
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11-Jul-2023 | djudoml 7217 | A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.) |
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11-Jul-2023 | omp1eomlem 7092 | Lemma for omp1eom 7093. (Contributed by Jim Kingdon, 11-Jul-2023.) |
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11-Jul-2023 | xp01disjl 6434 | Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.) |
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10-Jul-2023 | sbthom 14710 |
Schroeder-Bernstein is not possible even for ![]() ![]() |
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10-Jul-2023 | endjusym 7094 | Reversing right and left operands of a disjoint union produces an equinumerous result. (Contributed by Jim Kingdon, 10-Jul-2023.) |
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10-Jul-2023 | omp1eom 7093 |
Adding one to ![]() |
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9-Jul-2023 | refeq 14712 | Equality of two real functions which agree at negative numbers, positive numbers, and zero. This holds even without real trichotomy. From an online post by Martin Escardo. (Contributed by Jim Kingdon, 9-Jul-2023.) |
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9-Jul-2023 | seqvalcd 10458 |
Value of the sequence builder function. Similar to seq3val 10457 but the
classes ![]() ![]() |
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9-Jul-2023 | djuun 7065 | The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.) |
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9-Jul-2023 | djuin 7062 | The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.) |
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8-Jul-2023 | limcimo 14070 |
Conditions which ensure there is at most one limit value of ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
8-Jul-2023 | ennnfonelemh 12404 | Lemma for ennnfone 12425. (Contributed by Jim Kingdon, 8-Jul-2023.) |
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7-Jul-2023 | seqf2 10463 | Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 7-Jul-2023.) |
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3-Jul-2023 | limcimolemlt 14069 | Lemma for limcimo 14070. (Contributed by Jim Kingdon, 3-Jul-2023.) |
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28-Jun-2023 | dvfgg 14093 |
Explicitly write out the functionality condition on derivative for
![]() ![]() ![]() ![]() |
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28-Jun-2023 | dvbsssg 14091 | The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.) (Revised by Jim Kingdon, 28-Jun-2023.) |
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27-Jun-2023 | dvbssntrcntop 14089 | The set of differentiable points is a subset of the interior of the domain of the function. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.) |
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27-Jun-2023 | eldvap 14087 |
The differentiable predicate. A function ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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27-Jun-2023 | dvfvalap 14086 | Value and set bounds on the derivative operator. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.) |
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27-Jun-2023 | dvlemap 14085 | Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 27-Jun-2023.) |
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25-Jun-2023 | reldvg 14084 | The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 25-Jun-2023.) |
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25-Jun-2023 | df-dvap 14062 |
Define the derivative operator. This acts on functions to produce a
function that is defined where the original function is differentiable,
with value the derivative of the function at these points. The set
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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18-Jun-2023 | limccnpcntop 14080 |
If the limit of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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18-Jun-2023 | r19.30dc 2624 | Restricted quantifier version of 19.30dc 1627. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.) |
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17-Jun-2023 | r19.28v 2605 |
Restricted quantifier version of one direction of 19.28 1563. (The other
direction holds when ![]() |
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17-Jun-2023 | r19.27v 2604 |
Restricted quantitifer version of one direction of 19.27 1561. (The other
direction holds when ![]() |
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16-Jun-2023 | cnlimcim 14076 |
If ![]() |
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16-Jun-2023 | cncfcn1cntop 14017 | Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.) (Revised by Jim Kingdon, 16-Jun-2023.) |
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14-Jun-2023 | cnplimcim 14072 |
If a function is continuous at ![]() ![]() |
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14-Jun-2023 | metcnpd 13956 |
Two ways to say a mapping from metric ![]() ![]() ![]() |
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6-Jun-2023 | cntoptop 13969 | The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.) |
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6-Jun-2023 | cntoptopon 13968 | The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.) |
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3-Jun-2023 | limcdifap 14067 |
It suffices to consider functions which are not defined at ![]() ![]() ![]() ![]() |
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3-Jun-2023 | ellimc3ap 14066 | Write the epsilon-delta definition of a limit. (Contributed by Mario Carneiro, 28-Dec-2016.) Use apartness. (Revised by Jim Kingdon, 3-Jun-2023.) |
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3-Jun-2023 | df-limced 14061 | Define the set of limits of a complex function at a point. Under normal circumstances, this will be a singleton or empty, depending on whether the limit exists. (Contributed by Mario Carneiro, 24-Dec-2016.) (Revised by Jim Kingdon, 3-Jun-2023.) |
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30-May-2023 | modprm1div 12246 | A prime number divides an integer minus 1 iff the integer modulo the prime number is 1. (Contributed by Alexander van der Vekens, 17-May-2018.) (Proof shortened by AV, 30-May-2023.) |
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30-May-2023 | modm1div 11806 | An integer greater than one divides another integer minus one iff the second integer modulo the first integer is one. (Contributed by AV, 30-May-2023.) |
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30-May-2023 | eluz4nn 9567 | An integer greater than or equal to 4 is a positive integer. (Contributed by AV, 30-May-2023.) |
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30-May-2023 | eluz4eluz2 9566 | An integer greater than or equal to 4 is an integer greater than or equal to 2. (Contributed by AV, 30-May-2023.) |
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29-May-2023 | mulcncflem 14026 | Lemma for mulcncf 14027. (Contributed by Jim Kingdon, 29-May-2023.) |
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26-May-2023 | cdivcncfap 14023 | Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.) (Revised by Jim Kingdon, 26-May-2023.) |
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26-May-2023 | reccn2ap 11320 |
The reciprocal function is continuous. The class ![]() |
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23-May-2023 | iooretopg 13964 | Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.) (Revised by Jim Kingdon, 23-May-2023.) |
