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Theorem List for Metamath Proof Explorer - 24901-25000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcvmtop2 24901 Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmcn 24902 A covering map is a continuous function. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmcov 24903* Property of a covering map. In order to make the covering property more manageable, we define here the set of all even coverings of an open set in the range. Then the covering property states that every point has a neighborhood which has an even covering. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t               CovMap

Theoremcvmsrcl 24904* Reverse closure for an even covering. (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsi 24905* One direction of cvmsval 24906. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t        t t

Theoremcvmsval 24906* Elementhood in the set of all even coverings of an open set in . is an even covering of if it is a nonempty collection of disjoint open sets in whose union is the preimage of , such that each set is homeomorphic under to . (Contributed by Mario Carneiro, 13-Feb-2015.)
t t        t t

Theoremcvmsss 24907* An even covering is a subset of the topology of the domain (i.e. a collection of open sets). (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsn0 24908* An even covering is nonempty. (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsuni 24909* An even covering of has union equal to the preimage of by . (Contributed by Mario Carneiro, 11-Feb-2015.)
t t

Theoremcvmsdisj 24910* An even covering of is a disjoint union. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t

Theoremcvmshmeo 24911* Every element of an even covering of is homeomorphic to via . (Contributed by Mario Carneiro, 13-Feb-2015.)
t t        t t

Theoremcvmsf1o 24912* , localized to an element of an even covering of , is a bijection. (Contributed by Mario Carneiro, 14-Feb-2015.)
t t        CovMap

Theoremcvmscld 24913* The sets of an even covering are clopen in the subspace topology on . (Contributed by Mario Carneiro, 14-Feb-2015.)
t t        CovMap t

Theoremcvmsss2 24914* An open subset of an evenly covered set is evenly covered. (Contributed by Mario Carneiro, 7-Jul-2015.)
t t        CovMap

Theoremcvmcov2 24915* The covering map property can be restricted to an open subset. (Contributed by Mario Carneiro, 7-Jul-2015.)
t t        CovMap

Theoremcvmseu 24916* Every element in is a member of a unique element of . (Contributed by Mario Carneiro, 14-Feb-2015.)
t t               CovMap

Theoremcvmsiota 24917* Identify the unique element of containing . (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmopnlem 24918* Lemma for cvmopn 24920. (Contributed by Mario Carneiro, 7-May-2015.)
t t               CovMap

Theoremcvmfolem 24919* Lemma for cvmfo 24940. (Contributed by Mario Carneiro, 13-Feb-2015.)
t t                      CovMap

Theoremcvmopn 24920 A covering map is an open map. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmliftmolem1 24921* Lemma for cvmliftmo 24924. (Contributed by Mario Carneiro, 10-Mar-2015.)
CovMap               𝑛Locally                                           t t                             t

Theoremcvmliftmolem2 24922* Lemma for cvmliftmo 24924. (Contributed by Mario Carneiro, 10-Mar-2015.)
CovMap               𝑛Locally                                           t t

Theoremcvmliftmoi 24923 A lift of a continuous function from a connected and locally connected space over a covering map is unique when it exists. (Contributed by Mario Carneiro, 10-Mar-2015.)
CovMap               𝑛Locally

Theoremcvmliftmo 24924* A lift of a continuous function from a connected and locally connected space over a covering map is unique when it exists. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by NM, 17-Jun-2017.)
CovMap               𝑛Locally

Theoremcvmliftlem1 24925* Lemma for cvmlift 24939. In cvmliftlem15 24938, we picked an large enough so that the sections are all contained in an even covering, and the function enumerates these even coverings. So is a neighborhood of , and is an even covering of , which is to say a disjoint union of open sets in whose image is . (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem2 24926* Lemma for cvmlift 24939. is a subset of for each . (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem3 24927* Lemma for cvmlift 24939. Since is a neighborhood of , every element satisfies . (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem4 24928* Lemma for cvmlift 24939. The function will be our lifted path, defined piecewise on each section for . For , it is a "seed" value which makes the rest of the recursion work, a singleton function mapping to . (Contributed by Mario Carneiro, 15-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem5 24929* Lemma for cvmlift 24939. Definition of at a successor. This is a function defined on as where is the unique covering set of that contains evaluated at the last defined point, namely (note that for this is using the seed value ). (Contributed by Mario Carneiro, 15-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem6 24930* Lemma for cvmlift 24939. Induction step for cvmliftlem7 24931. Assuming that is defined at and is a preimage of , the next segment is also defined and is a function on which is a lift for this segment. This follows explicitly from the definition since is in for the entire interval so that maps this into and maps back to . (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem7 24931* Lemma for cvmlift 24939. Prove by induction that every function is well-defined (we can immediately follow this theorem with cvmliftlem6 24930 to show functionality and lifting of ). (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem8 24932* Lemma for cvmlift 24939. The functions are continuous functions because they are defined as where is continuous and is a homeomorphism. (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap                                                                       t