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23-May-2023 | minclpr 11244 |
The minimum of two real numbers is one of those numbers if and only if
dichotomy (![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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22-May-2023 | qtopbasss 13957 | The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Jim Kingdon, 22-May-2023.) |
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22-May-2023 | iooinsup 11284 | Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.) |
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21-May-2023 | df-sumdc 11361 |
Define the sum of a series with an index set of integers ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
19-May-2023 | bdmopn 13940 |
The standard bounded metric corresponding to ![]() ![]() |
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19-May-2023 | bdbl 13939 |
The standard bounded metric corresponding to ![]() ![]() ![]() |
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19-May-2023 | bdmet 13938 | The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.) |
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19-May-2023 | xrminltinf 11279 | Two ways of saying an extended real is greater than the minimum of two others. (Contributed by Jim Kingdon, 19-May-2023.) |
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19-May-2023 | clel5 2874 |
Alternate definition of class membership: a class ![]() ![]() ![]() ![]() |
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18-May-2023 | xrminrecl 11280 | The minimum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 18-May-2023.) |
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18-May-2023 | ralnex2 2616 | Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 18-May-2023.) |
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17-May-2023 | bdtrilem 11246 | Lemma for bdtri 11247. (Contributed by Steven Nguyen and Jim Kingdon, 17-May-2023.) |
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15-May-2023 | xrbdtri 11283 | Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.) |
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15-May-2023 | bdtri 11247 | Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.) |
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15-May-2023 | minabs 11243 | The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.) |
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11-May-2023 | xrmaxadd 11268 | Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.) |
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11-May-2023 | xrmaxaddlem 11267 |
Lemma for xrmaxadd 11268. The case where ![]() |
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10-May-2023 | xrminadd 11282 | Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.) |
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10-May-2023 | xrmaxlesup 11266 | Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 10-May-2023.) |
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10-May-2023 | xrltmaxsup 11264 | The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.) |
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9-May-2023 | bdxmet 13937 | The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.) |
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9-May-2023 | bdmetval 13936 | Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.) |
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7-May-2023 | setsmstsetg 13917 | The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Revised by Jim Kingdon, 7-May-2023.) |
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6-May-2023 | dsslid 12667 |
Slot property of ![]() |
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5-May-2023 | mopnrel 13877 | The class of open sets of a metric space is a relation. (Contributed by Jim Kingdon, 5-May-2023.) |
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5-May-2023 | fsumsersdc 11402 | Special case of series sum over a finite upper integer index set. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Jim Kingdon, 5-May-2023.) |
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4-May-2023 | blex 13823 | A ball is a set. (Contributed by Jim Kingdon, 4-May-2023.) |
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4-May-2023 | summodc 11390 | A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.) |
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4-May-2023 | summodclem2 11389 | Lemma for summodc 11390. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.) |
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4-May-2023 | xrminrpcl 11281 | The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.) |
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4-May-2023 | xrlemininf 11278 | Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 4-May-2023.) |
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3-May-2023 | xrltmininf 11277 | Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 3-May-2023.) |
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3-May-2023 | xrmineqinf 11276 | The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) (Revised by Jim Kingdon, 3-May-2023.) |
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3-May-2023 | xrmin2inf 11275 | The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.) |
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3-May-2023 | xrmin1inf 11274 | The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.) |
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3-May-2023 | xrmincl 11273 | The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.) |
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2-May-2023 | xrminmax 11272 | Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.) |
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2-May-2023 | xrnegcon1d 11271 | Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.) |
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2-May-2023 | infxrnegsupex 11270 |
The infimum of a set of extended reals ![]() |
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2-May-2023 | xrnegiso 11269 | Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.) |
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30-Apr-2023 | xrmaxltsup 11265 | Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.) |
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30-Apr-2023 | xrmaxrecl 11262 | The maximum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 30-Apr-2023.) |
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30-Apr-2023 | xrmax2sup 11261 | An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 30-Apr-2023.) |
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30-Apr-2023 | xrmax1sup 11260 | An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 30-Apr-2023.) |
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29-Apr-2023 | xrmaxcl 11259 | The maximum of two extended reals is an extended real. (Contributed by Jim Kingdon, 29-Apr-2023.) |
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29-Apr-2023 | xrmaxiflemval 11257 | Lemma for xrmaxif 11258. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.) |
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29-Apr-2023 | xrmaxiflemcom 11256 | Lemma for xrmaxif 11258. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.) |
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29-Apr-2023 | xrmaxiflemcl 11252 | Lemma for xrmaxif 11258. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.) |
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29-Apr-2023 | sbco2v 1948 | Version of sbco2 1965 with disjoint variable conditions. (Contributed by Wolf Lammen, 29-Apr-2023.) |
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28-Apr-2023 | xrmaxiflemlub 11255 |
Lemma for xrmaxif 11258. A least upper bound for ![]() ![]() ![]() ![]() ![]() |
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26-Apr-2023 | xrmaxif 11258 |
Maximum of two extended reals in terms of ![]() |
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26-Apr-2023 | xrmaxiflemab 11254 | Lemma for xrmaxif 11258. A variation of xrmaxleim 11251- that is, if we know which of two real numbers is larger, we know the maximum of the two. (Contributed by Jim Kingdon, 26-Apr-2023.) |
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26-Apr-2023 | xrmaxifle 11253 |
An upper bound for ![]() ![]() ![]() ![]() ![]() |
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25-Apr-2023 | xrmaxleim 11251 | Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.) |
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25-Apr-2023 | rpmincl 11245 | The minumum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 25-Apr-2023.) |
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25-Apr-2023 | mincl 11238 | The minumum of two real numbers is a real number. (Contributed by Jim Kingdon, 25-Apr-2023.) |
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24-Apr-2023 | psmetrel 13758 | The class of pseudometrics is a relation. (Contributed by Jim Kingdon, 24-Apr-2023.) |
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23-Apr-2023 | bcval5 10742 |
Write out the top and bottom parts of the binomial coefficient
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23-Apr-2023 | ser3le 10517 | Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Jim Kingdon, 23-Apr-2023.) |
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23-Apr-2023 | seq3z 10510 |
If the operation ![]() ![]() ![]() ![]() |
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23-Apr-2023 | seq3caopr 10482 | The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.) |
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23-Apr-2023 | seq3caopr2 10481 | The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.) |
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22-Apr-2023 | ser3sub 10505 | The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.) |
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22-Apr-2023 | seq3caopr3 10480 | Lemma for seq3caopr2 10481. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.) |
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22-Apr-2023 | ser3mono 10477 | The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.) |
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21-Apr-2023 | metrtri 13813 | Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Jim Kingdon, 21-Apr-2023.) |
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21-Apr-2023 | sqxpeq0 5052 | A Cartesian square is empty iff its member is empty. (Contributed by Jim Kingdon, 21-Apr-2023.) |
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20-Apr-2023 | xmetrel 13779 | The class of extended metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.) |
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20-Apr-2023 | metrel 13778 | The class of metrics is a relation. (Contributed by Jim Kingdon, 20-Apr-2023.) |
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19-Apr-2023 | psmetge0 13767 | The distance function of a pseudometric space is nonnegative. (Contributed by Thierry Arnoux, 7-Feb-2018.) (Revised by Jim Kingdon, 19-Apr-2023.) |
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18-Apr-2023 | xleaddadd 9886 |
Cancelling a factor of two in ![]() |
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17-Apr-2023 | xposdif 9881 | Extended real version of posdif 8411. (Contributed by Mario Carneiro, 24-Aug-2015.) (Revised by Jim Kingdon, 17-Apr-2023.) |
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17-Apr-2023 | nmnfgt 9817 | An extended real is greater than minus infinite iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.) |
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17-Apr-2023 | npnflt 9814 | An extended real is less than plus infinity iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.) |
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16-Apr-2023 | xltadd1 9875 | Extended real version of ltadd1 8385. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Jim Kingdon, 16-Apr-2023.) |
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13-Apr-2023 | xrmnfdc 9842 | An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.) |
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13-Apr-2023 | xrpnfdc 9841 | An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.) |
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11-Apr-2023 | dmxpid 4848 | The domain of a square Cartesian product. (Contributed by NM, 28-Jul-1995.) (Revised by Jim Kingdon, 11-Apr-2023.) |
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9-Apr-2023 | isumz 11396 | Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 9-Apr-2023.) |
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9-Apr-2023 | summodclem2a 11388 | Lemma for summodc 11390. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 9-Apr-2023.) |
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9-Apr-2023 | summodclem3 11387 | Lemma for summodc 11390. (Contributed by Mario Carneiro, 29-Mar-2014.) (Revised by Jim Kingdon, 9-Apr-2023.) |
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9-Apr-2023 | sumrbdc 11386 | Rebase the starting point of a sum. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Jim Kingdon, 9-Apr-2023.) |
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9-Apr-2023 | seq3coll 10821 |
The function ![]() ![]() ![]() ![]() ![]() ![]() |
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8-Apr-2023 | zsumdc 11391 | Series sum with index set a subset of the upper integers. (Contributed by Mario Carneiro, 13-Jun-2019.) (Revised by Jim Kingdon, 8-Apr-2023.) |
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8-Apr-2023 | sumrbdclem 11384 | Lemma for sumrbdc 11386. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Jim Kingdon, 8-Apr-2023.) |
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8-Apr-2023 | isermulc2 11347 | Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Jim Kingdon, 8-Apr-2023.) |
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8-Apr-2023 | seq3id 10507 |
Discarding the first few terms of a sequence that starts with all zeroes
(or any element which is a left-identity for ![]() |
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8-Apr-2023 | seq3id3 10506 |
A sequence that consists entirely of "zeroes" sums to
"zero". More
precisely, a constant sequence with value an element which is a ![]() ![]() |
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7-Apr-2023 | seq3shft2 10472 | Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.) |
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7-Apr-2023 | seq3feq 10471 | Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.) |
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7-Apr-2023 | r19.2m 3509 | Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1638). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.) |
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6-Apr-2023 | lmtopcnp 13686 | The image of a convergent sequence under a continuous map is convergent to the image of the original point. (Contributed by Mario Carneiro, 3-May-2014.) (Revised by Jim Kingdon, 6-Apr-2023.) |
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6-Apr-2023 | cnptoprest2 13676 | Equivalence of point-continuity in the parent topology and point-continuity in a subspace. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 6-Apr-2023.) |