Theoremcvmliftlem9 24933* Lemma for cvmlift 24939. The functions are defined on almost disjoint intervals, but they overlap at the edges. Here we show that at these points the functions agree on their common domain. (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem10 24934* Lemma for cvmlift 24939. The function is going to be our complete lifted path, formed by unioning together all the functions (each of which is defined on one segment of the interval). Here we prove by induction that is a continuous function and a lift of by applying cvmliftlem6 24930, cvmliftlem7 24931 (to show it is a function and a lift), cvmliftlem8 24932 (to show it is continuous), and cvmliftlem9 24933 (to show that different functions agree on the intersection of their domains, so that the pasting lemma paste 17312 gives that is well-defined and continuous). (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap                                                                       t        t

Theoremcvmliftlem11 24935* Lemma for cvmlift 24939. (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem13 24936* Lemma for cvmlift 24939. The initial value of is because is a subset of which takes value at . (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem14 24937* Lemma for cvmlift 24939. Putting the results of cvmliftlem11 24935, cvmliftlem13 24936 and cvmliftmo 24924 together, we have that is a continuous function, satisfies and , and is equal to any other function which also has these properties, so it follows that is the unique lift of . (Contributed by Mario Carneiro, 16-Feb-2015.)
t t                      CovMap

Theoremcvmliftlem15 24938* Lemma for cvmlift 24939. Discharge the assumptions of cvmliftlem14 24937. The set of all open subsets of the unit interval such that is contained in an even covering of some open set in is a cover of by the definition of a covering map, so by the Lebesgue number lemma lebnumii 18944, there is a subdivision of the unit interval into equal parts such that each part is entirely contained within one such open set of . Then using finite choice ac6sfi 7310 to uniformly select one such subset and one even covering of each subset, we are ready to finish the proof with cvmliftlem14 24937. (Contributed by Mario Carneiro, 14-Feb-2015.)
t t                      CovMap

Theoremcvmlift 24939* One of the important properties of covering maps is that any path in the base space "lifts" to a path in the covering space such that , and given a starting point in the covering space this lift is unique. The proof is contained in cvmliftlem1 24925 thru cvmliftlem15 24938. (Contributed by Mario Carneiro, 16-Feb-2015.)
CovMap

Theoremcvmfo 24940 A covering map is an onto function. (Contributed by Mario Carneiro, 13-Feb-2015.)
CovMap

Theoremcvmliftiota 24941* Write out a function that is the unique lift of . (Contributed by Mario Carneiro, 16-Feb-2015.)
CovMap

Theoremcvmlift2lem1 24942* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 1-Jun-2015.)

Theoremcvmlift2lem9a 24943* Lemma for cvmlift2 24956 and cvmlift3 24968. (Contributed by Mario Carneiro, 9-Jul-2015.)
t t        CovMap                                                                t

Theoremcvmlift2lem2 24944* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmlift2lem3 24945* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmlift2lem4 24946* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 1-Jun-2015.)
CovMap

Theoremcvmlift2lem5 24947* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmlift2lem6 24948* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap                                           t

Theoremcvmlift2lem7 24949* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmlift2lem8 24950* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 9-Mar-2015.)
CovMap

Theoremcvmlift2lem9 24951* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 1-Jun-2015.)
CovMap                                           t t                                    t        t                                    t               t

Theoremcvmlift2lem10 24952* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 1-Jun-2015.)
CovMap                                           t t                      t t

Theoremcvmlift2lem11 24953* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 1-Jun-2015.)
CovMap                                                                              t t

Theoremcvmlift2lem12 24954* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 1-Jun-2015.)
CovMap

Theoremcvmlift2lem13 24955* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap

Theoremcvmlift2 24956* A two-dimensional version of cvmlift 24939. There is a unique lift of functions on the unit square which commutes with the covering map. (Contributed by Mario Carneiro, 1-Jun-2015.)
CovMap

Theoremcvmliftphtlem 24957* Lemma for cvmliftpht 24958. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap

Theoremcvmliftpht 24958* If and are path-homotopic, then their lifts and are also path-homotopic. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap

Theoremcvmlift3lem1 24959* Lemma for cvmlift3 24968. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap        SCon       𝑛Locally PCon

Theoremcvmlift3lem2 24960* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap        SCon       𝑛Locally PCon

Theoremcvmlift3lem3 24961* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap        SCon       𝑛Locally PCon

Theoremcvmlift3lem4 24962* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap        SCon       𝑛Locally PCon

Theoremcvmlift3lem5 24963* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap        SCon       𝑛Locally PCon

Theoremcvmlift3lem6 24964* Lemma for cvmlift3 24968. (Contributed by Mario Carneiro, 9-Jul-2015.)
CovMap        SCon       𝑛Locally PCon                                          t t                                                                       t

Theoremcvmlift3lem7 24965* Lemma for cvmlift3 24968. (Contributed by Mario Carneiro, 9-Jul-2015.)
CovMap        SCon       𝑛Locally PCon                                          t t                                    t PCon