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5-Apr-2023 | cnptoprest 13675 | Equivalence of continuity at a point and continuity of the restricted function at a point. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Jim Kingdon, 5-Apr-2023.) |
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4-Apr-2023 | exmidmp 7154 | Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.) |
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2-Apr-2023 | sup3exmid 8913 | If any inhabited set of real numbers bounded from above has a supremum, excluded middle follows. (Contributed by Jim Kingdon, 2-Apr-2023.) |
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31-Mar-2023 | cnptopresti 13674 | One direction of cnptoprest 13675 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Jim Kingdon, 31-Mar-2023.) |
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30-Mar-2023 | cncnp2m 13667 | A continuous function is continuous at all points. Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Jim Kingdon, 30-Mar-2023.) |
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29-Mar-2023 | exmidlpo 7140 | Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.) |
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28-Mar-2023 | icnpimaex 13647 | Property of a function continuous at a point. (Contributed by FL, 31-Dec-2006.) (Revised by Jim Kingdon, 28-Mar-2023.) |
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28-Mar-2023 | cnpf2 13643 |
A continuous function at point ![]() |
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28-Mar-2023 | cnprcl2k 13642 | Reverse closure for a function continuous at a point. (Contributed by Mario Carneiro, 21-Aug-2015.) (Revised by Jim Kingdon, 28-Mar-2023.) |
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27-Mar-2023 | mptrcl 5598 | Reverse closure for a mapping: If the function value of a mapping has a member, the argument belongs to the base class of the mapping. (Contributed by AV, 4-Apr-2020.) (Revised by Jim Kingdon, 27-Mar-2023.) |
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25-Mar-2023 | lmreltop 13629 | The topological space convergence relation is a relation. (Contributed by Jim Kingdon, 25-Mar-2023.) |
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25-Mar-2023 | fodjumkv 7157 | A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.) |
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25-Mar-2023 | fodjumkvlemres 7156 |
Lemma for fodjumkv 7157. The final result with ![]() |
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25-Mar-2023 | fodju0 7144 |
Lemma for fodjuomni 7146 and fodjumkv 7157. A condition which shows that
![]() |
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25-Mar-2023 | fodjum 7143 |
Lemma for fodjuomni 7146 and fodjumkv 7157. A condition which shows that
![]() |
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25-Mar-2023 | fodjuf 7142 |
Lemma for fodjuomni 7146 and fodjumkv 7157. Domain and range of ![]() |
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23-Mar-2023 | restrcl 13603 | Reverse closure for the subspace topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Jim Kingdon, 23-Mar-2023.) |
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22-Mar-2023 | neipsm 13590 | A neighborhood of a set is a neighborhood of every point in the set. Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.) (Revised by Jim Kingdon, 22-Mar-2023.) |
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19-Mar-2023 | mkvprop 7155 |
Markov's Principle expressed in terms of propositions (or more
precisely, the ![]() ![]() ![]() |
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18-Mar-2023 | omnimkv 7153 |
An omniscient set is Markov. In particular, the case where ![]() ![]() |
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18-Mar-2023 | ismkvmap 7151 | The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 18-Mar-2023.) |
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18-Mar-2023 | ismkv 7150 | The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.) |
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18-Mar-2023 | df-markov 7149 |
A Markov set is one where if a predicate (here represented by a function
![]() ![]() ![]()
In particular, |
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17-Mar-2023 | finct 7114 | A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.) |
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16-Mar-2023 | ctmlemr 7106 | Lemma for ctm 7107. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.) |
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15-Mar-2023 | caseinl 7089 | Applying the "case" construction to a left injection. (Contributed by Jim Kingdon, 15-Mar-2023.) |
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13-Mar-2023 | enumct 7113 |
A finitely enumerable set is countable. Lemma 8.1.14 of [AczelRathjen],
p. 73 (except that our definition of countable does not require the set
to be inhabited). "Finitely enumerable" is defined as
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13-Mar-2023 | enumctlemm 7112 |
Lemma for enumct 7113. The case where ![]() |
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13-Mar-2023 | ctm 7107 | Two equivalent definitions of countable for an inhabited set. Remark of [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.) |
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13-Mar-2023 | 0ct 7105 | The empty set is countable. Remark of [BauerSwan], p. 14:3 which also has the definition of countable used here. (Contributed by Jim Kingdon, 13-Mar-2023.) |
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13-Mar-2023 | ctex 6752 |
A class dominated by ![]() |
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12-Mar-2023 | cls0 13569 | The closure of the empty set. (Contributed by NM, 2-Oct-2007.) (Proof shortened by Jim Kingdon, 12-Mar-2023.) |
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12-Mar-2023 | algrp1 12045 |
The value of the algorithm iterator ![]() ![]() ![]() ![]() ![]() ![]() |
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12-Mar-2023 | ialgr0 12043 |
The value of the algorithm iterator ![]() ![]() ![]() |
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11-Mar-2023 | ntreq0 13568 | Two ways to say that a subset has an empty interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.) |
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11-Mar-2023 | clstop 13563 | The closure of a topology's underlying set is the entire set. (Contributed by NM, 5-Oct-2007.) (Proof shortened by Jim Kingdon, 11-Mar-2023.) |
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11-Mar-2023 | ntrss 13555 | Subset relationship for interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.) |
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10-Mar-2023 | iuncld 13551 | A finite indexed union of closed sets is closed. (Contributed by Mario Carneiro, 19-Sep-2015.) (Revised by Jim Kingdon, 10-Mar-2023.) |
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5-Mar-2023 | 2basgeng 13518 | Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Jim Kingdon, 5-Mar-2023.) |
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5-Mar-2023 | exmidsssn 4202 | Excluded middle is equivalent to the biconditionalized version of sssnr 3753 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.) |
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5-Mar-2023 | exmidn0m 4201 | Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.) |
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4-Mar-2023 | eltg3 13493 | Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.) |
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4-Mar-2023 | tgvalex 12711 | The topology generated by a basis is a set. (Contributed by Jim Kingdon, 4-Mar-2023.) |
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4-Mar-2023 | biadanii 613 | Inference associated with biadani 612. Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.) (Proof shortened by BJ, 4-Mar-2023.) |
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4-Mar-2023 | biadani 612 | An implication implies to the equivalence of some implied equivalence and some other equivalence involving a conjunction. (Contributed by BJ, 4-Mar-2023.) |
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16-Feb-2023 | ixp0 6730 |
The infinite Cartesian product of a family ![]() ![]() ![]() ![]() |
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16-Feb-2023 | ixpm 6729 |
If an infinite Cartesian product of a family ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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16-Feb-2023 | exmidundifim 4207 | Excluded middle is equivalent to every subset having a complement. Variation of exmidundif 4206 with an implication rather than a biconditional. (Contributed by Jim Kingdon, 16-Feb-2023.) |
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15-Feb-2023 | ixpintm 6724 | The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.) |
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15-Feb-2023 | ixpiinm 6723 | The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.) |
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15-Feb-2023 | ixpexgg 6721 |
The existence of an infinite Cartesian product. ![]() ![]() |
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15-Feb-2023 | nfixpxy 6716 | Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Jim Kingdon, 15-Feb-2023.) |
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13-Feb-2023 | topnpropgd 12701 | The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Jim Kingdon, 13-Feb-2023.) |
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12-Feb-2023 | slotex 12488 | Existence of slot value. A corollary of slotslfn 12487. (Contributed by Jim Kingdon, 12-Feb-2023.) |
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11-Feb-2023 | topnvalg 12699 | Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.) |
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10-Feb-2023 | slotslfn 12487 | A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by Jim Kingdon, 10-Feb-2023.) |
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9-Feb-2023 | pleslid 12656 |
Slot property of ![]() |
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9-Feb-2023 | topgrptsetd 12653 | The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
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9-Feb-2023 | topgrpplusgd 12652 | The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
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9-Feb-2023 | topgrpbasd 12651 | The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
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9-Feb-2023 | topgrpstrd 12650 | A constructed topological group is a structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 9-Feb-2023.) |
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9-Feb-2023 | tsetslid 12642 | Slot property of TopSet. (Contributed by Jim Kingdon, 9-Feb-2023.) |
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8-Feb-2023 | ipsipd 12639 | The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.) |
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8-Feb-2023 | ipsvscad 12638 | The scalar product operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.) |
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8-Feb-2023 | ipsscad 12637 | The set of scalars of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 8-Feb-2023.) |
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7-Feb-2023 | ipsmulrd 12636 | The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.) |
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7-Feb-2023 | ipsaddgd 12635 | The additive operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
7-Feb-2023 | ipsbased 12634 | The base set of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
7-Feb-2023 | ipsstrd 12633 | A constructed inner product space is a structure. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 7-Feb-2023.) |
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7-Feb-2023 | ipslid 12628 |
Slot property of ![]() |
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7-Feb-2023 | lmodvscad 12625 | The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 7-Feb-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
6-Feb-2023 | lmodscad 12624 | The set of scalars of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
6-Feb-2023 | lmodplusgd 12623 | The additive operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
6-Feb-2023 | lmodbased 12622 | The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 6-Feb-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
5-Feb-2023 | lmodstrd 12621 | A constructed left module or left vector space is a structure. (Contributed by Mario Carneiro, 1-Oct-2013.) (Revised by Jim Kingdon, 5-Feb-2023.) |
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5-Feb-2023 | vscaslid 12620 |
Slot property of ![]() |
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5-Feb-2023 | scaslid 12610 | Slot property of Scalar. (Contributed by Jim Kingdon, 5-Feb-2023.) |
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5-Feb-2023 | srngplusgd 12605 | The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by Jim Kingdon, 5-Feb-2023.) |
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5-Feb-2023 | srngbased 12604 | The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.) |
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5-Feb-2023 | srngstrd 12603 | A constructed star ring is a structure. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Jim Kingdon, 5-Feb-2023.) |
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5-Feb-2023 | opelstrsl 12572 | The slot of a structure which contains an ordered pair for that slot. (Contributed by Jim Kingdon, 5-Feb-2023.) |
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4-Feb-2023 | starvslid 12598 |
Slot property of ![]() ![]() |
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3-Feb-2023 | rngbaseg 12593 | The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Jim Kingdon, 3-Feb-2023.) |
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3-Feb-2023 | rngstrg 12592 | A constructed ring is a structure. (Contributed by Mario Carneiro, 28-Sep-2013.) (Revised by Jim Kingdon, 3-Feb-2023.) |
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3-Feb-2023 | mulrslid 12589 |
Slot property of ![]() |
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3-Feb-2023 | plusgslid 12570 |
Slot property of ![]() |
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2-Feb-2023 | 2strop1g 12581 | The other slot of a constructed two-slot structure. Version of 2stropg 12578 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.) |
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2-Feb-2023 | 2strbas1g 12580 | The base set of a constructed two-slot structure. Version of 2strbasg 12577 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.) |
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2-Feb-2023 | 2strstr1g 12579 | A constructed two-slot structure. Version of 2strstrg 12576 not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 2-Feb-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
31-Jan-2023 | baseslid 12518 | The base set extractor is a slot. (Contributed by Jim Kingdon, 31-Jan-2023.) |
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31-Jan-2023 | strsl0 12510 | All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 31-Jan-2023.) |
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31-Jan-2023 | strslss 12509 |