Theoremcvmlift3lem8 24966* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 6-Jul-2015.)
CovMap        SCon       𝑛Locally PCon                                          t t

Theoremcvmlift3lem9 24967* Lemma for cvmlift2 24956. (Contributed by Mario Carneiro, 7-May-2015.)
CovMap        SCon       𝑛Locally PCon                                          t t

Theoremcvmlift3 24968* A general version of cvmlift 24939. If is simply connected and weakly locally path-connected, then there is a unique lift of functions on which commutes with the covering map. (Contributed by Mario Carneiro, 9-Jul-2015.)
CovMap        SCon       𝑛Locally PCon

19.4.10  Normal numbers

Theoremsnmlff 24969* The function from snmlval 24971 is a mapping from positive integers to real numbers in the range . (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremsnmlfval 24970* The function from snmlval 24971 maps to the relative density of in the first digits of the digit string of in base . (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremsnmlval 24971* The property " is simply normal in base ". A number is simply normal if each digit occurs in the base- digit string of with frequency (which is consistent with the expectation in an infinite random string of numbers selected from ). (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremsnmlflim 24972* If is simply normal, then the function of relative density of in the digit string converges to , i.e. the set of occurences of in the digit string has natural density . (Contributed by Mario Carneiro, 6-Apr-2015.)

19.4.11  Godel-sets of formulas

Syntaxcgoe 24973 The Godel-set of membership.

Syntaxcgna 24974 The Godel-set for the Sheffer stroke.

Syntaxcgol 24975 The Godel-set of universal quantification. (Note that this is not a wff.)

Syntaxcsat 24976 The satisfaction function.

Syntaxcfmla 24977 The formula set predicate.

Syntaxcsate 24978 The -satisfaction function.

Syntaxcprv 24979 The "proves" relation.

Definitiondf-goel 24980 Define the Godel-set of membership. Here the arguments correspond to vN and vP , so actually means v0 v1 , not . (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-gona 24981 Define the Godel-set for the Sheffer stroke NAND. Here the arguments are also Godel-sets corresponding to smaller formulae. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-goal 24982 Define the Godel-set of universal quantification. Here corresponds to vN , and represents another formula, and this expression is where is the -th variable, is the code for . Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-sat 24983* Define the satisfaction predicate. This recursive construction builds up a function over wff codes and simultaneously defines the set of assignments to all variables from that makes the coded wff true in the model , where is interpreted as the binary relation on . The interpretation of the statement is that for the model , is a valuation of the variables (v0 , v1 , etc.) and is a code for a wff using that is true under the assignment . The function is defined by finite recursion; only operates on wffs of depth at most , and operates on all wffs. The coding scheme for the wffs is defined so that
• vi vj is coded as ,
• is coded as , and
• vi is coded as .

(Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-sate 24984* A simplified version of the satisfaction predicate, using the standard membership relation and eliminating the extra variable . (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-fmla 24985 Define the predicate which defines the set of valid Godel formulas. The parameter defines the maximum height of the formulas: the set is all formulas of the form or (which in our coding scheme is the set ; see df-sat 24983 for the full coding scheme), and each extra level adds to the complexity of the formulas in . is the set of all valid formulas. (Contributed by Mario Carneiro, 14-Jul-2013.)

Syntaxcgon 24986 The Godel-set of negation. (Note that this is not a wff.)

Syntaxcgoa 24987 The Godel-set of conjunction.

Syntaxcgoi 24988 The Godel-set of implication.

Syntaxcgoo 24989 The Godel-set of disjunction.

Syntaxcgob 24990 The Godel-set of equivalence.

Syntaxcgoq 24991 The Godel-set of equality.

Syntaxcgox 24992 The Godel-set of existential quantification. (Note that this is not a wff.)

Definitiondf-gonot 24993 Define the Godel-set of negation. Here the argument is also a Godel-set corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-goan 24994* Define the Godel-set of conjunction. Here the arguments and are also Godel-sets corresponding to smaller formulae. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-goim 24995* Define the Godel-set of implication. Here the arguments and are also Godel-sets corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-goor 24996* Define the Godel-set of disjunction. Here the arguments and are also Godel-sets corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-gobi 24997* Define the Godel-set of equivalence. Here the arguments and are also Godel-sets corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-goeq 24998* Define the Godel-set of equality. Here the arguments correspond to vN and vP , so actually means v0 v1 , not . Here we use the trick mentioned in ax-ext 2385 to introduce equality as a defined notion in terms of . The expression max here is a convenient way of getting a dummy variable distinct from and . (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-goex 24999 Define the Godel-set of existential quantification. Here corresponds to vN , and represents another formula, and this expression is where is the -th variable, is the code for . Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)

Definitiondf-prv 25000* Define the "proves" relation on a set. A wff is true in a model if for every valuation , the interpretation of the wff using the membership relation on is true. (Contributed by Mario Carneiro, 14-Jul-2013.)

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