Propagate component extraction to a structure ![]() ![]() |
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31-Jan-2023 | strslssd 12508 | Deduction version of strslss 12509. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 31-Jan-2023.) |
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30-Jan-2023 | strslfv3 12507 | Variant on strslfv 12506 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.) |
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30-Jan-2023 | strslfv 12506 |
Extract a structure component ![]() ![]() ![]() ![]() |
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30-Jan-2023 | strslfv2 12505 |
A variation on strslfv 12506 to avoid asserting that ![]() ![]() |
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30-Jan-2023 | strslfv2d 12504 | Deduction version of strslfv 12506. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.) |
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30-Jan-2023 | strslfvd 12503 | Deduction version of strslfv 12506. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 30-Jan-2023.) |
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30-Jan-2023 | strsetsid 12494 | Value of the structure replacement function. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 30-Jan-2023.) |
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30-Jan-2023 | funresdfunsndc 6506 | Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself, where equality is decidable. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 30-Jan-2023.) |
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29-Jan-2023 | ndxslid 12486 | A structure component extractor is defined by its own index. That the index is a natural number will also be needed in quite a few contexts so it is included in the conclusion of this theorem which can be used as a hypothesis of theorems like strslfv 12506. (Contributed by Jim Kingdon, 29-Jan-2023.) |
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29-Jan-2023 | fnsnsplitdc 6505 | Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 29-Jan-2023.) |
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28-Jan-2023 | 2stropg 12578 | The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.) |
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28-Jan-2023 | 2strbasg 12577 | The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.) |
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28-Jan-2023 | 2strstrg 12576 | A constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.) |
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28-Jan-2023 | 1strstrg 12574 | A constructed one-slot structure. (Contributed by AV, 27-Mar-2020.) (Revised by Jim Kingdon, 28-Jan-2023.) |
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27-Jan-2023 | strle2g 12565 | Make a structure from a pair. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.) |
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27-Jan-2023 | strle1g 12564 | Make a structure from a singleton. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.) |
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27-Jan-2023 | strleund 12561 | Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.) |
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24-Jan-2023 | setsslnid 12513 | Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.) |
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24-Jan-2023 | setsslid 12512 | Value of the structure replacement function at a replaced index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.) |
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22-Jan-2023 | setsabsd 12500 | Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.) |
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22-Jan-2023 | setsresg 12499 |
The structure replacement function does not affect the value of ![]() ![]() |
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22-Jan-2023 | setsex 12493 | Applying the structure replacement function yields a set. (Contributed by Jim Kingdon, 22-Jan-2023.) |
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22-Jan-2023 | 2zsupmax 11233 | Two ways to express the maximum of two integers. Because order of integers is decidable, we have more flexibility than for real numbers. (Contributed by Jim Kingdon, 22-Jan-2023.) |
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22-Jan-2023 | elpwpwel 4475 | A class belongs to a double power class if and only if its union belongs to the power class. (Contributed by BJ, 22-Jan-2023.) |
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21-Jan-2023 | funresdfunsnss 5719 | Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in a subset of the function itself. (Contributed by AV, 2-Dec-2018.) (Revised by Jim Kingdon, 21-Jan-2023.) |
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20-Jan-2023 | setsvala 12492 | Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 20-Jan-2023.) |
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20-Jan-2023 | fnsnsplitss 5715 | Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 20-Jan-2023.) |
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19-Jan-2023 | strfvssn 12483 |
A structure component extractor produces a value which is contained in a
set dependent on ![]() ![]() |
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19-Jan-2023 | strnfvn 12482 |
Value of a structure component extractor ![]() ![]() ![]() ![]() ![]() ![]() Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strslfv 12506. (Contributed by NM, 9-Sep-2011.) (Revised by Jim Kingdon, 19-Jan-2023.) (New usage is discouraged.) |
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19-Jan-2023 | strnfvnd 12481 | Deduction version of strnfvn 12482. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 19-Jan-2023.) |
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18-Jan-2023 | isstructr 12476 |
The property of being a structure with components in ![]() ![]() ![]() |
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18-Jan-2023 | isstructim 12475 |
The property of being a structure with components in ![]() ![]() ![]() |
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18-Jan-2023 | isstruct2r 12472 |
The property of being a structure with components in
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
18-Jan-2023 | isstruct2im 12471 |
The property of being a structure with components in
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
18-Jan-2023 | sbiev 1792 | Conversion of implicit substitution to explicit substitution. Version of sbie 1791 with a disjoint variable condition. (Contributed by Wolf Lammen, 18-Jan-2023.) |
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16-Jan-2023 | toponsspwpwg 13458 | The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) (Revised by Jim Kingdon, 16-Jan-2023.) |
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14-Jan-2023 | istopfin 13436 |
Express the predicate "![]() |
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14-Jan-2023 | fiintim 6927 |
If a class is closed under pairwise intersections, then it is closed
under nonempty finite intersections. The converse would appear to
require an additional condition, such as ![]() ![]() ![]() This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use a pairwise intersection and some texts use a finite intersection, but most topology texts assume excluded middle (in which case the two intersection properties would be equivalent). (Contributed by NM, 22-Sep-2002.) (Revised by Jim Kingdon, 14-Jan-2023.) |
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9-Jan-2023 | divccncfap 14013 | Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Jim Kingdon, 9-Jan-2023.) |
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7-Jan-2023 | eap1 11792 |
![]() |
![]() ![]() ![]() | ||
7-Jan-2023 | eap0 11790 |
![]() |
![]() ![]() ![]() | ||
7-Jan-2023 | egt2lt3 11786 |
Euler's constant ![]() |
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6-Jan-2023 | eirr 11785 |
![]() ![]() |
![]() ![]() ![]() ![]() | ||
6-Jan-2023 | eirrap 11784 |
![]() ![]() ![]() |
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6-Jan-2023 | btwnapz 9382 | A number between an integer and its successor is apart from any integer. (Contributed by Jim Kingdon, 6-Jan-2023.) |
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6-Jan-2023 | apmul2 8745 | Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 6-Jan-2023.) |
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1-Jan-2023 | nnap0i 8949 | A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.) |
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31-Dec-2022 | 2logb9irrALT 14328 | Alternate proof of 2logb9irr 14325: The logarithm of nine to base two is not rational. (Contributed by AV, 31-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
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31-Dec-2022 | 2logb3irr 14327 | Example for logbprmirr 14326. The logarithm of three to base two is not rational. (Contributed by AV, 31-Dec-2022.) |
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31-Dec-2022 | logbprmirr 14326 |
The logarithm of a prime to a different prime base is not rational. For
example, ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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30-Dec-2022 | elpqb 9648 | A class is a positive rational iff it is the quotient of two positive integers. (Contributed by AV, 30-Dec-2022.) |
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29-Dec-2022 | sqrt2cxp2logb9e3 14329 |
The square root of two to the power of the logarithm of nine to base two
is three. ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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29-Dec-2022 | 2logb9irr 14325 | Example for logbgcd1irr 14321. The logarithm of nine to base two is not rational. Also see 2logb9irrap 14331 which says that it is irrational (in the sense of being apart from any rational number). (Contributed by AV, 29-Dec-2022.) |
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29-Dec-2022 | logbgcd1irrap 14324 |
The logarithm of an integer greater than 1 to an integer base greater
than 1 is irrational (in the sense of being apart from any rational
number) if the argument and the base are relatively prime. For example,
![]() ![]() ![]() ![]() ![]() ![]() |
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29-Dec-2022 | logbgcd1irr 14321 |
The logarithm of an integer greater than 1 to an integer base greater
than 1 is not rational if the argument and the base are relatively
prime. For example, ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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29-Dec-2022 | logbgt0b 14320 | The logarithm of a positive real number to a real base greater than 1 is positive iff the number is greater than 1. (Contributed by AV, 29-Dec-2022.) |
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29-Dec-2022 | cxpcom 14293 | Commutative law for real exponentiation. (Contributed by AV, 29-Dec-2022.) |
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29-Dec-2022 | elpq 9647 | A positive rational is the quotient of two positive integers. (Contributed by AV, 29-Dec-2022.) |
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26-Dec-2022 | apdivmuld 8769 | Relationship between division and multiplication. (Contributed by Jim Kingdon, 26-Dec-2022.) |
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25-Dec-2022 | tanaddaplem 11745 | A useful intermediate step in tanaddap 11746 when showing that the addition of tangents is well-defined. (Contributed by Mario Carneiro, 4-Apr-2015.) (Revised by Jim Kingdon, 25-Dec-2022.) |
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25-Dec-2022 | subap0 8599 | Two numbers being apart is equivalent to their difference being apart from zero. (Contributed by Jim Kingdon, 25-Dec-2022.) |
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23-Dec-2022 | 2irrexpq 14330 |
There exist real numbers ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
For a theorem which is the same but proves that |
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23-Dec-2022 | rpcxpsqrtth 14286 | Square root theorem over the complex numbers for the complex power function. Compare with resqrtth 11039. (Contributed by AV, 23-Dec-2022.) |
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23-Dec-2022 | sqrt2irr0 12163 | The square root of 2 is not rational. (Contributed by AV, 23-Dec-2022.) |
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22-Dec-2022 | tanval3ap 11721 |
Express the tangent function directly in terms of ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
22-Dec-2022 | tanval2ap 11720 |
Express the tangent function directly in terms of ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
22-Dec-2022 | tanclapd 11719 | Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016.) (Revised by Jim Kingdon, 22-Dec-2022.) |
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21-Dec-2022 | tanclap 11716 | The closure of the tangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.) |
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21-Dec-2022 | tanvalap 11715 | Value of the tangent function. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Jim Kingdon, 21-Dec-2022.) |
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20-Dec-2022 | reef11 11706 | The exponential function on real numbers is one-to-one. (Contributed by NM, 21-Aug-2008.) (Revised by Jim Kingdon, 20-Dec-2022.) |
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20-Dec-2022 | efltim 11705 | The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 20-Dec-2022.) |
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20-Dec-2022 | eqord1 8439 |
A strictly increasing real function on a subset of ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
14-Dec-2022 | iserabs 11482 | Generalized triangle inequality: the absolute value of an infinite sum is less than or equal to the sum of absolute values. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Jim Kingdon, 14-Dec-2022.) |
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12-Dec-2022 | efap0 11684 | The exponential of a complex number is apart from zero. (Contributed by Jim Kingdon, 12-Dec-2022.) |
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8-Dec-2022 | efcllem 11666 | Lemma for efcl 11671. The series that defines the exponential function converges. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.) |
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8-Dec-2022 | efcllemp 11665 | Lemma for efcl 11671. The series that defines the exponential function converges. The ratio test cvgratgt0 11540 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.) |
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8-Dec-2022 | eftvalcn 11664 | The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 8-Dec-2022.) |
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8-Dec-2022 | mertensabs 11544 |
Mertens' theorem. If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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3-Dec-2022 | mertenslemub 11541 |
Lemma for mertensabs 11544. An upper bound for ![]() |
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2-Dec-2022 | mertenslemi1 11542 | Lemma for mertensabs 11544. (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.) |
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2-Dec-2022 | fsum3cvg3 11403 | A finite sum is convergent. (Contributed by Mario Carneiro, 24-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.) |
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2-Dec-2022 | fsum3cvg2 11401 | The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.) |
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24-Nov-2022 | cvgratnnlembern 11530 | Lemma for cvgratnn 11538. Upper bound for a geometric progression of positive ratio less than one. (Contributed by Jim Kingdon, 24-Nov-2022.) |
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23-Nov-2022 | cvgratnnlemfm 11536 | Lemma for cvgratnn 11538. (Contributed by Jim Kingdon, 23-Nov-2022.) |
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23-Nov-2022 | cvgratnnlemsumlt 11535 | Lemma for cvgratnn 11538. (Contributed by Jim Kingdon, 23-Nov-2022.) |
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21-Nov-2022 | cvgratnnlemrate 11537 | Lemma for cvgratnn 11538. (Contributed by Jim Kingdon, 21-Nov-2022.) |
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21-Nov-2022 | cvgratnnlemabsle 11534 | Lemma for cvgratnn 11538. (Contributed by Jim Kingdon, 21-Nov-2022.) |
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21-Nov-2022 | cvgratnnlemseq 11533 | Lemma for cvgratnn 11538. (Contributed by Jim Kingdon, 21-Nov-2022.) |
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15-Nov-2022 | cvgratnnlemmn 11532 | Lemma for cvgratnn 11538. (Contributed by Jim Kingdon, 15-Nov-2022.) |
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15-Nov-2022 | cvgratnnlemnexp 11531 | Lemma for cvgratnn 11538. (Contributed by Jim Kingdon, 15-Nov-2022.) |
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12-Nov-2022 | cvgratnn 11538 |
Ratio test for convergence of a complex infinite series. If the ratio
![]() ![]() ![]() |
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12-Nov-2022 | fsum3cvg 11385 | The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 12-Nov-2022.) |
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12-Nov-2022 | seq3id2 10508 |
The last few partial sums of a sequence that ends with all zeroes (or
any element which is a right-identity for ![]() |
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11-Nov-2022 | cvgratgt0 11540 |
Ratio test for convergence of a complex infinite series. If the ratio
![]() ![]() ![]() ![]() |
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11-Nov-2022 | cvgratz 11539 |
Ratio test for convergence of a complex infinite series. If the ratio
![]() ![]() ![]() |
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4-Nov-2022 | seq3val 10457 | Value of the sequence builder function. This helps expand the definition although there should be little need for it once we have proved seqf 10460, seq3-1 10459 and seq3p1 10461, as further development can be done in terms of those. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 4-Nov-2022.) |
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4-Nov-2022 | df-seqfrec 10445 |
Define a general-purpose operation that builds a recursive sequence
(i.e., a function on an upper integer set such as ![]() ![]()
The first operand in the parentheses is the operation that is applied to
the previous value and the value of the input sequence (second operand).
The operand to the left of the parenthesis is the integer to start from.
For example, for the operation
Internally, the frec function generates as its values a set of
ordered pairs starting at (Contributed by NM, 18-Apr-2005.) (Revised by Jim Kingdon, 4-Nov-2022.) |
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3-Nov-2022 | seq3f1o 10503 |
Rearrange a sum via an arbitrary bijection on ![]() ![]() ![]() ![]() ![]() |
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3-Nov-2022 | seq3m1 10467 | Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 3-Nov-2022.) |
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29-Oct-2022 | absgtap 11517 | Greater-than of absolute value implies apartness. (Contributed by Jim Kingdon, 29-Oct-2022.) |
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29-Oct-2022 | absltap 11516 | Less-than of absolute value implies apartness. (Contributed by Jim Kingdon, 29-Oct-2022.) |
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29-Oct-2022 | 1ap2 9125 | 1 is apart from 2. (Contributed by Jim Kingdon, 29-Oct-2022.) |
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28-Oct-2022 | expcnv 11511 |
A sequence of powers of a complex number ![]() |
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28-Oct-2022 | expcnvre 11510 | A sequence of powers of a nonnegative real number less than one converges to zero. (Contributed by Jim Kingdon, 28-Oct-2022.) |
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27-Oct-2022 | ennnfone 12425 |
A condition for a set being countably infinite. Corollary 8.1.13 of
[AczelRathjen], p. 73. Roughly
speaking, the condition says that ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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27-Oct-2022 | ennnfonelemim 12424 | Lemma for ennnfone 12425. The trivial direction. (Contributed by Jim Kingdon, 27-Oct-2022.) |
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27-Oct-2022 | ennnfonelemr 12423 | Lemma for ennnfone 12425. The interesting direction, expressed in deduction form. (Contributed by Jim Kingdon, 27-Oct-2022.) |
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27-Oct-2022 | ennnfonelemnn0 12422 |
Lemma for ennnfone 12425. A version of ennnfonelemen 12421 expressed in
terms of ![]() ![]() |
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24-Oct-2022 | pwm1geoserap1 11515 |
The n-th power of a number decreased by 1 expressed by the finite
geometric series ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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24-Oct-2022 | geoserap 11514 |
The value of the finite geometric series ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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24-Oct-2022 | geosergap 11513 |
The value of the finite geometric series ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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23-Oct-2022 | expcnvap0 11509 |
A sequence of powers of a complex number ![]() |
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22-Oct-2022 | divcnv 11504 |
The sequence of reciprocals of positive integers, multiplied by the
factor ![]() |
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22-Oct-2022 | impcomd 255 | Importation deduction with commuted antecedents. (Contributed by Peter Mazsa, 24-Sep-2022.) (Proof shortened by Wolf Lammen, 22-Oct-2022.) |
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21-Oct-2022 | isumsplit 11498 |
Split off the first ![]() |
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21-Oct-2022 | seq3split 10478 | Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.) (Revised by Jim Kingdon, 21-Oct-2022.) |
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20-Oct-2022 | fidcenumlemrk 6952 | Lemma for fidcenum 6954. (Contributed by Jim Kingdon, 20-Oct-2022.) |
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20-Oct-2022 | fidcenumlemrks 6951 | Lemma for fidcenum 6954. Induction step for fidcenumlemrk 6952. (Contributed by Jim Kingdon, 20-Oct-2022.) |
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19-Oct-2022 | fidcenum 6954 |
A set is finite if and only if it has decidable equality and is finitely
enumerable. Proposition 8.1.11 of [AczelRathjen], p. 72. The
definition of "finitely enumerable" as
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19-Oct-2022 | fidcenumlemr 6953 | Lemma for fidcenum 6954. Reverse direction (put into deduction form). (Contributed by Jim Kingdon, 19-Oct-2022.) |
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